Stability and Sensitivity of Uncertain Linear Programs

The present paper deals with uncertain linear optimization problems where the objective function coefficient vector belongs to a compact convex uncertainty set and the feasible set is described by a linear semi-infinite inequality system (finitely many variables and possibly infinitely many constrainsts), whose coefficients are also uncertain. Perturbations of both, the objective coefficient vector set and the constraint coefficient set, are measured by the Hausdorff metric. The paper is mainly concerned with analyzing the Lipschitz continuity of the optimal value function, as well as the lower and upper semicontinuity in the sense of Berge of the optimal set mapping. Inspired by Sion’s minimax theorem, a new concept of weak optimal solution set is introduced and analyzed.


Introduction
In this paper our focus is on the continuity and Lipschitz behavior of uncertain linear optimization problems, where uncertainty appears in both the objective function and the constraint system. Formally, we consider Dedicated to Miguel A. Goberna on his 70th birthday where x ∈ R n is the vector of decision variables, and C ⊂ R n and U ⊂ R n+1 represent the uncertainty sets related to the objective function and the constraints, respectively. Along the work, for convenience, C is a nonempty compact convex set, while U is assumed to be closed. All elements in R n are regarded as column-vectors and y ′ denotes the transpose of y ∈ R n , so that y ′ x denotes the usual inner product in R n . Elements in R n+1 will be written in the form a b , where a ∈ R n and b ∈ R. Observe that we are dealing with the robust counterpart of the corresponding uncertain problem, where, regarding the objective function, a pessimistic approach is adopted as far as uncertainty on the 'primary' objective function x ↦ c ′ x (where c lies in C) is tackled through the 'worst-case' function x ↦ max c � x ∶ c ∈ C ; in this way, a typical minimax problem arises. Moreover, we comment that P(C, U) is a semi-infinite optimization problem since it has a finite amount of variables ( x ∈ R n ) and possibly infinitely many constraints (U may be infinite); see [4,Section 3.4.2] in the context of ordinary (finite) linear programming. The reader is addressed, for instance, to [19] and [20, Sections 1.2.1, 5.1. 4] for details on minimax problems appearing in stochastic programming.
At this moment, we point out that the main contributions of this paper consist of analyzing the lower and upper semicontinuity, as well as the Lipschitz continuity, of the optimal value function and the lower and upper semicontinuity in the sense of Berge of the optimal set mapping, both mappings associated with the parameterized problem (1). In a first stage, the lower and upper semicontinuity, as well as the Aubin property, of the feasible set mapping are analyzed in Section 3 (the semicontinuity and Lipschitz-type properties are recalled in Section 2). To formalize the parametric setting we borrow from [2] the following notation: CB(R n ) for the nonempty closed and bounded (i.e., compact) convex subsets of R n and CL R n+1 for the nonempty closed subsets in R n+1 . Both parameter spaces CB(R n ) and CL R n+1 are endowed with the Hausdorff distance (indeed, an extended distance -it might be infinite-for the latter); see Section 2 for details.
In this parametric setting, the feasible set mapping F ∶ CL R n+1 ⇉ R n is given by Observe that F could have been defined on the whole 2 R n+1 (under the convention F � = R n ), but for our purposes it is enough to consider CL R n+1 . Also, observe that in 2 R n+1 we would have F(U) = F(clU), where cl means closure. The optimal value function ∶ CB(R n ) × CL R n+1 ⟶ [−∞, +∞] and the optimal set mapping S ∶ CB(R n ) × CL R n+1 ⇉ R n assign to each (C, U) ∈ CB(R n ) × CL R n+1 respectively, (with inf � ∶= +∞ ) where, Function w C (the notation comes after 'worst case' function) is nothing else but the support function of C (cf. [17, p. 26]), and, therefore, it is sublinear (convex and positively (1) homogeneous) and finite-valued, hence continuous, on R n (see [17,Theorem 13.2]). Thus, S(C, U) is a convex set.
Remark 1.1 According to Sion's minimax theorem [21] (see [15] for an elementary proof), we have two alternative expressions for the optimal value: is not required to be bounded, S(C, U) may be empty even in the case when (C, U) is finite.
Looking at the 'maxinf' expression of (C, U) a new multifunction, the weak optimal set mapping, denoted by S w , is introduced in Section 5, where the relationship between S and S w is analyzed. We advance that S(C, U) is always contained in S w (C, U) and the inclusion may be strict. Moreover, a criterion for the nonemptiness of S(C, U) in terms of S w (C, U) is provided.
Appealing to the previous notation, the current work characterizes (in Section 4) the lower and upper semicontinuity of function , as well as also characterizes the lower semicontinuity in the sense of Berge of multifunction S and provides a sufficient condition for the corresponding upper semicontinuity (see Section 5). In a second step, in Section 6, we quantify the stability of by providing an upper bound on its Lipschitz modulus, which, roughly speaking, estimates the rate of variation of (C, U) with respect to perturbations of The immediate antecedents of this paper can be traced out from [3,5] and [6]. Papers [3] and [5] analyze the Aubin continuity and calmness of F , respectively, and some of the results established there (recalled in Section 3 for completeness) constitute key starting points of this work. Paper [6] contains the deterministic counterpart of current Sections 4 and 5, with a fixed linear objective function x ↦ c ′ x and in a different parametric context of linear systems with a fixed index set T, where the distance between parameters is measured with the supremum (Chebyshev) extended norm. Specifically, [6] deals with linear optimization problems of the form where c ∈ R n , T is an arbitrary (but fixed, possibly infinite) index set, and a t ∈ R n , b t ∈ R for all t ∈ T. We refer to the constraint system of (5) as which may be identified with the function a t b t t∈T ∈ R n+1 T , and perturbations on σ are measured by means of the supremum extended norm (see Section 2.1 for details).
Indeed, we can find in the literature numerous antecedents (at the moment we cite [10, Chapter 10]) in this Chebyshev setting. In contrast, the current paper removes the requirement of a fixed index set, so that it allows for both robust counterparts and discretization strategies. In other words, a 'nominal system' U may be approached by U + B n+1 , where B n+1 stands for the closed unit ball in R n+1 (with respect to any given norm) and ε > 0, and also by grids of stepsize ε, as U + B n+1 ∩ Z n+1 , where Z, as usual, stands for the integers.
Now we summarize the structure of this paper. Section 2 contains the necessary notation and preliminary results used throughout the paper. Specifically, we underline some key tools traced out from [5] which allow us to connect both Hausdorff and Chebyshev settings, regarding the constraints, by means of the so-called indexation strategies. Section 3 completes the picture from [3] by establishing the equivalence between the Aubin property of F and the (apparently weaker) lower semicontinuity of this mapping; the upper semicontinuity of F is also analyzed. Sections 4 and 5 are devoted to the continuity properties of and S, respectively, taking sometimes advantage of previous results given in [6] by using the referred indexation strategies. Section 6 applies the results of the previous sections to analyze the Lipschitzian behavior of .

