Piecewise parametric structure in the pooling problem: from sparse stronglypolynomial solutions to NPhardness
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Abstract
The standard pooling problem is a NPhard subclass of nonconvex quadraticallyconstrained optimization problems that commonly arises in process systems engineering applications. We take a parametric approach to uncovering topological structure and sparsity, focusing on the single quality standard pooling problem in its pformulation. The structure uncovered in this approach validates Professor Christodoulos A. Floudas’ intuition that pooling problems are rooted in piecewisedefined functions. We introduce dominant active topologies under relaxed flow availability to explicitly identify pooling problem sparsity and show that the sparse patterns of active topological structure are associated with a piecewise objective function. Finally, the paper explains the conditions under which sparsity vanishes and where the combinatorial complexity emerges to cross over the P / NP boundary. We formally present the results obtained and their derivations for various specialized single quality pooling problem subclasses.
Keywords
Standard pooling problem Global optimization Piecewise structure Sparsity Discretization P / NP boundary Stronglypolynomial algorithms1 Introduction
The standard pooling problem represents a NPhard subclass [3] of nonconvex quadraticallyconstrained optimization problems with bilinear terms and may have a multiplicity of local minima [33]. Pooling problems model the computational difficulties associated with intermediate blending of heterogeneous feedstocks and therefore have direct application in process system engineering [34, 56]. Specific application domains include: petroleum refining [7, 20, 49], mining [14], wastewater treatment [35, 42], crude oil scheduling [39], natural gas production [50], etc. We recently showed that standard pooling arises as a subproblem pattern in general mixedinteger nonlinear optimization (MINLP) [17].
Motivated by applications, Floudas and Visweswaran [22, 23, 57, 58] were the first to rigorously solve the pooling problem to global optimality. The Floudas and Visweswaran approach uses duality theory and Lagrangian relaxations. Subsequent global optimization contributions to solving the pooling problem include: making further Lagrangian relaxation contributions [1], developing alternative problem formulations [5, 11], augmenting models with reformulationlinearization cuts [42, 48, 51, 52] to create a provably dominant formulation [53], developing problemspecific polyhedral cuts based on small pooling networks [18, 19], and identifying the P / NP boundary with respect to the topological structure [3, 15, 31, 32]. More general techniques for nonconvex quadraticallyconstrained optimization problems with bilinear terms are also appropriate for the pooling problem. The more general methods include: using convex envelopes to formulate a linear relaxation [2, 24, 41], developing a general branchandcut method [6], applying a sumofsquares hierarchy [40], and using stateoftheart global optimization MINLP solver software [10, 12, 13, 37, 45, 46, 53, 54, 55]. Further details are available in reviews discussing the pooling problem [5, 16, 30, 43].
But, despite significant attention to the pooling problem, deterministic global optimization algorithms can have significant optimality gaps and impractical or unknown convergence times on largescale, industriallyrelevant instances. These impractical convergence properties are interesting because Beale et al. [9] report that a simple, piecewiselinear program serves as a practical heuristic for small, poolinglike instances. Meyer and Floudas [42] had a similar intuition that very large pooling problems may be approached via piecewiselinear relaxation schemes. This intuition, which also appeared in Karuppiah and Grossmann [35], suggests that the pooling problem, a continuous nonlinear optimization problem (NLP), may be effectively approximated as a mixedinteger linear optimization problem (MILP). Further evidence for this intuition appears in several effective algorithms optimizing industriallyrelevant pooling instances via piecewiselinear approaches, e.g. [26, 29, 36, 44, 47, 59]. Subsequent work used the standard pooling problem topology to develop a stateoftheart MILP discretization heuristic with a performance bound [21, 27, 28].
This paper validates and substantiates Professor Floudas’ intuition by formalizing and characterizing the piecewise structures arising in standard pooling subclasses. We build a bottomup, intuitive understanding of the P / NP boundary of the single quality standard pooling problem by taking a parametric view and relaxing the flow availability box constraints. The relaxations employed effectively remove the flow availability bounds on feeds and pools and fix the product demand at each output. In the semantics of Boland et al. [15], e.g. Fig. 1 of their manuscript, our approach unifies and generalizes the \(K = 1\) complexity results. Our parametric approach yields polynomialtime subclasses with a piecewiselinear or piecewise convex/concave monotone structure. We formalize these piecewise structures in single quality standard pooling subclasses that offer exact global solutions in polynomial time. The proofs lead to the unexpected outcome that the famous Haverly [33] pooling instances, i.e. the firstrecorded pooling instances, belong to a stronglypolynomial subclass! The stronglypolynomial result for the Haverly [33] instances is remarkable because these case studies have been used as test cases for exponential algorithms for more than 35 years.
