Minimality and comparison of sets of multi-attribute vectors

In a decision-making problem, there is often some uncertainty regarding the user preferences. We assume a parameterised utility model, where in each scenario we have a utility function over alternatives, and where each scenario represents a possible user preference model consistent with the input preference information. With a set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}A of alternatives available to the decision-maker, we can consider the associated utility function, expressing, for each scenario, the maximum utility among the alternatives. We consider two main problems: firstly, finding a minimal subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}A that is equivalent to it, i.e., that has the same utility function. We show that for important classes of preference models, the set of possibly strictly optimal alternatives is the unique minimal equivalent subset. Secondly, we consider how to compare \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}A to another set of alternatives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document}B, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}A and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document}B correspond to different initial decision choices. This is closely related to the problem of computing setwise max regret. We derive mathematical results that allow different computational techniques for these problems, using linear programming, and especially, with a novel approach using the extreme points of the epigraph of the utility function.


Introduction
In a decision-making problem, there can often be uncertainty regarding the user preferences. Suppose that, in a particular situation, A is the set of alternatives that are available to the decision-maker. This is interpreted in a disjunctive fashion, in that the user is free to choose any element a of A. However, as is common, we do not know precisely the user's preferences. The preference information available to the system is represented in terms of a set of user preference models, parameterised by a set (of scenarios) W where, associated with each scenario w 2 W, is a (real-valued) utility function f w over alternatives.
Each element w of W is viewed as representing a possible model of the user's preferences that is consistent with the preference information we know. If we knew that w were the true scenario, so that f w represents the user's preferences over alternatives, then we would be able to choose a best element of A with respect to f w , leading to a utility value Ut A ðwÞ ¼ max a2A f w ðaÞ. However, the situation will frequently be ambiguous given a nonsingleton set W of possible user models or scenarios.
The set W incorporates what we know about the user preferences; for example, if we have learned that the user regards alternative b as at least as good as alternative c, then W will only include scenarios w such that f w ðbÞ ! f w ðcÞ.
This framework is fairly general; for instance, the utility function f w may be based on a decomposition of utility, using, for example, an additive representation for a combinatorial problem (e.g., [37,42,58,62]). Also, f w ðaÞ could represent the expected utility of alternative a given that w is the correct user model, based on a probabilistic model with parameter w, for example in a multi-objective influence diagram [22,41,43], with a corresponding to a policy.
We consider, in particular, the following related pair of questions: (1) Are there elements of A that can be eliminated unproblematically? In particular, is there a strict subset A 0 of A that is equivalent to A? (2) Given a choice between one situation, in which the available alternatives are A, and another situation, in which alternatives B are available, is A at least as good as B in every scenario?
Regarding (1), we need to be able to eliminate unimportant choices, which can help to make the list of options manageable, in particular, if we want to display the alternatives to the user. We interpret this as finding a minimal subset A 0 of A such that Ut A ðwÞ ¼ Ut A 0 ðwÞ for every scenario w 2 W. In this situation, we say that A and A 0 are utility-equivalent. Question (2) concerns a case in which the user may have a choice between (I) being able to obtain any of the set of alternatives A, and (II) any alternative in B (and thus, the user could obtain any alternative in A [ B). Sets A and B may correspond to different choices X ¼ a and X ¼ b of a fundamental variable X, and determining that A dominates B may lead us to exclude X ¼ b, thus simplifying the problem. For instance, A might correspond to hotels in Paris, and B to hotels in Lisbon, for a potential weekend away. It can be useful to determine if one of these clearly dominates the other; if, for instance, A dominates B, then there may be no need for the system and the user to further consider B, and may therefore focus on Paris rather than Lisbon. We interpret this task as determining if in every scenario the utility of A is at least that for B, i.e., Ut A ðwÞ ! Ut B ðwÞ for all w 2 W. We can this relation utility-dominance.
The focus of this paper is to determine important properties of the utility-dominance and utility-equivalence relations, and to derive computational procedures, in order to find a minimal equivalent subset, and for testing dominance between A and B; we also determine properties and a computational technique for a form of maximum regret, that can be viewed as a degree of dominance, and which corresponds to setwise max regret defined in [62,65], and relates to the value of a query. The main computational procedures are based on linear programming (LP), or, alternatively, a novel method using the extreme points of the epigraph of the utility function (which we abbreviate to EEU). These procedures have been compared and evaluated in [59].
From the computational perspective we focus especially on the case in which each alternative a is associated with a multi-attribute utility vectorâ, based on a weighted average user preference model. Each utility vector is then an element of IR p , representing a number p of scales of utility (or objectives); each scenario w is a (typically normalised non-negative) vector in IR p , with a< w b if and only if the weighted sum ofâ with respect to w is at least that ofb. An input preference of a over b then leads to a linear constraint on the weights vector w, and we can define the set of consistent preference models W as the convex polytope generated by a set of input preferences of this form.
Given a finite set of alternatives A, we define the notion of setwise-minimal equivalent subset, and we show relationships with the set PSO W ðAÞ of possibly strictly optimal elements, where W is the set of parameters relating to a family of user preference models. It suffices to consider sets A of alternatives that are equivalence-free, i.e., such that no two alternatives in A have identical utility in all scenarios. For equivalence-free A, the set PSO W ðAÞ consists of those alternatives a 2 A that are such that there exists some scenario in W for which a is the only optimal alternative in A.
The main contributions of the paper are as follows: we give sufficient conditions for the set PSO W ðAÞ of possibly strictly optimal alternatives to be utility-equivalent with A. These apply for many natural situations including when the user preferences models are linear, and the set W of parameters defining the set of user preference models is convex. -We show that then PSO W ðAÞ is the unique minimal equivalent subset of A, for equivalence-free A (i.e., when no two alternatives in A have identical utility in all scenarios in W). Furthermore, the PSO W operator can be used to filter query sets to avoid the potential of a partially inconsistent answer. -We derive sophisticated computational methods for computing utility dominance, setwise max regret and for computing the unique minimal equivalent subset. These include both linear programming methods, and extreme point methods using the epigraph of the utility functions. To increase the efficiency, a number of pre-processing methods are developed, using the properties we derive for dominance relations between sets. Our algorithms are experimentally tested based on randomly generated instances.
This paper extends an earlier work [60], including further theoretical results with proofs of theorems, propositions and lemmas, and new experimental results related to a new implementation of our algorithms fixing some runtime exceptions related to precision issues (see Sect. 13). Part of this work is also included in the first author's PhD thesis [56]. Section 2 discusses related work. Section 3 gives the formal setup, defining dominance relations between sets of alternatives, and giving basic properties. Section 4 discusses different ways of defining optimal alternatives in a set, including the possibly strictly optimal set. Section 5 defines the setwise minimal equivalent set, for a set of alternatives, and explores the relationship between such sets and the possibly strictly optimal set. Section 6 considers continuous spaces of scenarios, and gives a sufficient condition for equivalence of the possibly strictly optimal set with the input set of alternatives. Section 7 considers the problem of reducing the size of a set A, whilst maintaining equivalence. Section 8 defines a form of maximum regret in this context, shows how it relates to dominance, and gives properties that will be useful for computation. Section 9 discusses the importance of the possibly optimal and possibly strictly optimal alternatives in incremental preference elicitation. Section 10 describes the EEU method. Section 11 brings together the computational techniques for the weighted multi-attribute utility case. Sections 12 and 13 describe the implementation and experimental testing, and Sect. 14 concludes the paper.

Related work
Multiattribute utility theory (MAUT) [37] involves numerical representations of user preferences with respect to alternatives evaluated over multiattribute spaces. Imprecisely specified multiattribute utility theory (ISMAUT) [68] is one of the earliest attempt to deal with parameterised utility information representing user preferences with linear inequalities and reducing the set of alternatives to those that are not dominated by any other alternative. Related research such as [33] and [67], deals with similar issues.
A major division in recent work on parameterised user preference models is whether a Bayesian model is assumed over the scenarios (corresponding to the different user preference models), or if there is a purely qualitative (logical) representation of the uncertainty over scenarios, where all we represent is that the scenario is in a set W. Bayesian approaches include [12,13,20,64]. Work involving a qualitative uncertainty representation includes [7,14,18,42,62]. Linear imprecise preference models, including those based on a simple form of MAUT model, have been considered in work such as [18,36,41,49] including in a conversational recommender system context [19,62,63].
Parameterised preference models are commonly used with interactive preference elicitation approaches (see, e.g., [9,30,38,54,57]). In this context, the purpose is to explore the alternatives based on different interactions with the decision-maker and without listing all the available alternatives. The parameterisation represents different decision-maker's preference scenarios, and the likelihood or the restrictions on the parameters represent the information obtained by the interaction with the decision-maker. Methods that iteratively interact with the decision-maker to reduce the uncertainty about a parameterised preference models are also called Incremental [3,10,40]. In general, the purpose of such methods is to recommend alternatives to the decision-maker without defining a precise utility function using methods such as the minimax regret criterion (see Sect. 8). A typical interactive approach consists of the following steps: 1. Computation: generate some undominated solutions. 2. Interaction: show to the decision-maker some of the generated solutions asking to input new preference information. 3. Termination: interruption of the elicitation process by the decision-maker or if some specified stopping criterion has been satisfied.
Different approaches have been explored to generate new queries for the decision-maker (see, e.g., [52]). A classical interactive approach is based on a comparison of alternatives (see, e.g., [53,75]), i.e., the preference elicitation system asks the decision-maker to specify their preference between a set of alternatives, and the response is used to reduce the uncertainty of the preference model. Ideally, the uncertainty of the preference model should be reduced to a scenario in which we have a unique optimal alternative according to the preference information collected. However, often this is unfeasible or requires too many interactions with the decision-maker. Because of this, the parameterisation of utility functions leads to different notions of optimality that can be used to classify alternatives (see, e. g., [71]). In Sect. 4 we consider a number of operators representing different notions of optimality. The set UD W ðAÞ is a subset of A not including strictly dominated alternatives, a natural generalisation of the Pareto-optimal elements, appears in many contexts, e.g., [36,42]. Possibly optimal PO W ðAÞ (also known as potentially optimal) elements have been considered in many publications, such as [6,10,28,29,33,72]. See Sect. 11 for details and references about the computation of UD W ðAÞ and PO W ðAÞ for linear and convex utility functions. The possibly strictly optimal set PSO W ðAÞ and the maximally possibly optimal set MPO W ðAÞ have been considered much less [46,70,71]. Regret-based decision making has a long history, with recent work in AI including [7,14,18]. We describe in Sect. 8 the relationship between the dominance relation < W 889 and setwise max regret [62,65]. See Sect. 11 for details and references about the computation of the setwise max regret and the setwise minimax regret for linear and convex utility functions.

Basic terminology and dominance relations between sets
In this section we give some basic definitions, in particular regarding utility-dominance and related dominance relations, and utility-equivalence. Section 3.1 describes the basic set up, involving a parameterised family of utility functions over a set X of alternatives. There are natural dominance and equivalence relations induced between alternatives, as described in Sect. 3.2. The utility of a finite subset of alternatives is defined to be the maximum utility over the alternatives in the set. This leads to a natural (utility-)dominance relation between finite sets of alternatives, and the corresponding utility-equivalence relation, as described in Sect. 3.3, where A utility-dominates B if and only if in every compatible scenario, the utility of A is at least as great as the utility of B. We also define two computationally simpler dominance relations between sets, which are useful as sufficient conditions for utilitydominance.