Preliminaries
Throughout this paper we use the following notation: Given X ⊂ R p (with p being either n or n + 1), convX and coneX represent the convex hull and the conical convex hull of X, respectively, with the convention cone� = 0 p (the zero vector in R p ); while intX, clX, and bdX stand for the interior, the closure, and the boundary of X, respectively.
Along this paper R n is endowed with an arbitrary norm ‖⋅‖, whose dual norm is given by whereas R n+1 is endowed with the norm From now on, for simplicity, we will denote our parameter space by and ∶= (C, U) ∈ Π is regarded as our perturbation parameter, whereas we consider a nominal (fixed) parameter at which we perform our stability analysis. As commented in Section 1, both spaces CB(R n ) and CL R n+1 are endowed with the Hausdorff distance (extended distance in the In the case of CB(R n ) we consider R n endowed with the dual norm ‖⋅‖ * since vectors c ∈ C are seen as functionals. For more details about the Hausdorff distance, the reader is addressed to [2, Section 3.2]. Our parameter space Π is endowed with the metric given by Regarding the linear semi-infinite optimization problem dealt in [6], recalled in (5), observe that C is a singleton there and constraint systems are indexed by a fixed index set T. Accordingly, the parameter space of problems (5) is R n × R n+1 T , where R n+1 T is endowed with the uniform convergence topology by means of the Chebyshev (extended) distance