This manuscript also justifies the Beale et al. [9] observation that the linear approximation is most effective when only a few variables are active at once. Using patterns of dominating topologies, we explicitly identify pooling problem sparsity, i.e. a limited number of active flow variables. We show that these sparse patterns of active topological structure are associated with a piecewise objective function and we take advantage of these structures. Lastly, we explain the conditions under which such sparsity vanishes by reintroducing constraints on flow availability and, together with them, the combinatorial complexity needed to cross over the P / NPtime boundary.
2 Standard pooling pformulation and assumptions used
Standard pooling problem notation [43]
Type  Notation  Description 

Indices  \(i \in \{i \ \ (i,\cdot )\in T_X \cup T_Z\}\)  Input streams (raw materials or feed stocks) 
\(l \in \overline{1,L}\)  Pools (blending facilities)  
\(j \in \overline{1,J}\)  Output streams (end products)  
\(k \in \overline{1,K}\)  Attributes (qualities monitored)  
Sets  \(T_X\)  (i, l) pairs for which input to pool connection exists, \(T_X=I\) 
\(T_Z\)  (i, j) pairs for which input to output connection exists, \(T_Z=H\)  
\(T_Y\)  (l, j) pairs for which pool to output connection exists  
Problem type  \(I{+}H{}L{}J{}K\)  \(I{+}H\) feeds (inputs + directs), L pools, J outputs and K qualities 
\(I{+}H{}L{}J\)  \(I{+}H\) feeds (inputs + directs), L pools, J outputs and one quality  
\(I{}L{}J\)  (No directs) I inputs, L pools, J outputs and one quality  
\(H{}L{}J\)  (No inputs) H directs, L pools, J outputs and one quality  
Variables  \(x_{i,l}\)  Flow from input i to pool l 
\(y_{l,j}\)  Flow from intermediate pool node l to output j  
\(z_{i,j}\)  Bypass flow directly from input feed stock i to product j  
\(p_{l,k}\)  Level/concentration of quality attribute k in pool l  
Parameters  f  The objective function of the problem 
\(\gamma _i\)  Unit cost of raw material feed stock i  
\(d_j\)  Unit revenue for product j  
\(A_i^LA_i^U\)  Availability bounds (required usage to max. availability) of input i  
\(S_l\)  Volumetric size capacity of pool l  
\(D_j^LD_j^U\)  Demand bounds (required to limit demand) for product j  
\(C_{i,k}\)  Level of quality k in raw material feed stock i  
\(P_{j,k}^LP_{j,k}^U\)  Acceptable composition range of quality k in product j 
Remark 2.1
In Problem P–2.1, the upper hard bounds on variable sets \(\{x\},\ \{y\},\ \{z\}\) are redundant and can be dropped. These upper hard bounds are implicitly met by simultaneously enforcing the constraints on feed availability, pool capacity, product demand and material balance. Similarly, the hard bounds on \(p_{l,k}\ \forall l,k\) can be dropped, as they are implicitly met by replacing all y variables in the quality balance with x variables from the material balance.
Assumption 2.2
Problem P–2.1 is restricted to a single quality and assumed feasible, with dropped constraints on feed availability and pool capacity and fixed product demands \(D_j>0,\ \forall j\).
Note: Remarks 3.3.10, 4.6 and 5.6 explain how Assumption 2.2 provides tight bounds for sparsity and polynomialtime solvability. Remark 3.3.10 shows that the sparse, piecewise monotone structure of subclass \(\mathrm{I}{+}\mathrm{H}{}1{}1\) is tightly conditioned on Assumption 2.2. Remark 4.6 shows that the polynomialtime solvability of subclass \(\mathrm{I}{+}\mathrm{H}{}1{}\mathrm{J}\) is also tightly conditioned on Assumption 2.2. Remark 5.6 justifies these observations for the \(\mathrm{I}{+}\mathrm{H}{}\mathrm{L}{}1\) subclass.