Uncertain preference structures
We consider a (possibly infinite) set X of alternatives, and another set U, the elements of which we call scenarios, that corresponds with a set of user preference models. With each scenario w 2 U is associated a utility function f w on X, i.e., a function from X to IR; this gives rise to a total pre-order < w on X given by a< w b () f w ðaÞ ! f w ðbÞ, for a; b 2 X.
For each w 2 U, we also define associated relations 1 w and w in the standard way: a1 w b if and only if a< w b and :ðb< w aÞ, which is if and only if f w ðaÞ [ f w ðbÞ. We define a w b if and only if a< w b and b< w a, which is if and only if f w ðaÞ ¼ f w ðbÞ.
Notation M: We use the symbol M to represent the set of finite non-empty subsets of the set of alternatives X.
Each A 2 M gives rise to a function giving values of utility for each element of U. This expresses how good the set A is, with respect to different user models.
Definition 1 (Utilities associated with A.) We define, for w 2 U, Ut A ðwÞ to be max a2A f w ðaÞ.
We do not a priori assume anything about the functions f w ; however certain mathematical results make additional assumptions, such as continuity with respect to w. Of particular interest in this paper is the case when f w ðaÞ is a linear function of w, where U ¼ IR p for some p, and so f w ðaÞ can be written as P p i¼1 a i wðiÞ ¼ ða 1 ; . . .; a p Þ Á w, for some reals a i , with a i representing how good alternative a is with respect to objective/criterion i. We then write the vector ða 1 ; . . .; a p Þ asâ. Thus, in the linear case, for each a 2 X there exists an associated vectorâ 2 IR p , and f w ðaÞ ¼â Á w for all w 2 U.
If a 2 IR p then we could setâ ¼ a, giving a simple weighted sum utility function. However, this linear function case also covers more complex preference models, including GAI representations [16,31], Ordered Weighted Averages [73], and preference models based on Choquet integrals [7,32], since the utility functions in these cases are linear in the parameter w (although not linear with respect to a in the latter two cases). For the GAI and Choquet representations, there are a larger number of parameters, so the dimension p of the utility and parameter spaces is larger.

Dominance and equivalence between alternatives
Very often we will have information that restricts the set of scenarios (i.e., user models). In particular, the user may previously have answered some queries, and we then only consider the set W of user models compatible with their answers.
For W U we define relation < W on X by a< W b () for all w 2 W, a< w b. Thus, a< W b holds if and only if a is at least as good as b in every scenario in W. We define 1 W to be the strict part of < W , i.e., for a; b 2 X, a1 W b if and only if a< W b and b 6 < W a. Thus, a1 W b if and only if a is at least as good as b in every scenario in W, and strictly better in at least one scenario in W. Relation 1 W is transitive and acyclic. We define equivalence relation W to be the symmetric part of < W , given by a W b if and only if a< W b and b< W a.
We define the notion of being equivalence-free. Alternatives a and b are equivalent, i.e., a W b, if and only if a w b for all w 2 W. It can be unnecessary to include two equivalent alternatives a and b in a set of alternatives A, since a should be acceptable if and only if b is acceptable. In an equivalence-free set of alternatives no two alternatives are equivalent: Definition 2 (Equivalence-free) We say that A (2 M) is W -free (or equivalence-free) if for all a; b 2 A, we have a 6 W b.
One can reduce any A to an equivalence-free set A 0 by including exactly one element in A 0 of each W -equivalence class in A.

Dominance relations between sets
Given a set of scenarios W, we will define three relations, < W 889 , < W 898 and < W 988 , between sets of alternatives A and B in M, that specify when A is better than B. These relations are based on a set W of scenarios, and the corresponding set of relations, < w , for w 2 W. In this paper we focus especially on the relation < W 889 , with the other two relations being useful computationally. A< W 889 B holds if and only if in every scenario w 2 W, there's an element of A that is at least as good as any element of B. The relation thus relates to Question (2) in the introduction. This section gives some basic properties of these three dominance relations between sets, that are useful for our algorithmic methods. In the notation in the subscript for the three relations (such as 898 in relation < W 898 ), the first 8 symbol relates to the quantification for all b 2 B, the second 8 symbol relates to for all w 2 W, and the 9 symbol relates to there exists a 2 A.
Relation ; 5Þg be sets of utility vectors of hotels in Paris and Lisbon respectively. For example, the first value of each utility vector could be a score for the location and the second value could be a score for cleanliness, where the higher the score, the better. We assume linear utility functions of the form representing different normalised weightings of the two criteria. We assume that the user has an associated weights vector that is unknown and we want to recommend to the user a trip to Paris or Lisbon based on her preferences on the available hotels. Suppose then that we ask the user for her preference between the hotel with utility vector (10,4) and the hotel with utility vector (11,2). An input preference of (10, 4) over (11,2) implies w Á ð10; 4Þ ! w Á ð11; 2Þ and so 2w 2 ! w 1 and thus, w 1 2 3 , leading to the set of scenarios W ¼ fðw 1 ; w 2 Þ : w 1 þ w 2 ¼ 1 & 0 w 1 2 3 g. This example is illustrated in Fig. 1 4 Some choice functions associated with the set W of scenarios Given a finite set A of alternatives, and a set of scenarios (i.e., user models) W, some alternatives may very well be of less interest than others. It is therefore often desirable to define a reduced, i.e., filtered, set OP W ðAÞ of alternatives, by eliminating elements that are considered to be non-optimal. OP W will be a choice function, i.e., it maps a finite set A of alternatives to a subset of A. There are a variety of the different natural ways of defining such a choice function OP W . We consider UD W ðAÞ, which removes strictly dominated alternatives from A, and PO W ðAÞ, which removes alternatives that are not possibly optimal, i.e., not optimal with respect to any scenario in W, and two refined variations, MPO W ðAÞ and PSO W ðAÞ. MPO W ðAÞ consists of alternatives in A that are optimal in a maximal set of scenarios.
Our focus is especially on PSO W ðAÞ, the set of possibly strictly optimal alternatives; for equivalence-free A, this is the set of alternatives that are uniquely optimal in some scenario, i.e., they are (strictly) better than any other element of A. Our interest in PSO W ðAÞ is particularly because of the close relationship with minimal equivalent subsets (shown in later sections such as Sects. 5 and 6). The operators PO W , UD W and MPO W can all be useful in the efficient computation of PSO W ðAÞ. This section gives basic properties of these operators, and their relationships with utility-dominance and utility-equivalence. PO W , UD W and MPO W all always maintain utility-equivalence, i.e., for any A 2 M, PO W ðAÞ, UD W ðAÞ and MPO W ðAÞ are all utility-equivalent with A. Although PSO W does not maintain utility-equivalence in general, we will later show that for many natural forms of uncertain preference structure, PSO W does maintain utility-equivalence.
The operators UD W , PO W , MPO W and PSO W defined below, respect the equivalence W in that if OP W is any of these operators and a; b 2 A are such that a W b then OP W ðAÞ 3 a () OP W ðAÞ 3 b.
We first define the set UD W ðAÞ of undominated alternatives in A.
Definition 5 (The Undominated set UD W ðAÞ.) For A 2 M we define UD W ðAÞ to be the set of a 2 A such that there does not exist c 2 A such that c1 W a. Thus, the element a of A is not in UD W ðAÞ if and only if there exists some c 2 A such that c is at least as good as a in every scenario, and strictly better in at least one scenario. The set UD W ðAÞ is a natural generalisation of the Pareto-optimal elements, and is sometimes referred to as the set of undominated elements in A.
Proposition 5 gives some basic properties of the relationships between function UD W and the three dominance relations; in particular, part (ii) shows that it preserves equivalences between sets of alternatives, and implies that UD W ðAÞ is non-empty for non-empty A. We next define the Possibly Optimal Set PO W ðAÞ.
Definition 6 (O w ðAÞ and Possibly Optimal Set PO W ðAÞ.) For each w 2 U and A 2 M we define O w ðAÞ to be all elements a of A that are optimal in A in scenario w, i.e., such that for all b 2 A, a< w b. For W U we define PO W ðAÞ to be S w2W O w ðAÞ, the set of alternatives that are optimal in some scenario, i.e., optimal for some consistent user preference model.
Next we define the possibly strictly optimal Set PSO W ðAÞ: Definition 7 (SO W w ðAÞ and Possibly Strictly Optimal Set PSO W ðAÞ.) We define SO W w ðAÞ to be all elements a of A such that a1 w b, for all b 2 A with b 6 W a. These elements a are said to be strictly optimal in scenario w. We then define PSO W ðAÞ, the set of possibly strictly optimal elements, to be S w2W SO W w ðAÞ, i.e., all the elements that are strictly optimal in some scenario in W.
For equivalence-free A 2 M, the set PSO W ðAÞ consists of all alternatives a 2 A that are uniquely optimal in some scenario w 2 W (i.e., O w ðAÞ ¼ fag). It can be easily seen that PSO W ðAÞ PO W ðAÞ \ UD W ðAÞ.
It is convenient to have a notation for the set of scenarios in which a is optimal in A: We now define the maximally possibly optimal elements to be those that are optimal in a maximal set of scenarios. In other words, if alternative a in A is not maximally possibly optimal in A then there exists an alternative b in A that is optimal in every scenario that a is optimal (and at least one more scenario  Fig. 1 we can see a graphical interpretation of Opt A W ðð4; 7ÞÞ ¼ ½0; 1 3 , i.e., w 1 2 ½0; 1 3 is an interval in which there is no line strictly above the line associated to (4,7). We have ð4; 7Þ 2 PSO W ðAÞ because for any w 1 2 ½0; 1 3 Þ the line associated to (4, 7) is (strictly) above all the other lines, and ð6; 6Þ 2 PO W ðAÞ because at w ¼ 1 3 there is no line (strictly) above the line associated to (6,6).
h We now give some basic properties of the operator PSO W . Part (i) shows that for any utility-equivalent subset B of an equivalence-free set A contains any possibly strictly optimal elements. Part (ii) is used to show part (iii), which implies that the relation < W 889 could be used for computing PSO W ðAÞ.   Example 2 Let W ¼ fw 1 ; w 2 ; w 3 g, let A ¼ fa; b; cg and suppose that a w1 b1 w1 c; and b w2 c1 w2 a; and c w3 a1 w3 b. A is equivalence-free, and no alternative dominates any other alternative; e.g., a 6 1 W b because b1 w 2 a. Thus, UD W ðAÞ ¼ A. We have fa; bg W

889
A since in every scenario either a or b is optimal in A. Similarly, fa; cg and fb; cg are utilityequivalent to A. However, it is not the case that fa; bg W 898 A because neither a or b dominate c. We have Opt A W ðaÞ ¼ fw 1 ; w 3 g because a is optimal in scenarios w 1 and w 3 , and we have Opt A W ðbÞ ¼ fw 1 ; w 2 g, and Opt A W ðcÞ ¼ fw 2 ; w 3 g. Since these three sets are nonempty, each alternative is possibly optimal, and because none contains any other, each alternative is maximally possibly optimal. Hence, PO W ðAÞ ¼ MPO W ðAÞ ¼ A. However, there are no possibly strictly optimal alternatives, i.e. PSO W ðAÞ ¼ ;. The reason is that we have O w 1 ¼ fa; bg, and O w 2 ¼ fb; cg, and O w 3 ¼ fb; cg, so there are no uniquely optimal alternatives. Hence, PSO W ðAÞ is not utility-equivalent to A, i.e., PSO W ðAÞ 6 W 889 A.