Indexation Strategies
In [5], with the aim of analyzing the calmness of F by applying previous results in the context of systems (6), a certain indexation strategy was introduced; see (9) below. This strategy connects both Hausdorff ('non-indexed') and Chebyshev ('indexed') perturbation settings. Note that we can assign to any indexed system ∈ R n+1 T a non-indexed U ∈ CL R n+1 by taking U ∶= cl rge( ), where rge stands for range (image). The following immediate result shows that the Hausdorff distance provides in some sense the tightest uniform approximation between indexed systems.

Remark 2.1
For any index set T ≠ ∅ and any 1 , 2 ∈ R n+1 T one has Formally, [5] shows that there is an injection from with T = R n+1 (the possibility of restricting to a smaller T is discussed in the same paper), which preserves distances to the nominal system U 0 ∈ CL R n+1 . Specifically, such an injection assigns to each set U ∈ CL R n+1 an element U ∈ R n+1 R n+1 (an indexed system, according to the previous comments) given by where P U is an arbitrary selection of the metric projection mapping on U; i.e., In this way, for t ∉ U we first project on U 0 and then this projection is projected on U; [5, Section 3.1] shows that just projecting on U does not preserve distances in the sense of (11) below. Observe that and, therefore, U and σ U produce the same constraint set, although in σ U there may be repeated constraints.
Theorem 2.1 [5,Theorem 3.1] Keeping the previous notation, we have As commented in [3], the indexation strategy recalled in (9) is appropriate for studying the calmness of F , but it is no longer convenient for the analysis of the Aubin property (definitions are gathered in the next subsection), for which a new indexation strategy involving pairs of systems around U 0 was introduced in [3, Lemma 3.1].

Continuity and Lipschitzian Properties for Generic Multifunctions
Let M ∶ Y ⇉ X be a set-valued mapping between metric spaces (both distances denoted by d). M is said to be (Berge-) lower semicontinuous (lsc) M is closed at y 0 ∈ Y if for each pair of sequences y r r∈N ⊂ Y and x r r∈N ⊂ X such that x r ∈ M y r , for all r, lim r y r = y 0 and lim r x r = x 0 for some x 0 ∈ X, it follows that Now we recall some stability properties of quantitative nature: M has the Aubin property (also called pseudo-Lipschitz -cf. [14]-or Lipschitz-like -cf. [16]-) at y 0 , x 0 ∈ gphM (the graph of M ) if there exist a constant κ ≥ 0 and neighborhoods W of x 0 and V of y 0 such that The infimum of constants κ over all ( , W, V) satisfying (12) is called the Lipschitz modulus of M at y 0 , x 0 , denoted by lipM y 0 , x 0 , and it is defined as +∞ when the Aubin property fails at y 0 , x 0 . The particularization of (12) to y 2 = y 0 yields the definition of calmness of M at y 0 , x 0 , whose associated calmness modulus, clmM y 0 , x 0 , is defined analogously.
For details on the Aubin property, calmness, and other topics -as metric regularity notions-of variational analysis, the reader is addressed to the monographs [8,13,14,16,18].