Removing the feasibility assumption is not discussed, at it serves only to remove the check for an infeasible/unprofitable problem with all feeds inactive and \(f^*=0\).
3 Subclass \(\mathrm{I}{+}\mathrm{H}{}1{}1\): one pool, one output
Active sets, dominance relations and breakpoints are essential building blocks to find the structure of \(f^*(p)\) in Problem P–3.4 and are all introduced in Definitions 3.1–3.4.
Definition 3.1

An input active set if \( A\subseteq T_X,\qquad \ x_i=0\ \forall i \in T_X \setminus A,\ z_i=0\ \forall i \in T_Z\).

A direct active set if \(\ A\subseteq T_Z,\qquad \ x_i=0\ \forall i \in T_X,\qquad \,\, z_i=0\ \forall i \in T_Z \setminus A\).

A mixed active set if \( A\subseteq T_X\cup T_Z,\ x_i=0\ \forall i \in T_X\setminus A,\ z_i=0\ \forall i \in T_Z \setminus A,\ A\setminus T_X\notin \{A,\emptyset \}\).
Definition 3.2
(Feasibility with respect to product quality constraints) A Problem P–3.4 active set is feasible if the product quality bounds \([P^L,P^U]\) are met, i.e. the second constraint holds. An infeasible active set is not a valid Problem P–3.4 solution and is therefore strictly dominated by any feasible active set (see Definition 3.3).
Definition 3.3
 Set \(A_1\) dominates \(A_2\) at p (in the sense of maximized objective function profitability) when,$$\begin{aligned} {\varvec{A}}_1 {\varvec{\succeq }}_p {\varvec{A}}_2 \quad \Leftrightarrow \quad f^*_{A_1}(p) \ge f^*_{A_2}(p) \quad \Leftrightarrow \quad h^*_{A_1}(p) \le h^*_{A_2}(p). \end{aligned}$$(2)
 Pool concentration p is a breakpoint between \(A_1\) and \(A_2\) if:$$\begin{aligned} {\varvec{A}}_1 {\varvec{\asymp }}_p {\varvec{A}}_2 \quad \Leftrightarrow \quad f^*_{A_1}(p) = f^*_{A_2}(p) \quad \Leftrightarrow \quad h^*_{A_1}(p) = h^*_{A_2}(p). \end{aligned}$$(3)

The dominance relation also extends to direct active sets, but in this case f is not parametric on p. Consequently, when comparing two direct active sets, dominance is established similarly via Eq. (2) but independent of p, and as such no breakpoints exist. Thus, for fixed p, a total order can be established over the set of all possible active sets.
Definition 3.4

\(\mathcal {A}_I\) is the dominant input active set at p if \(\mathcal {A}_I(p) = \mathop {\mathrm{arg\,max}}\limits _{A\subseteq T_X}f^*_{A}(p)\).

\(\mathcal {A}_M\) is the dominant mixed active set at p if \(\mathcal {A}_M(p)= \mathop {\mathrm{arg\,max}}\limits _{A\subseteq T_X\cup T_Z,\ A\setminus T_X\notin \{A,\emptyset \}}f^*_{A}(p)\).

\(\mathcal {A}_D\) is the dominant direct active set if \(\mathcal {A}_D= \mathop {\mathrm{arg\,max}}\limits _{A\subseteq T_Z}f^*_{A}\).
The input, direct and mixed active sets have different dominance properties and thus the analysis proceeds in Sects. 3.1–3.3 by active set type. Section 3.1 ignores directs and product quality constraints and focuses only on inputs. Since directs are ignored, the pool concentration p represents the output concentration, and hence p is assumed free of product quality bounds. The analysis of the pparametric optimal objective \(f^*(p)\) reveals a piecewiselinear structure associated with pairs of inputs acting as the dominant input active set. Section 3.2 treats the complementary case, ignoring inputs and focusing only on directs while assuming product quality constraints. Since inputs and therefore the pool are assumed to send no flow in this case, the optimal objective \(f^*\) and the dominant direct active set are found independently of p. Finally, Sect. 3.3 integrates the Sects. 3.1–3.2 results, combining both inputs and directs under assumed product quality constraints to reveal a sparse, piecewisemonotone structure of the pparametric optimal objective \(f^*(p)\).