Setwise-minimal equivalent subsets and the possibly strictly optimal elements
In this section we consider the issue of replacing A with an (utility-)equivalent subset of A that is minimal. This relates with Question (1) in the introduction. We show that, for equivalence-free set A of alternatives, the set of possibly strictly optimal elements PSO W ðAÞ is the intersection of all the minimal equivalent subsets. Also, we give a sufficient condition for PSO W ðAÞ to be utility-equivalent with A (W being A-extendable), and in this case, PSO W ðAÞ is the unique minimal utility-equivalent subset of A. Setwise-minimal equivalent subsets: We may want to reduce A to a utility-equivalent subset that cannot be reduced any further, i.e., there is no proper subset of it that is also utility equivalent to A. Theorem 1 determines when there is a unique such subset.
Definition 10 (SME W ðAÞ.) We define SME W ðAÞ to be the set of subsets B of A that are setwise-minimal equivalent to A, i.e., such that B W 889 A and there does not exist any strict subset C of B such that C W 889 A.
In Sect. 7.1 we give a simple method for finding setwise-minimal equivalent subsets.

5.1
Relating SME W ðAÞ and PSO W ðAÞ in the general case Theorem 1 below gives some relationships between PSO W ðAÞ, SME W ðAÞ and the dominance relation < W 889 , for equivalence-free A. Any setwise-minimal equivalent subset of A contains PSO W ðAÞ, the set of possibly strictly optimal elements. The latter set is equal to the intersection of all the setwise-minimal equivalent subsets, and is equivalent to A if and only if there is a unique minimal equivalent subset, which is thus equal to PSO W ðAÞ.
The condition that PSO W ðAÞ is equivalent to A holds in the linear multi-objective case considered in Sect. 11 below (see Theorem 3), and so then PSO W ðAÞ is the unique minimal equivalent subset of A.
Theorem 1 Let W U be a set of scenarios and assume that A (2 M) is W -free. Then the following hold:  [1,21]. The path independence property allows the computation of the best elements to be performed in a modular way [45,47,72]

The case of A-extendable W
In this section we consider a condition, that W is A-Extendable, which is sufficient for the function PSO W to maintain utility-equivalence (i.e., for the equivalence PSO W ðBÞ W 889 B to hold for all subsets B of A); thus, by Theorem 1, given this condition, PSO W ðAÞ is equal to the unique setwise-minimal equivalent subset for equivalence-free A. Because of Proposition 8(iii) above, the sufficient condition also implies that PSO W , viewed as a function on 2 A , satisfies path independence. In Sect. 6 we show that this sufficient condition holds for a large class of utility functions, including the linear case. Our algorithms for computing the minimal equivalent subset for A make use of the utility-equivalence property, i.e., that PSO W ðAÞ W 889 A. Loosely speaking, W is A-Extendable if for every element w of W there exists an element w 0 whose preference ordering is more precise than (or equal to) that for w, and such that w 0 totally orders non-equivalent elements of A. If this latter condition holds we say that w 0 is total over A given W, and W 6 ¼ A is defined to be the set of all such w 0 .
Definition 11 [Total w over W, and W 6 ¼ A .] Given w 2 W U and A 2 M, we say that w is total over A given W if for all a; b 2 A, either a1 w b or b1 w a or a W b. Let W 6 ¼ A be the set of all w 2 W that are total over A given W.
We say that W is A-Extendable if for all w 2 W there exists w 0 2 W that is total over A given W (i.e., w 0 2 W 6 ¼ A ) and that extends w over A.
Consider a set of scenarios W like in Example 1, and consider an arbitrary finite set A of pairs of real numbers. Not every element of W is necessarily total over A. For instance, if A contains the pairs (3,2) and (0, 4) then w ¼ ð0:4; 0:6Þ ranks (3, 2) the same as it ranks (0, 4), i.e., ð3; 2Þ w ð0; 4Þ, even though ð3; 2Þ 6 W ð0; 4Þ. However, W is A-Extendable, i.e., any scenario in W can be extended to one that is total over A. To illustrate this for w, consider positive [ 0; then w 0 ¼ ð0:4 þ ; 0:6 À Þ will have ð3; 2Þ1 w 0 ð0; 4Þ, and, if we choose to be sufficiently small we will have for all a; b 2 A, (i) if a1 w b then a1 w 0 b; and (ii) if a w 0 b then a ¼ b. (The fact that A is finite is crucial for this to hold.) By (i), w 0 extends w over A, and by (ii), w 0 is total over W, i.e., w 0 2 W 6 ¼ A . The results of Sect. 6 show that the A-Extendable property holds very often for continuous sets of scenarios W; roughly speaking, it holds if f w ðaÞ is a completely smooth function of w. Extendability also holds for classes of discrete W such as for the sets of lexicographic preference models defined from the compositional preference languages in [70].
We give three lemmas that enable the proof of Theorem 2 below. The following result, which follows almost immediately from the definitions, shows that the distinction between PSO and PO disappears when all scenarios are total.
Lemma 2 Suppose that subset T of W only contains w that are total over A given W. Then for all B A, PSO T ðBÞ ¼ PO T ðBÞ.
Note that for any w 2 W 6 ¼ A , relations w and W are equal (when viewed as relations on A) and so, W 6 ¼ A ¼ W ; we therefore have the following result:

Lemma 4 shows that, under the assumption that
Then, for any B; C A, we have The theorem below shows that PSO W maintains utility-equivalence over subsets of A if W is A-Extendable. 6 PSO W ðAÞ as unique minimal equivalent set in the continuous case

Theorem 2 Suppose that A 2 M and that the set of scenarios
The results in this section show that for continuous sets W of user models, under natural assumptions on the utility functions, we have that the set of possibly strictly optimal alternatives PSO W ðAÞ is the unique minimal equivalent set for A. In particular, we show that a particular property, the Identity property, is sufficient for W being A-Extendable (see Sect. 5.2) and hence for PSO maintaining utility-equivalence. The Identity property, which holds for important classes of function, states essentially that if two utility functions over W are equal locally, i.e., within a neighbourhood of a point in W, then they are globally equal. This property holds for linear utility functions (where f w ðaÞ ¼ w Áâ) and other polynomial (and analytic) functions. The results also imply that in the linear case and when W is convex, PSO W ðAÞ ¼ MPO W ðAÞ; this enables then a method for computing the minimal equivalent set by computing MPO W ðAÞ.
In this section we will be considering, for a given alternative a, how f w ðaÞ varies as a function of w. It is then convenient to make the following definition: For each a 2 X, we define function f a on U by f a ðwÞ ¼ f w ðaÞ, for each w 2 U. For a; b 2 X, and W U we define W a6 ¼b ¼ fw 2 W : f a ðwÞ 6 ¼ f b ðwÞg. These are the scenarios for which a has different utility from b. Recalling the definition of W 6 ¼ A from Sect. 5.2, we have, for A 2 M, that W 6 ¼ A is equal to the set of all elements w of W such that f a ðwÞ 6 ¼ f b ðwÞ holds for all a; b 2 A with a 6 W b. Thus, Subset W 0 of W is said to be closed if its complement W n W 0 in W is an open set in W. For example, it can be shown that Opt A W ðaÞ (the set of elements in W that make a optimal in A) is always a closed set in W. For arbitrary subset W 0 of W, its closure ClðW 0 Þ is defined to be the intersection of all closed sets in W that contain W 0 , which is the unique smallest closed set containing W 0 . If ClðW 0 Þ ¼ W then we say that W 0 is dense in W; this means that for every element of W there exists an arbitrarily close element of W 0 .

The Identity property
We first define the Identity property, which in Sect. 6.2 we show is sufficient for W being A-Extendable and hence for PSO W maintaining utility-equivalence (by Theorem 2). We go on to show that the Identity property holds for sets of linear functions of the scenario parameter w, as well as for sets of multivariate polynomials.
Definition 13 (Identity Property.) Let F be a set of continuous real-valued functions on a subset W of IR p (and similarly, for an arbitrary metric space W). We say that F satisfies the Identity property if for every f ; g 2 F, if f and g agree on any non-empty open subset of W then they agree on W. In other words, if there exists a non-empty open subset T of W such that for all w 2 T , f ðwÞ ¼ gðwÞ then for all w 2 W, f ðwÞ ¼ gðwÞ.
Thus, F satisfies the Identity property if and only if when two of the functions are locally equal then they are globally equal. Loosely speaking, it holds for classes of functions that are completely smooth.
Observation: If the Identity property holds for F and G F then the Identity property holds for G.

The Identity property for linear functions of w
The Identity property holds for classes of natural functions, in particular, for linear functions.
Proposition 9 Assume that each a in finite set A is associated with a vectorâ 2 IR p . Let W be a convex subset of IR p , and define, for each a 2 A, the function f a by f a ðwÞ ¼â Á w, representing the utility of alternative a in scenario w. Then the set of functions ff a : a 2 Ag satisfies the Identity Property.
Note that we're not making any assumptions at all on the form of the function that maps a tô a. (It can also be shown that the assumption that W is convex can be considerably weakened.)

The Identity Property for multivariate polynomials
Consider, for instance, a convex set W IR p , and where, for each a 2 A, the function f a is a multivariate polynomial function of the components of w. For example, the function g given as follows is a multivariate polynomial in p ¼ 3 variables w 1 , w 2 and w 3 (with these three variables being the components of w): gðwÞ ¼ 3w 1 À 4:5w 2 þ 2w 1 w 2 À À3:2w 1 w 2 w 2 3 þ w 3 2 w 2 3 þ 9w 2 1 w 3 . More generally for the multivariate polynomial case, f a ðwÞ can be written as P s r s ðaÞ G s ðwÞ where the sum is finite and each s corresponds with a p-dimensional vector of non-negative integer indices, and G s ðwÞ is the corresponding product w sð1Þ 1 Á Á Á w sðpÞ p , and each r s is a function on A.
If W is a convex set (or similarly, a finite union of convex sets of the same dimension) and for each a 2 A we have that f a is some multivariate polynomial function of the components of w (2 W) then the set of functions ff a : a 2 Ag satisfies the Identity Property.
To see that this is the case, the first step is to rewrite the multivariate polynomial in terms of k components w i , where k is the dimension of W. This is always possible; for example, if k ¼ p À 1 then we just have the constraint P i w i ¼ 1, so we can replace w p by 1 À ðw 1 þ Á Á Á þ w pÀ1 Þ, and multiply out to obtain a multivariate polynomial in p À 1 variables. To show the Identity Property we need to show that if two multivariate polynomials are equal in a neighbourhood of u 2 W then they are equal on the whole of W. By taking the difference between the multivariate polynomials this is equivalent to: if a multivariate polynomial is equal to zero in a neighbourhood of u then it is equal to zero on all of W, i.e., it is the zero function. Now, this is the case since if a multivariate polynomial P s r s G s ðvÞ equals zero for all v 2 IR k close to u then each r s has to be zero. The latter fact can be shown by reasoning as follows. For any fixed values of variables w 1 ; . . .; w kÀ1 close to values u 1 ; . . .; u kÀ1 we can consider the multivariate polynomial as a (univariate) polynomial in w k , which is zero in an interval around u k , and thus, by a basic classical result, is equal to the zero polynomial. Hence, for each power of u k , the corresponding coefficient is zero, where the corresponding coefficient is a multivariate polynomial in the k À 1 variables fw 1 ; . . .; w kÀ1 g. Iterating this for k À 1; k À 2; . . .; 1 implies that each coefficient r s is equal to zero, as required.