Continuity and Aubin Property of F
For our feasible set mapping F the concept of Hausdorff lower semicontinuity is too restrictive, since it fails to hold in very simple cases with F(U) unbounded, as when U = a b with a≠ 0 n and n ≥ 2. Concerning the Berge lower semicontinuity of the feasible set mapping in the parameter space R n+1 T with the metric (8) -taking into account that all norms in R n+1 are equivalent-, we can find different characterizations in [10, Theorem 6.1] (see also [11,Theorem 3.1]). One of these characterizations is the so-called strong Slater condition (SSC), which is held at a t b t t∈T ∈ R n+1 T if there exists a feasible point x and a positive slack ρ satisfying a � tx ≥ b t + for all t ∈ T. The immediate translation of this property to our parameter U ∈ CL R n+1 is the existence of x and ρ > 0 such that Such a x is called a strong Slater element of U. In the following results domF stands for the domain of F; i.e., (12) d x 1 , M y 2 ≤ d y 1 , y 2 for all y 1 , y 2 ∈ V and all x 1 ∈ M y 1 ∩ W.
Moreover, for any index set T, let F T ∶ R n+1 T ⇉ R n be the feasible set mapping associated with systems (6). Specifically, Observe that F T ( ) = F(cl rge( )) for all ∈ R n+1 T .
Proof Assume that F is lsc at U 0 and take any open set W ⊂ R n such that Condition (ii) in the next theorem appeals to indexation function (9). Theorem 3.1 Let U 0 ∈ domF. The following conditions are equivalent: is lsc at U 0 ; (iii) The SSC holds at U 0 ; (iv) U 0 ∈ int domF (v) 0 n+ 1 ∉ clconvU 0 (vi) F has the Aubin property at U 0 , x 0 for any x 0 ∈ F U 0 .
Additionally, if a ∈ R n ∶ (a, b) ∈ U 0 is bounded, then for x 0 ∈ F U 0 we have where d * stands for the distance associated with ‖⋅‖ * and Proof (i) ⇒ (ii) . It follows from applying the previous lemma to 0 = U 0 , taking (10) into account.
(ii) ⇒ (i). Take now any open set W ⊂ R n such that F U 0 ∩ W ≠ � and consider δ > 0 such that with d ∞ , U 0 < . Take, for the same δ, U ∈ CL R n+1 such that d H U, U 0 < . Now, Theorem 2.1 ensures that d ∞ U , U 0 < and, so, follows straightforwardly from [10, Theorem 6.1 (i) ⇔ (vi) ] since the SSC at U 0 is equivalent to the same property at 0 = U 0 .
(ii) ⇔ (iv). Appealing to [10, Theorem 6.1 (i) ⇔ (ii) ], (ii) turns out to be equivalent to U 0 ∈ int domF R n+1 and this can be easily shown to be equivalent to (iv) by using again Remark 2.1 and Theorem 2.1.

Proof
(i) Take a pair of sequences U r r and x r r converging to U 0 and some x 0 ∈ R n , respectively, and such that x r ∈ F U r for all r. From Theorem 2.1 it is clear that U r ⊂ R n+1 R n+1 converges to U 0 and, from (10), x r ∈ F R n+1 U r for all r. It is well-known that F R n+1 is closed (see, e.g., [10, Section 6.1]) and,

Continuity of
Regarding function w C defined in (4), with C ∈ CB(R n ), we have the following result, which follows directly from [1, Theorem 4.2.2] (adapted to max-type problems) by considering (C, x) ↦ w C (x) as the optimal value function φ with the notation therein, where the feasible set mapping is M(C, x) = C (again with the notation of [1] ), which is trivially Hausdorff continuous. Nevertheless, for the sake of completeness we give a self-contained proof, whose argument will also serve in the proof of Theorem 4.1 below.

Proposition 4.1 The function CB(R
An analogous estimation works for w C 0 (x) − w C (x), and hence for Finally, by using the triangle inequality, we have Hereafter in this paper, recalling (3) and (7), we use the following notation for the sets of consistent, bounded , and solvable problems, respectively: The following theorem establishes the counterpart result of [10, Theorem 10.1 (i), (ii) ] (see also [6,Theorem 4.2]) to our current uncertainty setting. The proof of (ii) is inspired in that of [10, Theorem 10.1 (ii) ], although it presents notable differences as far as our current F and S allow for a more flexible approach as a consequence of removing the assumption of a fixed index set. Observe that is trivially lsc at any π 0 ∈ Π c ∖ Π b . Theorem 4.1 Given 0 = C 0 , U 0 ∈ Π c , the following statements hold: is usc at π 0 if and only if the SSC holds at U 0 ; (ii) Assume π 0 ∈ Π b . Then, is lsc at π 0 if and only if S 0 is nonempty and bounded. (iii) π 0 ∈ int Π s if and only if SSC holds at U 0 and S 0 is nonempty and bounded.