Sections 3.1–3.3 analytically and parametrically identify all (dominance) breakpoints, sparse dominant active sets and associated pparametric solutions for Problem P–3.4. This analysis leads to a stronglypolynomial algorithm in Sect. 3.3 for solving the \(\mathrm{I}{+}\mathrm{H}{}1{}1\) subclass formalized in Problem P–3.4. Furthermore the full structure of the pparametric optimal objective function \(f^*(p)\) developed in Sect. 3.3 is vital for Sects. 4–5.
Remark 3.5
For any \(i,j\in T_X\cup T_Z\) with \(i\not =j,\ C_i=C_j\), if \(\gamma _i\le \gamma _j\) precedence is given to the node i with cheaper flow, or at cost equality a random choice is made. After prefiltering all feeds of equal concentrations on cost criteria, we are assured \(\forall i, j\in T_X\cup T_Z, i\not =j\) that \(C_i\not = C_j\). For any \(i,j\in T_X\cup T_Z\) with \(i\not =j,\ C_i < C_j\) if \(\gamma _i=\gamma _j\) precedence is given to i if \(C_i\in [P^L,P^U],\ C_j\notin [P^L,P^U]\) and vice versa. We apply the enumerated precedence rules throughout Sect. 3. This prefiltering avoids undefined expressions, e.g. denominators with value zero in the subsequent sections.
3.1 Inputsonly analysis (I\({}1{}1\) subcase)
Remark 3.1.1
(Dominant input active set at p of cardinality 2) For fixed p, Problem P–3.1.5 can be rewritten as a standard LP in the x variables with two active constraints at the optimal basis, implying the dominant input active set has cardinality 2, i.e. \(\mathcal {A}_I(p) = 2\) (Fig. 2).
Lemma 3.1.2
Proof
The flows in Eq. (7) result from \(x_i+x_j=D,\ p=(C_ix_i+C_jx_j)/(x_i+x_j)\). Eq. (6) follows by substituting Eq. (7) flows into \(f_A(p)=dD\gamma _ix_i\gamma _j x_j\) and differentiating w.r.t. p. \(\square \)
Definition 3.1.3
Proposition 3.1.4
Proof
W.l.o.g. \(C_i\ge p\ge C_j\) and \(C_k\ge p\ge C_l\). Eq. (2) implies \(ij\succeq _p kl\Leftrightarrow \gamma _ix_i+\gamma _jx_j\le \gamma _kx_k+\gamma _lx_l.\) Eq. (9b) follows from Definition 3.1.3 after substituting Eq. (7) for flows in the previous condition. Eq. (9a) follows from separating out the terms with factor/slope p in Eq. (9b). \(\square \)
Proposition 3.1.5
Theorem 3.1.6
 (i)Input dominance breakpoints can occur only at input concentrations \(C_i,\ i\in T_X\), hence,which requires I (number of inputs) evaluations.$$\begin{aligned} f^*=\max _{i\in T_X}f^*(C_i)=\max _{i\in T_X}D(d\gamma _i), \end{aligned}$$
 (ii)A full description of \(f^*(p)\) can be obtained in stronglypolynomial time \(O(I^3)\), with the set \(\mathcal {B}_I\) of input dominance breakpoints,Between any two consecutive elements of \(\mathcal {B}_I\), the dominant input active set remains constant, i.e.$$\begin{aligned} \mathcal {B}_I = \left\{ C_i \left \ i\in T_X,\ \gamma _i< \mathop {\mathrm{arg\,min}}\limits _{\{k,l\}\subseteq T_X\setminus \{i\}} \gamma _{kl}(C_i)\right\} \right. . \end{aligned}$$(11a)$$\begin{aligned} C_i,C_j\in \mathcal {B}_I,\ \mathcal {B}_I\cap (C_i,C_j)=\emptyset \quad \Rightarrow \quad \mathcal {A}_I(p)=\{i,j\}\ \forall p\in [C_i,C_j] \end{aligned}$$(11b)
Proof
 (i)
Since Proposition 3.1.5 implies \(\mathcal {A}_I(p)=2\), let two such dominant active input pairs, \(\{i,j\}\) and \(\{k,l\}\), and w.l.o.g. assume \(C_i<C_j\), \(C_k<C_l\). Assume, to achieve a contradiction, that an input dominance breakpoint occurs at b, where \(b\in (C_i,C_j)\cap (C_k,C_l)\). Consequently, again w.l.o.g. assume \(\{k,l\}\succeq _p \{i,j\}\) \(\forall p\in (b,C_l)\). As a result, in the geometric construction of Fig. 3, \(i{}l{}j{}k\) forms a quadrilateral with \(f^*(b)\) at the intersection of its diagonals. Notice, \(\forall p\in (C_i,C_l)\), the \(f_{\{i,l\}}^*(p)\) values obtained on the side \(il\) (dashed green) are higher than the optimal objective values obtained by going through the breakpoint b (lines in bold blue), contradiction. Therefore no input dominance breakpoint can occur at a pool concentration \(b\notin \{C_i\ i\in T_X \}\). Since Lemma 3.1.2 implies \(f^*(p)\) is linear between any two input dominance breakpoints when an input pair is active, the assertion made follows.