An example when the identity property does not hold
We have shown that the Identity property holds some for some important classes of smooth functions. Here we give an example of a very small set of functions where the Identity property fails to hold.

Example 3 Consider an example based on Example 2 of Sect. 4, again with
As in Example 2 we have PSO W ðAÞ being empty. The Identity property does not hold for the set of functions ff a ; f b ; f c g because, for instance, f a ðw 1 Þ ¼ f b ðw 1 Þ for all w 1 in any open ball contained in W 1 , so the two functions f a are f b are locally equal, but they are not globally equal, since f a ðw 2 Þ 6 ¼ f b ðw 2 Þ for w 2 2 W 2 . The functions cannot all be smooth, and in fact there must be discontinuities in the functions at the boundaries between the regions. Even if we retract the assumption that the union of the three regions covers W, we will still have that the Identity property fails, for the same reason.

Consequences of the identity property
The lemma below establishes equivalent forms of the Identity property. For our purposes the key implication is (a) ) (c), i.e., that the Identity property implies that W 6 ¼ A is dense in W. The Identity property (a) implies the property (b) that for every non-equivalent a; b 2 A, W a6 ¼b is dense in W, i.e., for every element w in W, there is an arbitrarily close element in W that distinguishes a and b. If this were not the case, then there would be an open set (a neighbourhood) containing w in which every scenario makes a and b equal, i.e., f a ðw 0 Þ ¼ f b ðw 0 Þ for all w 0 in the open set; but the Identity property would then imply that f a ¼ f b , i. e., a and b are equivalent, contradicting the assumption. From (b) it can be shown that (c) Lemma 5 Let A 2 M and assume, for each a 2 A, that the function f a is a real-valued continuous function on the metric space W. The three conditions below are equivalent, i.e., if one holds then the other two also hold.
(a) The set of functions ff a : a 2 Ag on W satisfies the Identity property.
If the topological closure ClðW 6 ¼ A Þ of W 6 ¼ A is equal to W then for any element w in W, we can find arbitrarily close other elements w 0 that totally order (non-equivalent elements of) A. Because of the finiteness of A and the continuity of the functions f a this implies that we can then find such an element w 0 2 W that extends w, showing that W is A-Extendable.
Lemma 6 Let A 2 M and assume, for each a 2 A, that the function f a is a real-valued continuous function on the metric space W. If W equals the topological closure Putting Lemmas 5 and 6 together we obtain the following result showing that the Identity property is a sufficient condition for W to be A-Extendable.
Proposition 10 Let A 2 M and assume, for each a 2 A, that the function f a is a realvalued continuous function on the metric space W. If the set of functions ff a : a 2 Ag on W satisfies the Identity property then W is A-Extendable.
Proposition 10, together with Theorem 2, implies that the Identity property is sufficient for PSO W to maintain utility-equivalence, and the existence of unique minimal equivalent subset for a set A.
Theorem 3 Let C 2 M and assume, for each a 2 C, that the function f a is a real-valued continuous function on the metric space W, and that the set of functions ff a : a 2 Cg satisfies the Identity property. Then the following hold, for all non-empty A C.
(i) PSO W ðAÞ W 889 A. (ii) If A is W -free then there exists a unique setwise-minimal equivalent subset for A, i. e., SME W ðAÞ is a singleton, and this equals PSO W ðAÞ.
Proof (i): Since ff a : a 2 Cg satisfies the Identity property, by Proposition 10, W is A-Extendable, and thus, PSO W ðAÞ W 889 A, using Theorem 2. (ii): If A is W -free then, by part (i) and Theorem 1, PSO W ðAÞ is the unique setwiseminimal equivalent subset for A. h Theorem 3 immediately implies that PSO W maintains utility-equivalence for the linear case (by Proposition 9), as well as for other cases such as for multivariate polynomial utility functions. The assumption that W is convex can be substantially weakened, e.g., to W being a finite union of convex sets of the same dimension.
Corollary 2 Assume that each a in finite set A is associated with a vectorâ 2 IR p . Let W be a convex subset of IR p , and define, for each a 2 A, the function f a by f a ðwÞ ¼â Á w, representing the utility of alternative a in scenario w. Then we have PSO W ðAÞ W 889 A.
We observed earlier that an alternative a is possibly optimal (a 2 PO W ðAÞ) if and only if the set of scenarios Opt A W ðaÞ ( W) is non-empty (see Definition 8 in Sect. 4). When the Identity property holds (including the cases of linear and polynomial utility functions), we have a related statement regarding possibly strict optimality, namely: a is possibly strictly optimal if and only if Opt A W ðaÞ has a non-empty interior, which corresponds, at least in the linear case, with Opt A W ðaÞ having the same dimension as W (this follows from Auxiliary Lemma 15 in the appendix, since Opt A W ðaÞ is then convex). In the running example (see the continuation of Example 1 in Sect. 4), the alternative (6, 6) is possibly optimal but not possibly strictly optimal, which is reflected by Opt A W ð6; 6Þ ¼ f 1 3 g, being a non-empty set of smaller dimension than W.
Proposition 11 Let C 2 M and assume, for each a 2 C, that the function f a is a realvalued continuous function on the metric space W, and that the set of functions ff a : a 2 Cg satisfies the Identity property. Suppose that A C. For a 2 A, we have a 2 PSO W ðAÞ if and only if Opt A W ðaÞ contains a non-empty open set, i.e., has a non-empty interior.
We show below that for the linear case with convex W, we have that the maximally possible optimal elements are the same as the possibly strictly optimal elements, so that MPO W ¼ PSO W . We use this property as a basis of our algorithm for computing the set of possibly strictly optimal elements (which equals the minimal equivalent set) in Sect. 12.1.

Corollary 3
Assume that W is a convex subset of IR p , and consider A 2 M and assume that for each a 2 A there existsâ 2 IR p such that for all w 2 IR p , f w ðaÞ ¼ w Áâ. Then MPO W ðAÞ ¼ PSO W ðAÞ.
7 Filtering algorithms for minimal equivalent subsets, PSO W ðAÞ and MPO W ðAÞ this section includes two simple kinds of filtering to reduce the input set of alternatives, both of which can be used for the computation of the possibly strictly optimal elements. In Sect. 7.1 we use a simple filtering method to compute a setwise-minimal equivalent set, and thus, in certain circumstances, PSO W ðAÞ (such as when the hypotheses of Theorem 3 or Corollary 2 hold). We use this in a linear programming algorithm for computing the set of Possibly Optimal elements in Sect. 11.3 below, which is used in the method (a) in Sect. 12.1. In Sect. 7.2 we show how the set of maximally possibly optimal elements MPO W ðAÞ can be computed using a certain dominance relation between alternatives in A. This therefore gives another method for computing PSO W ðAÞ when PSO W ðAÞ ¼ MPO W ðAÞ (cf. Corollary 3 above). We use this in the (b) algorithm in Sect. 12.1 below.

Filtering with relations on sets
A simple way of generating a minimal equivalent subset of A is to sequentially delete elements a of A that are not needed for maintaining equivalence, i.e., such that Let us label A as a 1 ; . . .; a n , where n ¼ jAj. Formally, the labelling is a bijection r from f1; . . .; ng to A (so that rðiÞ ¼ a i ), and let K be the set of all n! labellings. We define Filter r ðA; < W 889 Þ iteratively as follows. We set We then define Filter r ðA; < W 889 Þ to be A n , i.e., the set remaining after iteratively deleting elements from A that are dominated with respect to relation < W 889 . As the proposition below states, when applying the filtering operation Filter r ðA; < W 889 Þ, (i) equivalence is always maintained; and (ii) we always obtain a minimal equivalent subset, and any such subset can be achieved for some ordering. Part (iii) implies that for any labelling r we have Filter r ðA; Proposition 12 Let W ( U) be a set of scenarios, let A 2 M and let r be any labelling of A. Then we have: Hence, we can generate a setwise minimal equivalent subset of A by choosing any labelling r and computing Filter r ðA; < W 889 Þ. Furthermore, this will be equal to PSO W ðAÞ, and be the unique setwise minimal equivalent subset, if the operator PSO W maintains utility-equivalence, such as in the linear case (see Corollary 2).

A structure for computation of MPO W ðAÞ
The maximally possibly optimal elements of A are those that are undominated with respect to a certain strict partial order (Opt A W -dominance) as defined below. This leads to a simple method for computing MPO W ðAÞ, by iteratively deleting elements, if one has a method for testing the Opt A W -dominance relation.
The fact that Opt A W -dominance is a strict partial order means that one can iteratively delete Opt A W -dominated elements from A until one reaches the set of Opt A W -undominated elements, i.e., MPO W ðAÞ. The result below formally expresses this fact.
Proposition 13 Let W ( U) be a set of scenarios and let A 2 M. Let A 1 ; . . .; A k be a sequence of sets with (i) In the context of artificial intelligence, it can be used to recommend an alternative that minimises the max regret (i.e., the worst-case loss) with respect to a utility function and all the available alternatives [13,14,17,50]. Applications include, for example, the elicitation of multi-attribute utilities (see, e.g., [10,18,66]), or the elicitation of preferences for ranking and voting problems (see, e.g., [7,8,40]). The utility-dominance condition A< W 889 B states that in every scenario, the set of alternatives A is at least as good as the set B; or, equivalently, Ut A ðwÞ ! Ut B ðwÞ for all w 2 W, by Proposition 3. A natural related numerical measure is setwise max regret SMR W ðA; BÞ defined below which is a generalisation of the max regret and expresses how much worse A could be than B, i.e., the maximum regret of choosing A over B [59,62,65]. For example, a set minimising the SMR could be used as a recommendation set in a decision-making problem.
Recommendation sets can also be used in elicitation, where they are treated as choice queries (i.e., queries of the kind "Among alternatives a, b, and c, which one do you prefer?") with the goal of reducing uncertainty to improve the quality of future recommendations; that is, reducing minimax regret. To minimise the number of interactions with the decision-maker, we need to carefully choose the queries to reduce the uncertainty as fast as possible. Ideally, evaluating a question at a given iteration should take into account all future questions and possible responses (e.g., [13,34]). However, in practice, this evaluation is often carried out myopically. It turns out [65] that optimal recommendation sets with respect to SMR are also myopically optimal in an elicitation sense, as they ensure the highest worst-case (with respect to the possible query's responses) reduction of minimax regret a posteriori.
Definition 15 ( SMR W ðA; BÞ.) For W U and finite subsets A and B of the set of alternatives, X we define the setwise max regret SMR W ðA; BÞ to be sup w2W Ut B ðwÞ À Ut A ðwÞ.
When A B, SMR W ðA; BÞ is non-negative and equals the setwise max regret SMRðA; WÞ defined in [62]; that paper defines a method that involves finding a subset A of B (among a particular class of subsets, e.g., all those of a fixed cardinality k) that minimises SMR W ðA; BÞ. 2 A can then be considered as a maximally informative query, to be used in an incremental elicitation process for finding an optimal element of B. SMR W ðA; BÞ is closely related also to the notion of setwise max regret defined in [5].
Regarding Ut A ðwÞ as the utility achieved from set A in scenario w (and similarly, for Ut B ðwÞ), we have that SMR W ðA; BÞ is the worst-case loss of utility (or maximum regret) if we choose set A instead of set B. For instance if A is a subset of B, and SMR W ðA; BÞ is very close to zero, then we might consider that A is a sufficiently close approximation of B, simplifying the set of choices for the user. We have SMR W ðA; BÞ 0 if and only if A< W 889 B (see Proposition 14 below). The problem of computing SMR W ðA; BÞ is thus strongly related to that of determining A< W 889 B. The definitions and results from earlier sections (apart from Sect. 6), regarding < W 889 , SME, PO, PSO and UD, depended only on the orderings < w , for w 2 W, and so were ordinal, in the sense that they are not affected by any strictly monotonic transformations of each function f w (where the monotonic transformations can be different for each w). However, this is not the case for SMR, which has much weaker invariance properties.
We say that SMR W ðA; BÞ is achieved if there exists w 2 W such that Ut B ðwÞ À Ut A ðwÞ ¼ SMR W ðA; BÞ, so that then SMR W ðA; BÞ is equal to max w2W Ut B ðwÞ À Ut A ðwÞ. We will mainly be interested in situations in which SMR W ðA; BÞ is achieved; this always happens, for instance, if for each a 2 X, f w ðaÞ is a continuous function of w, and W is compact.
Proposition 14 shows the connections between setwise max regret and (i) utility dominance, (ii) the possibly optimal alternatives and (iii) the possibly strictly optimal alternatives. (i) relates the function SMR W with the relation < The following result shows that we can pre-process A and B using UD W and < W 898 without changing the value of setwise max regret. We use this property in Sect. 9, and in the algorithmic methods in Sects. 11.3 and 12.2.