Proof (i) The 'if' condition is an immediate consequence of [1, Theorem 4.2.2 (1)] together
with Theorem 3.1 and Proposition 4.1; whereas the converse implication is trivial by taking the equivalence (iii) ⇔ (iv) in Theorem 3.1 into account. (ii) Let us see that the lower semicontinuity of at π 0 entails the nonemptiness and boundedness of S 0 . For this, by a standard argument of continuity and compactness, it is sufficient to show the existence of a nonempty and bounded sublevel set x ∈ F U 0 | w C 0 (x) ≤ . Taking into account that, for all > C 0 , U 0 , the aforementioned sublevel sets have the same recession cone (cf. [17, Section 8]), assume reasoning by contradiction that for some 0 > C 0 , U 0 the corresponding sublevel set is unbounded, so that it has a recession direction z 0 ≠ 0 n . In such a case, consider the perturbed problem ∶= C , U 0 , with C ε := C 0 − εz 0 , for an arbitrarily small ε > 0. Then, for any x ∈ F U 0 with w C 0 (x) ≤ 0 and any λ > 0 we have where ‖ ‖ z 0 ‖ ‖2 stands for the Euclidean norm of z 0 . Accordingly, = −∞ for all ε > 0, contradicting the lower semicontinuity of at π 0 .
Conversely, assume that S 0 is nonempty and bounded. Write According to Theorem 3.2, F is usc at Ũ 0 . Fix arbitrarily ε > 0 and let us see that ( ) ≥ v 0 − for π close enough to π 0 . To do this, take ρ > 0 such that ‖x‖ < for all x ∈ S 0 and let us consider the open set Accordingly, Now consider any x ∈ F(U) and let us see that w C (x) > v 0 − . In the nontrivial case when w C (x) ≤ v 0 (i.e., x ∈ F Ũ ) we have, recalling (16) but switching the roles of C and C 0 there, (iii) If π 0 ∈ int Π s , it is clear that SSC holds at U 0 (by applying Theorem 3.1). Moreover, S 0 has to be bounded, since otherwise we can find a recession direction of S 0 , d≠ 0 n , and by taking x r = x 0 + rd, with x 0 ∈ S 0 and r = 1,2,..., we conclude therefore, we attain the contradiction ∶= C 0 − d, U 0 ∉ Π s for any ε > 0. In order to prove the converse implication, assume that SSC holds at U 0 and S 0 ≠ ∅ is bounded. Let x ∈ F U 0 be a strong Slater element and take ρ > 0 such that a �x ≥ b + for all a b ∈ U 0 . Take x 0 ∈ S 0 ; i.e., such that a ′ x 0 ≥ b, for all a b ∈ U 0 , and c � x 0 ≤ v 0 ∶= C 0 , U 0 . One easily checks the existence of 0 < α < 1 such that x ∶= x + (1 − )x 0 verifies c � x ≤ v 0 + for all c ∈ C 0 (we appeal here to the compactness of C 0 ). Moreover, Hence x α is a strong Slater element for system , c ∈ C 0 } with slack αρ; so, moreover, observe that F(Ũ) is bounded (since it has the same recession cone as S 0 ), which entails the upper semicontinuity of F at Ũ . Now, fo r U 1 , C 1 c l o s e e n o u g h t o U 0 , C 0 , we h ave t h a t Ũ 1 ∶= U 1 ∪ (−C 1 × {− v 0 + (1 + ) }) ∈ domF and F(Ũ 1 ) is also bounded. Therefore, S U 1 , C 1 ≠ ∅ since F(Ũ 1 ) = x ∈ F U 1 | w C 1 (x) ≤ v 0 + (1 + ) (i.e., F(Ũ 1 ) is a bounded sublevel set of the convex function w C 1 ). So, we have Remark 4.1 Our problem (1) can be seen as a particular case of that considered in [19,Section 4] where the SSC at U 0 yields the so-called lopside convergence of problems approaching π 0 in our setting. If, moreover, S 0 is nonempty and bounded, then, our argument in the proof of Theorem 4.1 (ii) about the uniform boundedness of sublevel sets yields that such lopside convergence is tight in the terminology of [19,Section 4], which implies the continuity of at π 0 . This approach provides an alternative way of tackling the continuity of the optimal value function.