Implications for incremental preference elicitation
In recent years there has been considerable focus in the AI preference community on incremental preference elicitation techniques, a form of active learning, see e.g., [6,10,11,14,20,62,64]. We argue that the notion of being possibly strictly optimal is important here.
Let a and b be alternatives. Preference model w is said to satisfy a preference statement a ! b if f w ðaÞ ! f w ðbÞ, i.e., a is at least as good as b given w. For set of alternatives A the preference statement a ! A means a ! b for all b 2 A. Thus, for a 2 A, a scenario w satisfies the preference statement a ! A if and only if (given w) a is a most preferred element in A, a 2 O w ðAÞ, i.e., w makes a optimal in A. This holds if and only if w 2 Opt A W ðaÞ. In incremental elicitation a common strategy is to generate a small set of alternatives A, and to ask the user which element of A is most preferred. If they reply "a" then this is interpreted as a ! A. We will then update W to the set of all w 2 W such that a is a most preferred option in A, i.e., we update W to Opt A W ðaÞ. There can be forms of inconsistency, of different kinds, between the user answers and the model we have of the user. We say that, given set of preference models W, alternative a is a feasible answer to query A if Opt A W ðaÞ is non-empty, i.e., there exists some user preference model in W under which a is optimal in A.
For W IR p we say that a is a strongly feasible answer to query A (given W) if Opt A W ðaÞ has a non-empty interior (with respect to the induced topology for W). For standard cases, e. g., when W is convex, this holds if and only if Opt A W ðaÞ has the same dimension as W. In the example in Fig. 1, with the query fð11; 1Þ; ð10; 4Þ; ð7; 5Þ; ð6; 6Þ; ð4; 7Þg, the elements (11, 1) and (7,5) are infeasible answers; e.g., (10,4) is strongly feasible because the dimension of Opt A W ð10; 4Þ ¼ ½ 1 3 ; 2 3 is 1, i.e., the same dimension as W. Alternative (6, 6) is feasible but not strongly feasible, because Opt A W ð6; 6Þ ¼ f 1 3 g and so has smaller dimension than W.
The following result, which is an immediately consequence of Proposition 11, characterises feasible and strongly feasible answers to queries.
Proposition 16 Consider A 2 M and W IR p .
(i) a is a feasible answer to query A given W if and only if a 2 PO W ðAÞ.
(ii) If the set of functions ff a : a 2 Ag satisfies the Identity property, we have that a is a strongly feasible answer to query A given W if and only if a 2 PSO W ðAÞ.
Thus, the feasible answers are the possibly optimal elements, and the strongly feasible answers are exactly the possibly strictly optimal elements, in cases where the Identity property holds, such as for utility values that are linear or polynomial in w.
If the user chooses a from A, and a is not a feasible answer to A, then we get an inconsistency, since the updated W will be empty. Suppose now, on the other hand, a is not a strongly feasible answer to A. We can still consistently update W, so this is a less strong kind of inconsistency; however, such an answer would be seriously troubling. For instance, suppose W IR p , and consider any probability distribution over W, regarding which is the true user model w, such that (as one would expect) the probability distribution is compatible with the measure of the sets. If a is not a strongly feasible answer to query A then the probability that w is such that a ! A holds would be zero (since Opt A W ðaÞ has then measure zero in W, being of lower dimension than W). A choice, by the user, of a would hence correspond with an event of probability zero.
To ensure that every answer to a query A is feasible, we thus require that PO W ðAÞ ¼ A. And, to ensure that every answer to A is strongly feasible, we require that PSO W ðAÞ ¼ A, i. e., that every element of A is strictly possibly optimal in A.
We thus argue that the standard methods for generating queries in incremental preference learning should be modified to ensure that every element in the query set is strictly possibly optimal. 3 Since Theorem 3 implies that PSO W ðAÞ is non-empty, (and indeed equivalent to A) we can therefore replace a potential query A by PSO W ðAÞ.
It is shown in [62,63,65] that choosing a subset A, of the set of available alternatives B, that maximises setwise regret SMR W ðA; BÞ (among small subsets) is a desirable and wellfounded choice for an informative query. However, it can easily happen that, for such a query A, we have PSO W ðAÞ 6 ¼ A and even PO W ðAÞ 6 ¼ A. Such a choice of A is then in danger of leading to an inconsistency, as described above. Fortunately, one can easily solve this problem by replacing A by PSO W ðAÞ, since if A maximises setwise regret then PSO W ðAÞ also maximises setwise regret (assuming the Identity property, as holds for linear functions or polynomial functions of w) because SMR W ðPSO W ðAÞ; BÞ ¼ SMR W ðA; BÞ, by Theorem 3 and Lemma 8(i). Continuing the running example, it can be seen from Fig. 1 that the set ExtðCðW; AÞÞ of the extreme points of the epigraph is equal to fð0; 7Þ; ð 1 3 ; 6Þ ð 2 3 ; 8Þg, where we are again abbreviating w to just its first component w 1 , so that e.g., ð 1 3 ; 6Þ represents the pair ðw; Ut A ðwÞÞ with w ¼ ð 1 3 ; 2 3 Þ. Then, using Theorem 4(iv), SMR W ðA; fð5:5; 6:5ÞgÞ is equal to maxðÀ0:5; 1 6 ; À2 1 6 Þ ¼ 1 6 [ 0; for instance, the middle term in the max equals f w ðð5:5; 6:5ÞÞ À 6 ¼ 1 3 Á 5:5 þ 2 3 Á 6:5 À 6 ¼ 1 6 . Because SMR W ðA; fð5:5; 6:5ÞgÞ is strictly positive, we have that A 6 < W 889 fð5:5; 6:5Þg, using Proposition 14(i). Note that the key extreme point ð 1 3 ; 6Þ of the epigraph does not involve any extreme point of W, so this example illustrates the fact that it is not sufficient to just consider the extreme points of W.

The linear and convex case
Here we focus on the case in which the utility f w ðaÞ is a linear function of w, and when the set of scenarios W is a compact and convex subset of IR p . The results are of most interest when W is also a convex polytope, and thus expressible in terms of linear constraints (so is equal to the intersection of closed half-spaces).
Section 11.1 considers the computation of dominance with respect to the relations < W 898 and < W 988 , making use of the extreme points of W. These are useful as sufficient conditions for utility-dominance for the convex polytope case, since then the set of extreme points is finite. In Sect. 11.2 we give a result that, for the convex polytope case, leads to a straightforward linear programming method for computing setwise max regret and hence (because of the relationship between the two shown in Proposition 14(i)) utility-dominance.
In Sect. 10 we showed how the extreme points of the epigraph of the utility function can be used to compute utility-dominance. Section 11.4 shows how the extreme points of the epigraph can be used to compute the minimal equivalent subset. In particular, it is shown how to compute Opt A W -dominance, and thus, by the method of Sect. 7.2, the set of maximally possibly optimal elements; the results of Sect. 6 (Theorem 3, Corollary 2 and Corollary 3) imply that this is equal to the minimal equivalent subset (of an equivalence-free set of alternatives).
We now consider the situation in which we are especially interested, where an alternative a in X corresponds with a multi-attribute utility vectorâ, and the utility functions are linear in the parameter w, with f w ðaÞ ¼â Á w, i.e., P p i¼1 w iâi , and W is a compact convex subset of IR p . We therefore have, for a; b 2 X, a< w b if and only if ðâ ÀbÞ Á w ! 0. Also, Ut A ðwÞ ¼ max a2A w Áâ.
Of particular interest in the case in which W is also a convex polytope, being defined by a finite number of linear inequalities. Given a finite set K ¼ fk i : i ¼ 1; . . .; kg of vectors in IR p , and corresponding real numbers r i , we can define W to be the set of w 2 U such that for all i ¼ 1; . . .; k, w Á k i ! r i . In particular, such linear inequalities can arise from input preferences of the form a is preferred to b, leading to the constraint w Á ðâ ÀbÞ ! 0.
This form of preferences has been studied a great deal; for instance, UD W ðAÞ consists of the non-dominated alternatives in A for a multiobjective program (MOP) given a cone (with the cone generated as the dual of W) [25,69,74]. Often the elements of W are assumed to be non-negative and normalised, so for each w 2 W we have for all i ¼ 1; . . .; p, w i ! 0, and P p i¼1 w i ¼ 1. Then, without any additional preferences (so that W is just the unit ðp À 1Þsimplex), relation < W is the Pareto ordering on alternatives, and UD W ðAÞ is set of Paretooptimal alternatives, with the supported alternatives being also in PO W ðAÞ. UD W ðAÞ can be computed by discarding the alternatives b 2 A if there exists a 2 A such that f w ðaÞ ! f w ðbÞ for all the extreme points w of W, and f w ðaÞ [ f w ðbÞ for at least one extreme points w of W (see, e.g., [36]). PO W ðAÞ can be computed by discarding all alternatives b 2 A for which there does not exists w 2 W such that f w ðbÞ ! f w ðaÞ for all a 2 A n fbg, which can be checked by testing the satisfiability of the constraints with a linear programming solver (see, e.g., [2,4] In Sect. 10 we gave a method, based on the extreme points of the epigraph of the utility function, for computing SMR W and testing dominance; in Sect. 11.2 we give a straightforward LP method related to the approaches used in [5,10,62]. In Sect. 11.4 we give a result that enables one to compute the minimal equivalent subset using the extreme points of the epigraph.  An optimal recommendation set of a given size k with respect to SMR is also myopically optimal in an elicitation sense [65]. A straightforward approach to compute a subset A of B with jAj ¼ k with minimum setwise max regret is computing SMR W ðA; BÞ for all the subsets A with jAj ¼ k. However, this approach is very computational demanding, and in the literature we can find alternative heuristic strategies based on the max regret to compute queries for elicitation purposes (see, e.g., [14,62,65]). In [59], we proposed an efficient branch and bound method to compute the set with minimum setwise max regret, which allows avoiding the computation of the setwise max regret for some subsets. In the latter work, we used the novel algorithm to compute SMR W ðA; BÞ presented in this paper (see Sect. 12.2) showing better time performance with respect to the standard method based on linear programming for the values k and p considered. ( [62,65] gives also extra useful sufficient conditions for an element a to be Opt A Wdominated. In those cases, the proofs that a is Opt A W -dominated makes use of the equality between MPO W ðAÞ and PSO W ðAÞ. (To use these sufficient conditions, we do not, of course, need to find any Opt A W -dominating element b of a-it is sufficient to know implicitly that such an element b has to exist.)

The structure of the algorithms
In this section we make use of mathematical results in previous sections in developing computational methods for computing the minimal equivalent set PSO W ðAÞ and testing dominance between sets, for the case of multi-attribute utility vectors, with the set of scenarios W being a convex polytope, and with linear utility functions. 4

Computing the minimal equivalent set and the set of Possibly Optimal elements
Given A 2 M, we aim to generate a subset A 0 A with A 0 W 889 A, and such that for any strict subset A 00 of A 0 , A 00 6 W 889 A. Theorem 3 implies that there exists a unique minimal equivalent set, i.e., SME W ðAÞ has a unique element, say, A 0 , and this equals PSO W ðAÞ. To compute PSO W ðAÞ, first we pre-process by eliminating elements of A not in UD W ðAÞ since from Proposition 7 it follows that PSO W ðAÞ UD W ðAÞ. At the same time we can make A equivalence-free. Then we have two different methods for computing PSO W ðAÞ.
(a) Multiple (i.e., jAj) tests of the form A n fag< W 889 fag, which can be achieved using a linear programming approach as described in Sect. 11 Regarding the computation of the set PO W ðAÞ of possibly optimal elements of A in W, also in this case we can first compute UD W ðAÞ and make A equivalence free. Then we can use one of the following two methods.
(i) A well-known approach in the literature (see, e.g., [4]) consists into iteratively excluding elements a from A if a 6 2 PO W ðAÞ. This iterative procedure is similar to that described in Sect. 7 We focus first on testing the dominance condition A< W 889 B, for given finite sets A; B of alternatives, i.e., A; B 2 M. Our algorithm includes three steps of increasing complexity, with the first two steps being pre-processing that helps the efficiency of the algorithm.
The first stage of the algorithm is a pre-processing step, using a necessary and a sufficient condition for dominance between sets. Proposition 22(i) below shows that A< W 0 889 B is a necessary condition for A< W 889 B, and A< W 0 988 B is a sufficient condition. Part (i) follows using Proposition 17 and nestedness (Proposition 1), and monotonicity with respect to W (Proposition 4). Parts (ii) and (iii), which follow from Proposition 3, give efficient methods of computing the necessary condition and the sufficient condition.

Proposition 22
Assume that for w 2 IR p ; a 2 IR p , f w ðaÞ ¼ w Áâ. Let W be a compact and convex subset of IR p and let W 0 ¼ ExtðWÞ be the set of extreme points of W. Then The second stage of the algorithm is another pre-processing step, whose correctness is shown by the following result, which easily follows using Proposition 5 and the definitions.

Experimental testing
In this section we show some experimental results, and we analyse the computational cost of the procedures presented in Sect. 12 for filtering a set of alternatives maintaining equivalence and for testing the dominance between sets. We considered linear utility functions f w ðaÞ ¼ w Áâ in all our experiments, with the set W of feasible scenarios w being a subset of the unit ðp À 1Þ-simplex defined by the intersection of q randomly generated half-spaces representing input user preferences. Specifically, we choose q (consistent) random user preferences of the form aw i þ bw j ! cw k (meaning that the user prefers a units of w i and b units of w j to c units of w k ), like in [41]. The sets A and B of utility vectors used in our experiments are randomly generated. See Appendix B for details about our random problem generator. All experiments were performed on computer facilitated by an Intel(R) Xeon(R) E5620 2.40 GHz processor with 32 GB of RAM. We used CPLEX 12.8 [35] as the linear programming solver, and we used the Python library pycddlib [61] for computing the extreme points of a convex polytope. CPLEX is an industrial tool highly optimised which has been commercialised for the first time more than 30 years ago and continuously improved. Pycddlib is a wrapper for the Komei Fukuda's cddlib library [26] based on the double description method [27,44]. It is worth noticing that the comparison of our algorithms may not be fair since pycddlib may not be as highly optimised as CPLEX. For example, we noticed that we could incur runtime exceptions related to precision issues if we do not approximate real numbers with fractions, and the use of fractions slowed down our algorithms up to 5 times for the CPLEX-based implementations and up to 15 times for the pycddlib-based implementations. We first show the experimental results for our main procedures as a whole, i.e., including the preliminary steps such us the UD W filtering. After that, we will show the performances of some specific operations.

Comparison of algorithms for computing PSOðA,WÞ
Here we describe some experimental results for the two methods presented in Sect. 12.1 to compute PSOðA; WÞ; the first based on a linear programming solver (PSO LP ) (see Sect. 12.1(a)), and the second based on the evaluation of the extreme points of the epigraph CðW; AÞ (PSO EP ) (see Sect. 12.1(b)). The results are an average over 100 experiments.
In Table 1 we show the performances of the methods PSO LP and PSO EP with respect to dimðWÞ and q ¼ 3 randomly generated user preferences. As expected, increasing dimðWÞ the computation time and the size of the output set increased. PSO EP was faster than PSO LP for dimðWÞ 4. However, PSO LP scaled better with respect to dimðWÞ; in fact, with dimðWÞ ¼ 7, PSO LP was around 1.5 times slower than its average execution time with dimðWÞ ¼ 6, on the other hand, PSO EP was 4 times slower. This is presumably related to the exponential growth of the number of extreme points of the epigraph with respect to dimðWÞ (see Table 5).
In Table 2 we show the performances of the methods PSO LP and PSO EP with respect to the number of user preferences q and considering fixed dimðWÞ ¼ 4. The two methods performed similarly under this configuration. As expected, by increasing the number of user preferences, the size of PSOðA; WÞ decreases, and the execution time tends to reduce.
In Fig. 2 we can see the execution time with dimðWÞ ¼ 3 and q ¼ 0 of the methods PSO LP and PSO EP with respect to the size of the input set A. As we can see, in our experiments the two methods scaled very roughly linearly in this setting.

Computational cost
Let n ¼ jAj be the size of the input set and q the number of inequalities representing user preferences. The computational cost of PSO LP is then OðnC LP ðn þ qÞÞ, where C LP ðn þ qÞ is the computational cost of the linear programming solver with n þ q constraints.
Let p ¼ dimðWÞ þ 1 be the size of the utility vectors in A, m ¼ jC 0 ðW; AÞj the number of extreme points of CðW; AÞ, and C EP ðCðW; AÞÞ the computational cost of computing the Table 1 Average execution time of the methods PSO LP and PSO EP for computing the minimal equivalent subset with respect to dimðWÞ. The last column shows the average size of the sets filtered with the PSO operator The experiments relates to randomly generated sets A with size jAj ¼ 100 and q ¼ 3 randomly generated user preferences. The timings include the UD filtering on the input set   ); however, if we suppose that the number jE A W ðaÞj of extreme points associated with an element a is 1 n the number of extreme points of CðW; AÞ, i.e., Oð m n Þ, the computational cost of this operation is reduced to OðnmpÞ. Thus, the whole computational cost of PSO EP can then be approximated as OðC EP ðCðW; AÞÞ þ nmpÞ.

Algorithms comparison for testing A< W 889 B
In this section we show some experimental results for the three methods presented in Sect. 12.2 to determine whether the condition A< W 889 B holds, for given sets A and B of alternatives. The first makes use of a linear programming solver (T LP ) (Sect. 12.2(a)), the second compares the extreme point sets of the two epigraph CðW; AÞ and CðW; A [ BÞ (T EU ) (Sect. 12.2(b)), and the third evaluates the value function f w ðbÞ ¼ w Á b for all b 2 B and the the extreme points w of the epigraph CðW; AÞ (T EE ) (Sect. 12.2(b)). In our experimental results, testing the necessary condition and the sufficient condition (Sect. 12.2 (1)) and filtering out elements of B dominated by elements of A (Sect. 12.2(2)) was enough to evaluate A< W 889 B for the majority of our experiments (see Sect. 13.4). Thus, to assess the performances of the methods (a), (b) and (c) of Sect. 12.2, we considered an average of 100 experiments where the necessary condition succeeded, the sufficient condition failed and the filtering of B didn't reduce its size to zero.
As we can see from Table 3 and Table 4, T LP was in general the fastest method and scaled better, and T EE performed slightly better than T EU . The percentage of dominance is on average above 50%. This means that given two input sets A and B generated with our random problem generator, B is likely to be dominated by A once the necessary condition is satisfied. This may not be surprising since the necessary condition is satisfied if and only if Ut A ðwÞ ! Ut B ðwÞ for all w 2 W 0 .
In Fig. 3 we can see the execution time with dimðWÞ ¼ 3 and q ¼ 0 of the methods T LP , T EU and T EE with respect to the size of the input set A. As we can see, in our experiments the two methods scaled roughly linearly when fixing dimðWÞ and q. The peak for jAj ¼ 700 is due to the randomness of the experiments; in this case, the UD filtering was on average less effective in reducing the size of A. We can see the same anomaly in Fig. 4 which reports the timing of the UD filtering.   OðC EP ðCðW; AÞÞ þ nm A pÞ, and supposing that the sets ExtðCðW; AÞÞ and ExtðCðW; A [ BÞÞ are ordered, the computational cost of T EU is OðC EP ðCðW; A [ BÞÞ þ m A[B pÞ. For a generic evaluation of the computational cost of the above methods, we are supposing A ¼ B ¼ n, but the size of B can be much smaller than the size of A (see Table 7 and Table 8).

Computation of the extreme points
For the preliminaries step we need first to compute the extreme points W 0 of the convex polytope W. The extreme points of the epigraph CðW; AÞ and CðW; BÞ are instead computed after the preliminaries step for the methods T EU ðA; B; WÞ and T EE ðA; B; WÞ to test the dominance and the method PSO EP ðA; WÞ to compute the minimal equivalent subset. Tables 5 and 6 show the number of extreme points of W, the number of extreme points of the epigraph CðW; AÞ and the corresponding computational time with respect to the dimension of W (Table 5) and the number of user preferences ( Table 6). The results are an average of 100 instances with three randomly generated user preferences for the results in Table 5, and dimðWÞ ¼ 5 for Table 6. The number of extreme points and the computational time with respect to dimðWÞ increase roughly linearly for W and exponentially for CðW; AÞ. Each user preference is a half-space with a corresponding hyper-plane that may redefine the boundaries of the convex polytope W, thus increasing the number of extreme points of W and then the corresponding computational time. On the other hand, the number of extreme points of the epigraph and the corresponding computational time both decrease, as one increases the number of user preferences (Table 6). This is because the user preferences reduce the hyper-volume of W; therefore, since the epigraph is built over W, the intersection of W with the half-spaces representing the user preferences can exclude some of the extreme points of the epigraph. For details regarding the computational cost of computing the extreme points of a convex polytope see, e.g., [24].