Continuity of S and Related Optimality Notions
The following theorem gathers some stability properties of S, whose counterpart in the context of problems (5) can be found in [6, Theorem 5.1]. Again, the absence of a fixed index set allows for a more flexible approach. Specifically, the proof of [6, Theorem 5.1] (and others in [6]) requires to appeal to systems indexed by T ∪ t 0 , with t 0 ∉T, in order to enlarge the constraint system. Formally, the associated F T∪{t 0 } is a different mapping from F T . Now we do not need such a trick: the same mapping F (or S ) works for systems with different cardinalities.
Theorem 5.1 Let 0 = C 0 , U 0 ∈ Π s and assume that SSC holds at U 0 . Then, we have: (i) S is closed at π 0 ; (ii) If S 0 is bounded, then S is usc at π 0 ; (iii) S is lsc at π 0 if and only if S 0 is a singleton.

Proof
(i) Under the current assumption is usc at π 0 (recall Theorem 4.1 (i)). Then, the closedness of S at π 0 follows from [1, Theorem 4. Because of boundedness of S 0 , which entails the upper semicontinuity of F at Pick a strong Slater element x 0 of U 0 with a positive slack ρ. If w C 0 x 0 < 0 , then x 0 turns out to be a strong Slater element of system Ũ 0 , whose feasible set is x ∈ F U 0 | w C 0 (x) ≤ 0 . Accordingly F is both lower and upper semicontinuous at Ũ 0 (by Theorems 3.1 and 3.2). As a consequence of this, there exists a neighborhood V of π 0 such that = (C, Hence, problem π has a non-empty and bounded sublevel set contained in W, and, therefore, S( ) ⊂ W.
In the remaining case w C 0 x 0 ≥ 0 , take any x 1 ∈ S 0 and define x ∶= (1 − )x 0 + x 1 for 0 < λ < 1. It can be easily checked that x λ is a strong Slater element of U 0 with slack (1 − ) and that w C 0 x < 0 for λ close enough to 1. Then, we can reproduce the previous argument with x λ instead of x 0 . (iii) Assume that S is lsc at π 0 and, arguing by contradiction, suppose that there exist at least two different points x 1 , x 2 ∈ S 0 . Take, for each ε > 0, C ∶= C 0 − x 2 − x 1 and observe that, for any u ∈ R n such that provided that x 1 + u ∈ F U 0 , where in the last inequality we have taken into account that we have S C , U 0 ∩ W = � for any ε > 0, which yields the unfulfilment of the lower semicontinuity of S at π 0 .
To prove the converse implication, assume that S 0 is a singleton, in particular, from the previous statement, S is usc at π 0 . Moreover, it is clear that π 0 ∈ int Π s (see Theorem 4.1 (iii)). Under these facts the lower and the upper semicontinuity of S turn out to be equivalent properties.

Weak Optimal Solutions
This subsection is devoted to discuss a new possible notion of optimal solution for our problem (1) inspired by the 'maxinf' expression of the optimal value coming from Sion's minimax theorem [21] (recall Remark 1.1), Let us denote by v U ∶ R n → [−∞, +∞] the function defined by which is nothing else but the usual optimal value for the 'deterministic' objective function x ↦ c ′ x over the feasible set F(U). Consider the new multifunction, called here the weak optimal set mapping S w ∶ Π ⇉ R n by where arg max C v U stands for an abbreviation of arg max c∈C v U (c). For a geometrical interpretation of S( ) and S w ( ), with = (C, U), we appeal to the sub-level set mapping L ∶ C × R ⇉ R n , given by On the other hand, arg max C v U if and only if L(c, ( )) = arg min x∈F(U) c � x.
The following example shows that both solution sets may be empty for π ∈ Π b . Next we provide an example in which S( ) = � and S w ( ) ≠ �. Hence, the notion of weak solvability, understood as the nonemptiness of S w ( ), is strictly weaker than the 'ordinary' solvability, S( ) ≠ �.