Preliminaries step
The preliminary steps are the operations executed on the input sets before testing the dominance A< W 889 B or computing the SME, i.e., testing the necessary and the sufficient condition (Sect. 12.2(1)), and the < W 0 898 filtering (Sect. 12.2(2)) for the dominance, and the UD W filtering for both dominance and SME. In Table 8, Table 7 and Fig. 4, we show some experimental results of the preliminary steps. The results are an average of 100 experiments in which the necessary condition is true, the sufficient condition is False, and the < W 0 898 filtering did not reduce the size of B to zero.
Testing the necessary and the sufficient condition was the fastest operation performed before evaluating A< W 889 B. The computational cost of this operation is OðnjW 0 jpÞ. The necessary condition failed or the sufficient condition succeeded in most experiments, allowing the algorithm to stop early. This happened from 98:5% to 94% (depending on dimðWÞ) of the random problems generated with the set-up of the experiments of Table 7, and from 96% to 99:2% (depending on the number of user preferences) of the random problems generated with the set-up of the experiments of Table 8. Therefore, since the execution time was a small fraction of the time spent by UD W (see Table 7 and Table 8), it looks like that this is a very worthwhile check for testing A< W 889 B. We used the UD W filtering to reduce the size of the input sets A and B before evaluating UD W ðAÞ< W 0 898 UD W ðBÞ when testing the dominance, and to reduce the size of the input set A before computing PSO W ðAÞ. The computational cost of this operation is Oðn 2 jW 0 jpÞ and it seems to speed up the overall execution for both testing A< W 889 B and computing PSO W ðAÞ. For example, the UD W filtering on 100 randomly generated input set A with jAj ¼ 100, dimðWÞ ¼ 4 and three user preferences, reduced the input set size to an average of 24 elements with an average execution time of 0.3 s. The execution time of PSO W ðAÞ The < W 0 898 filtering is part of the pre-processing to further reduce the size of B after the UD W filtering and before evaluating the dominance. The computational cost of this operation is Oðn 2 jW 0 jpÞ. This filtering improved the overall execution time of our experiments since it reduced the size of the set UD W ðBÞ by more than half in average (see jB 0 j and jB 00 j of Table 7 and Table 8). For example, the < W 0 898 filtering on 100 randomly generated input set filtered by UD W filtering, with initial input sets size jAj ¼ jBj ¼ 100, dimðWÞ ¼ 4 and three user preferences, reduced the average size of UD W ðBÞ from 23.46 to 8.8 elements with an average execution time of 0.17 s. The execution time of testing testing A< W 889 B over the same experimental set-up with and without the set UD W ðBÞ filtered by the < W 0 898 filtering was on average 0.04 and 0.17 s respectively for T LP , 0.18 and 0.21 s for T EU , and 0.12 and 0.24 s for T EE . In some of the experiments with the necessary condition true and the sufficient condition false, the < W 0 898 filtering has been enough for testing A< W 889 B since it reduced the size of B to zero. This happened from 22:5% to 1% (depending on dimðWÞ) of the random problems generated for the results in Table 8, and from 18% to 11:5% (depending on the number of user preferences) of the random problems generated for the results in Table 7.
The UD W filtering and the < W 0 898 filtering can be executed in any order. However, the overall execution time of our experiments was faster executing first the UD W filtering. This may be because the A< W 0 898 B filtering compares every element of B with every element of A in the worst case, and thus there may be several redundant comparisons if A 6 ¼ UD W ðAÞ since for a; b 2 A with a< W 0 898 b we have that if b< W 0 898 c with c 2 B, then a< W 0 898 c. As we can see in Fig. 3, the < W 0 898 filtering seems to scale better and be faster than the UD W filtering, but this is because it is executed after the UD W filtering. In fact, the UD W filtering would be faster than the < W 0 898 filtering if the filtering order was inverted.

POðA,WÞ and SMRðA,B,WÞ
Here we describe some experimental results for the computation of the set of possibly optimal alternatives POðA; WÞ and the setwise max regret SMR W ðA; B; WÞ using well-  In Tables 9 and 10 we show some experimental results for the computation of POðA; WÞ with respect to dimðWÞ and the number q of user preferences. Our method PO EP ðA; WÞ performed better for dimðWÞ 4. However, PO LP ðA; WÞ scaled better with respect to dimðWÞ. This may be because of the exponential grow of the number of extreme points of the epigraph with respect to dimðWÞ.
In Fig. 5 we show how PO LP ðA; WÞ and PO EP ðA; WÞ scaled with respect to the size of the input sets. The timings include the UD filtering and it seems that the overall execution time scaled very roughly linearly. Table 11 and Table 12 show the execution time of our experiments for the computation of SMRðA; B; WÞ with B A with respect to dimðWÞ and the size of B. Table 11 shows the timings of SMR LP ðA; B; WÞ and Table 12 those of SMR EP ðA; B; WÞ. Also in this case the method based on linear programming scaled better. However, SMR EP ðA; B; WÞ was in average the fastest. Table 9 Average execution time of the methods PO LP and PO EP for computing the minimal equivalent subset with respect to dimðWÞ. The last column shows the average size of the filtered sets. The experiments relates to randomly generated sets A with size jAj ¼ 100 and q ¼ 3 randomly generated user preferences. The timings include the UD filtering on the input set

Discussion
We defined natural notions of equivalence and dominance for a general model of sets of multi-attribute utility, and proved general properties. Computationally we focused especially on the linear (weighted sum) case and we proved that there is a unique setwise-minimal equivalent subset of any (equivalence-free) set of utility vectors A. This set then equals the set of possibly strictly optimal alternatives PSOðAÞ, and is a compact representation of the utility function for A, giving the utility achievable with A for each scenario. We show that filtering a query with the PSO operator avoids the potential of inconsistency in the user response. Along with pre-processing techniques we developed a linear programming method for generating PSOðAÞ, and a method based on computing the extreme points of the epigraph of the utility function (EEU), as well as related methods for testing dominance. We implemented the approaches and our testing on random problems showed that both methods  Bold values highlight better average timings with respect to the results of Table 12 scaled to substantially sized problems, with the EEU method being better for lower dimensions. Our methods can be directly applied to reduce the set of utility vectors derived for a multi-objective influence diagram [41] or a multi-objective optimisation problem [42].
Our experimental testing assumed inputs in which the sets A are represented explicitly. In some situations, the sets are more naturally represented combinatorially, as a set of constraints or a SAT formula. However, the fundamental properties that are the bases of our algorithms still apply, and these can enable the methods to be adapted for the linear convex case. Specifically, the minimal equivalent set corresponds with the set of possibly strictly optimal elements, and PSO W is equal to MPO W and satisfies Path Independence (by Proposition 8 and Corollaries 2 and 3). In particular, it would interesting to explore the use of AND/OR Branch-and-Bound algorithms, similar to those used for calculating the possibly optimal alternatives in [72]. The fact that PSO W satisfies Path Independence and translation invariance means that it satisfies the additive decomposition property (see Proposition 1 of [72]), which is fundamental for the AND/OR B &B algorithms.
A further natural application of our model and methods is for computing the Value of Information [23] for a multi-objective influence diagram. Each observable variable generates a Value of Information function which is a utility function Ut A , so different observable variables can be compared using the relation < W 889 . Although we focus especially on the case where f w ðaÞ is linear in w, which covers a wide range of important preference models, it would also be interesting to develop computational procedures for non-linear cases (such as quadratic utility models) based on our more general characterisation results, such as Theorems 3 and 4.

A proofs appendix
This appendix includes all the proofs of the results in the paper that do not appear in the main body of the paper, and includes also auxiliary lemmas that are used to prove these results.

Results in Sect. 3
For the convenience of the reader, we recall the definitions of the three dominance relations from Definition 3. Consider any W U and A; B 2 M. We give a simple fundamental property of the set UD W ðAÞ, which is used to prove e.g., Proposition 5 below.
Auxiliary Lemma 1 Consider any A 2 M.
(i) If a 2 A n UD W ðAÞ then there exists c 2 UD W ðAÞ such that c1 W a.
(ii) If a 2 A then there exists c 2 UD W ðAÞ such that c< W a.
Proof (i): Consider any a 2 A n UD W ðAÞ. By the definition of UD W ðAÞ, for any b 2 A n UD W ðAÞ there We construct a sequence a 1 ; a 2 ; . . ., where for each i ¼ 1; 2; . . ., we have a iþ1 1 W a i , where we stop the sequence when we reach an element a i such that either (a) a i has appeared earlier in the sequence, or (b) a i 2 UD W ðAÞ. Because A is finite, there must be a last element a k in the sequence. Transitivity of 1 W implies that if 1 i\k then a k 1 W a i , so, in particular, a k 1 W a. If (a) a k ¼ a i for some i\k then a k 1 W a k which contradicts the fact that 1 W is irreflexive. Thus, we have (b) a k 2 UD W ðAÞ and a k 1 W a, showing part (i Proof (i) ): Assume B< V 889 C, and consider arbitrary c 2 C and w 2 W. Since W is A-Extendable, there exists w 0 that is total over A given W (so w 0 2 V) that extends w over A. Since we have B< V 889 C, there exists b 2 B such that b< w 0 c. But, b< w 0 c implies that b< w c, showing that B< W 889 C. The ( part of (i) is immediate from monotonicity with respect to W (see Proposition 4). (ii): First assume that b 2 PSO V ðBÞ, so there exists w 2 V such that b 2 SO V w ðBÞ. But since, by Lemma 3, b V c () b W c for b; c 2 A, we have SO V w ðBÞ ¼ SO W w ðBÞ, and thus, b 2 SO W w ðBÞ, so b 2 PSO W ðBÞ. Conversely, assume that b 2 PSO W ðBÞ, so there exists w 2 W such that b 2 SO W w ðBÞ. Since W is A-Extendable, there exists w 0 2 V that extends w over A. Consider any c 2 B such that c 6 V b, and so c 6 W b. Since b 2 SO W w ðBÞ, b1 w c, and thus, b1 w 0 c, since w 0 extends w. This shows that b 2 SO V w 0 ðBÞ, and thus, b 2 PSO V ðBÞ, as required. h

Results in Sect. 6
We define for W U and a; b 2 X, For instance, W a ! b is the set of scenarios w 2 W in which a is at least as good as b.  The small result below regarding closure of topological spaces will be useful in proving the equivalence expressed in Lemma 5. For the converse, we need to show that for all i ¼ 1; . . .; m, ClðT i Þ ¼ X implies that ClðT 1 \ Á Á Á \ T m Þ ¼ X.
It is sufficient to prove this for the case when m ¼ 2, since we can then apply this iteratively, to show that first ClðT 1 \ T 2 Þ ¼ X and then ClððT 1 \ T 2 Þ \ T 3 Þ ¼ X, using the fact that T 1 \ T 2 is an open set, and so on, to prove that ClððT 1 \ Á Á Á \ T mÀ1 Þ \ T m Þ ¼ X. So, assume that ClðT 1 Þ ¼ ClðT 2 Þ ¼ X; we need to show that which is an open set. We have ðS \ T 1 Þ \ T 2 ¼ ;, and S \ T 1 is an open set, so ClðT 2 Þ ¼ X implies that S \ T 1 ¼ ; (else X n ðS \ T 1 Þ is a closed set between T 2 and X). Similarly, ClðT 1 Þ ¼ X then implies that S ¼ ;, so ClðT 1 \ T 2 Þ ¼ X.
Since v 2 T we have ðâ ÀbÞ Á v ¼ 0. Thus, 0 ¼ ðâ ÀbÞ Á v ¼ ðâ ÀbÞ Á w þ ð1 À Þðâ ÀbÞ Á u ¼ ðâ ÀbÞ Á w, which implies that ðâ ÀbÞ Á w ¼ 0, and therefore, f a ðwÞ ¼ f b ðwÞ. h  Proof Assume that ClðW 6 ¼ A Þ ¼ W, and consider any w 2 W. To show that W is A-Extendable, we need to show that there exists w 0 that is total over A given W that extends w over A. Let d be the minimum value of jf w ðaÞ À f w ðbÞj over all a; b 2 A such that f w ðaÞ 6 ¼ f w ðbÞ. Because A is finite, we have d [ 0. Since the topological of closure of W 6 ¼ A equals W, there exists a sequence of elements of W 6 ¼ A that tend to w. In particular, there exists some w 0 2 W 6 ¼ A such that jf w ðaÞ À f w 0 ðaÞj\d=3 for all a 2 A. By definition of d, we have, for any a; b 2 A, that f w ðaÞ À f w ðbÞ [ 0 implies f w ðaÞ À f w ðbÞ ! d. Then, f w 0 ðaÞ À f w 0 ðbÞ ¼ ðf w 0 ðaÞ À f w ðaÞÞ þ ðf w ðbÞ À f w 0 ðbÞÞþ Thus, f w 0 ðaÞ [ f w 0 ðbÞ. We have shown that, for any a; b 2 A if a1 w b then a1 w 0 b; this implies that w 0 extends w over A, where w 0 is total over A given W. h Proposition 11 Let C 2 M and assume, for each a 2 C, that the function f a is a real-valued continuous function on the metric space W, and that the set of functions ff a : a 2 Cg satisfies the Identity property.
Suppose that A C. Proof Part (i) follows immediately using Proposition 2. (ii) follows from (i). (iii) follows from iterative application of (ii). Regarding (iv) and (v): suppose that a i 2 A n Filter r ðA; < W 898 Þ. Then, using the notation above, A i 6 3 a i , i.e., a i 6 2 FilterðA iÀ1 ; a i ; < W 898 Þ, and thus, there exists c 2 A iÀ1 with c< W a. If a i 2 UD W ðAÞ this implies that c W a. Applying this iteratively, we see that if a i 2 UD W ðAÞ n Filter r ðA; < W 898 Þ then there exists c 2 Filter r ðA; < W 898 Þ with c W a i . Now assume that a i 2 A n UD W ðAÞ; then, by Auxiliary Lemma 1, there exists b 2 UD W ðAÞ with b1 W a i ; by the above argument, there exists c 2 Filter r ðA; < W 898 Þ with c W b and thus, c1 W a i . We have c 2 Filter r ðA; < W 898 Þ A iÀ1 , which implies that a i 6 2 A i , and thus, a i 6 2 Filter r ðA; < W 898 Þ. This proves (iv) Filter r ðA; < W 898 Þ UD W ðAÞ. We showed that for a 2 UD W ðAÞ there exists c 2 Filter r ðA; < W 898 Þ with c W a; and also, Filter r ðA; < W 898 Þ UD W ðAÞ. Together these imply (v): Filter r ðA; < W 898 Þ W UD W ðAÞ. (vi) Firstly, we observe that if a 2 Filter r ðA; <Þ then Filter r ðA; <Þ n fag 6 <fag. (This follows using the fact that if a i ¼ a then A i ¼ FilterðA iÀ1 ; fa i g; <Þ 3 a i in the sequence of sets, i.e., A iÀ1 n fa i g 6 <fa i g, which, by monotonicity, implies Filter r ðA; <Þ n fa i g 6 <fa i g, since Filter r ðA; <Þ A iÀ1 .) This implies, using Proposition 2, that no strict subset of Filter r ðA; <Þ is equivalent to A. In particular, for the case in which < equals < W 889 , we obtain that Filter r ðA; < W 889 Þ 2 SME W ðAÞ. Conversely, for B 2 SME W ðAÞ; to complete the proof of (vi) we will show that there exists r 2 K such that  Proof Consider any w 2 W. By definition, ðw; Ut A ðwÞÞ 2 CðW; AÞ. Choose any N [ MaxUt W A ; by Auxiliary Lemma 11, for all w 2 W, N [ Ut A ðwÞ, and ðw; Ut A ðwÞÞ 2 C N ðW; AÞ. Using Auxiliary Lemma 11(ii), CHðExtðC N ðW; AÞÞÞ 3 ðw; Ut A ðwÞÞ so we can write ðw; Ut A ðwÞÞ as P J j¼1 s j q j where q j 2 ExtðC N ðW; AÞÞ, and the s j are non-negative reals that sum to 1. We will show that s j ¼ 0 unless q j 2 ExtðCðW; AÞÞ. This then implies that ðw; Ut A ðwÞÞ 2 CHðExtðCðW; AÞÞÞ, proving Auxiliary Lemma 12. So, suppose that there exists k with s k [ 0 and q k 2 CHðExtðC N ðW; AÞÞÞ nExtðCðW; AÞÞ. By Auxiliary Lemma 11(vi), q k ¼ ðw 0 ; N Þ for some w 0 2 W and N [ Ut A ðw 0 Þ. Let q 0 ¼ P J j¼1 s j q 0 j , where q 0 k ¼ ðw 0 ; Ut A ðw 0 ÞÞ, and q 0 j ¼ q j for j 6 ¼ k. Then for all j, q 0 j 2 CðW; AÞ and so, by convexity of CðW; AÞ, q 0 2 CðW; AÞ. Also, q 0 can be written as ðw; r 0 Þ for some r 0 , and r 0 \Ut A ðwÞ (see below). The definition of CðW; AÞ implies that ðw; r 0 Þ 6 2 CðW; AÞ, giving the required contradiction. In more detail, we have ðw; Ut A ðwÞÞ À q 0 ¼ P J j¼1 s j ðq j À q 0 j Þ ¼ s k ðq k À q 0 k Þ ¼ s k ððw 0 ; N ÞÀ ðw 0 ; Ut A ðw 0 ÞÞÞ ¼ s k ð0; N À Ut A ðw 0 ÞÞ. Then, q 0 ¼ ðw; r 0 Þ where r 0 ¼ Ut A ðwÞ À s k ðN À Ut A ðw 0 ÞÞ\Ut A ðwÞ. h The following result states that CðW; AÞ is determined by its extreme points, even though it is not compact.
Auxiliary Lemma 13 Consider any finite subsets A and B of IR p , and any compact and convex subset W of IR p , and assume that for all a 2 A [ B, f w ðaÞ is a convex and continuous function of w 2 W. Then CðW; AÞ ¼ CðW; BÞ ( ) ExtðCðW; AÞÞ ¼ ExtðCðW; BÞÞ.
Proof If CðW; AÞ ¼ CðW; BÞ then obviously ExtðCðW; AÞÞ ¼ ExtðCðW; BÞÞ. Regarding the converse, assume that ExtðCðW; AÞÞ ¼ ExtðCðW; BÞÞ. For any w 2 W, ðw; Ut A ðwÞÞ 2 CðW; AÞ and ðw; rÞ 2 CðW; AÞ if and only if r ! Ut A ðwÞ. Now, Auxiliary Lemma 12 implies that ðw; Ut B ðwÞÞ is in the convex hull of ExtðCðW; BÞÞ, and thus, in the convex hull of ExtðCðW; AÞÞ; hence, by convexity of CðW; AÞ, we have ðw; Ut B ðwÞÞ 2 CðW; AÞ, which shows that Ut B ðwÞ ! Ut A ðwÞ, which holds for an arbitrary element w of W. This implies CðW; AÞ CðW; BÞ. Switching the roles of A and B in the argument shows also CðW; AÞ CðW; BÞ, and thus, CðW; AÞ ¼ CðW; BÞ. h Proof Let W 00 ¼ CHðWÞ ¼ CHðW 0 Þ. We will show that < W equals < W 00 ; the same proof will show < W 0 equals < W 00 , and thus, < W ¼ < W 0 . Since W 00 W, we have < W < W 00 ; to prove the converse, suppose that a< W b and consider any w 2 W 00 . Then, by definition of convex hull, there exists w i 2 W and strictly positive reals r i , for i ¼ 1. . .; k, such that P k i¼1 r i ¼ 1 and w ¼ P k i¼1 r i w i . Because a< W b, for each i ¼ 1. . .; k, w i Á ðâ ÀbÞ ! 0, and thus w Á ðâ ÀbÞ ! 0, showing that w Áâ ! w Áb. This shows that a< W 00 b, and thus < W equals < W 00 , and hence, < W ¼ < W 0 . Since W is compact and W 0 is the set of extreme points of W we have CHðW 0 Þ ¼ W and so < W ¼ < containing w, with w not being an endpoint, i.e., there exists some w 1 ; w 2 2 Opt A W ðbÞ and s 2 ð0; 1Þ such that w ¼ sw 1 þ ð1 À sÞw 2 . Because w 1 ; w 2 2 Opt A W ðbÞ, for i ¼ 1; 2, w i Áb ! w i Áâ, i.e., w i Á ðb ÀâÞ ! 0. Also, since w 2 Opt A W ðaÞ and w 2 Opt A W ðbÞ we have w Á ðb ÀâÞ ¼ 0. Thus, 0 ¼ w Á ðb ÀâÞ ¼ sw 1 Á ðb ÀâÞ þ ð1 À sÞw 2 Á ðb ÀâÞ, and so, since both terms are non-negative, they are both zero: w 1 Á ðb ÀâÞ ¼ w 2 Á ðb ÀâÞ ¼ 0. Because w 1 ; w 2 2 Opt A W ðbÞ (b is optimal in A with respect to both w 1 and w 2 ), this implies that a is optimal in A with respect to both w 1 and w 2 , i.e., w 1 ; w 2 2 Opt A W ðaÞ. Thus, there exists a line segment in the convex set Opt A W ðaÞ containing w, where w is not one of the endpoints of the line segment, so w is not an extreme point of Opt A W ðaÞ. h Auxiliary Lemma 14 Assume that W is a convex subset of IR p , and that for w 2 IR p ; a 2 IR p , f w ðaÞ ¼ w Áâ. Consider A 2 M, a 2 A, w 2 W. Let I a ¼ fðw; rÞ 2 IR p Â IR : r ¼ w Áâg. For K IR p Â IR we write K # for the projection of K to IR p , i.e., K # ¼ fw 2 IR p : ðw; rÞ 2 Kg.