Proposition 5.2 S(C, U)
is nonempty if any of the following conditions holds: is nonempty and bounded for some ∅ ≠ C 1 ⊂ C and some ∈ R; (ii) U is finite, C is a polytope, and (C, U) is finite.

Proof
(i) First note that ⋂ c∈C 1 L(c, ) may be written as whose recession cone, provided that the set is nonempty, is given by and S (C, U) = S w (C, U) = (1, β) � .
x ∈ R n | c � x ≤ and a � x ≥ b for all c ∈ C 1 and all a b ∈ U , d ∈ R n | c � d ≤ 0 and a � d ≥ 0 for all c ∈ C 1 and all a b ∈ U .
Observe that the latter set is independent on α. Then, according to [17,Theorem 8.4], our current assumption (i) means that ⋂ c∈C 1 L(c, ) is bounded for all α such that set is nonempty; in particular for any 1 > (C, U). For such α 1 we have Since the last set is bounded, the continuous function w C attains its global minimum on the compact set x ∈ F(U) | w C (x) ≤ 1 . In other words, S(C, U) ≠ �.
(ii) Write C = convC 0 for some finite set C 0 . Let v 0 ∶= (C, U) and assume, reasoning by contradiction, that S(C, U) = �. This means that the system is inconsistent (infeasible), and we may apply [10,Theorem 4.4] to conclude Due to the finiteness of U and C 0 , the previous cone is already closed, and then we may write for some a 0 b 0 ∈ coneU, some c 0 ∈ C (here we use the convexity of C), and some μ ≥ 0. In fact, we have μ > 0, since otherwise, appealing again to [10,Theorem 4.4], the system associated with U would be inconsistent (i.e., F (U) = �, yielding (C, U) = +∞ ). Now pick arbitrarily x ∈ F(U) and multiply both sides of (17) by x −1 to obtain which implies c � 0 x ≥ v 0 + 1∕ , and, therefore, w C (x) ≥ v 0 + 1∕ . Since x was arbitrarily chosen in F(U), this contradicts the fact that v 0 = (C, U) = inf x∈F(U) w C (x).
As an immediate consequence of (i) in the previous proposition (by just taking arg max C v U and recalling Remark 5.1 and Proposition 5.1), we have the following corollary.
Corollary 5.1 If S w ( ) is nonempty and bounded, then S( ) also is.

Lipschitz Continuity of
The sensitivity analysis of problem (1) is tackled in the following result by means of an upper bound on lip 0 . To the authors' knowledge, this result is new even in the case when C 0 is a singleton; in other words, it does not have a direct counterpart in the setting of problems (5). From a different approach, related to distance to ill-posedness, more specifically, distance to inconsistency, [7,Theorem 4.3] provides a Lipschitz constant for when C 0 is a singleton. A recent upper bound (exact value under suitable hypotheses) on the Lipschitz modulus of the optimal value function for finite linear programs from a primal-dual approach (again with C 0 being a singleton) is given in [9,Theorem 5.2]. In contrast, the current paper follows a primal approach. Here we appeal to where C U 0 ,x is defined as in (15). Theorem 6.1 Let 0 = C 0 , U 0 ∈ Π. Assume that SSC holds at U 0 and that S 0 and a ∶ a b ∈ U 0 are nonempty and bounded in R n . Then Proof Let us see that for every ε > 0 there exists a neighborhood C × V ⊂ Π of π 0 such that where Then the result follows. The proof is carried out in four steps.
Step 1: 0 n ∉ C 0 , i.e., d * 0 n , C 0 > 0 due to the closedness of C 0 . Assume, reasoning by contradiction, that Finally, by symmetry, we have that Then, (18) is a direct consequence of the previous expression.