 (ii)
To fully describe \(f^*(p)\), if \(C_i\) for fixed \(i\in T_X\) is an input dominance breakpoint, then, according to Eq. (10), node i must strictly dominate at \(C_i\) any input pair not containing it. Eq. (11b) follows via the definitions of \(\mathcal {B}_I, \mathcal {A}_I\). \(\square \)
Remark 3.1.7
If product quality constraints are readded to Problem P–3.1.5, then Theorem 3.1.6 still applies, with valid input dominance breakpoints \((\mathcal {B}_I \cap (P^L,P^U))\cup \{P^L,P^U\}\).
3.2 Directsonly analysis (H\({}1{}1\) subcase)
Remark 3.2.1
(Dominant direct active set of maximum cardinality 2) Problem P–3.2.6 can be rewritten as a standard LP in the z variables with at most two active constraints at the optimal basis, implying the dominant direct active set has at most cardinality 2, i.e. \(\mathcal {A}_D \le 2\) (Fig. 5).
Lemma 3.2.2
 (i)
A feasible direct active set (solution) exists \(\Leftrightarrow \exists i\in T_Z\) s.t. \(P^L \le C_i \le P^U\) or \(\exists i,j\in T_Z\) s.t. \(C_i<P^L, C_j > P^U\).
 (i)
A direct active pair {i, j} is feasible \(\Leftrightarrow \) i and j are feasible or \((P^UC_i)(P^UC_j)<0\) or \((P^LC_i)(P^LC_j)<0\).
Proof
(i) If \(\exists i\in T_Z\) s.t. \(P^L \le C_i \le P^U\) then Problem P–3.2.6 is obviously feasible. Alternatively, w.l.o.g., if \(C_i<P^L\), there must \(\exists j\in T_Z\) with \(C_j > P^L\) so an output concentration within \([P^L,P^U]\) can be obtained. However, this case implicitly assumed \(\not \exists i\in T_Z\) s.t. \(P^L \le C_i \le P^U\), and consequently \(C_j > P^U\), concluding the proof.
(ii) Both \((P^UC_i)(P^UC_j)<0\) or \((P^LC_i)(P^LC_j)<0\) imply \(C_i\) and \(C_j\) are on opposite sides of a product quality bound and thus the linear combination of their concentrations implied by active \(\{i,j\}\) can be within \([P^L,P^U]\), making \(\{i,j\}\) feasible. \(\square \)
Lemma 3.2.3
Proof
’\(\Rightarrow \)’: If both \(C_i,C_j\in [P^L,P^U]\), then \({{\mathrm{arg\,min}}}_{i,j}(\gamma _i,\gamma _j) \succeq \{i,j\}\), contradiction, so we can assume w.l.o.g. \(C_i\not \in [P^L,P^U]\). If also \(C_j\not \in [P^L,P^U]\) both i and j are infeasible alone, but \(\{i,j\}\) can be feasible if the second condition in Lemma 3.2.2.i is met. In the previous case, one of the nodes i, j has the properties needed. Else if \(C_j\in [P^L,P^U]\) and \(\gamma _i>\gamma _j\), then \(j\succeq \{i,j\}\)  therefore \(\gamma _i<\gamma _j\). The reverse proof ’\(\Leftarrow \)’ is trivial. \(\square \)
Proposition 3.2.4
Proof
Proof in “Appendix A”. Note that due to Remark 3.5 and Lemmas 3.2.2–3.2.3 we have \((C_iC_j)(\gamma _i\gamma _j)\not =0\). \(\square \)
The result implies that for a feasible direct active pair \(\{i,j\}\) with \(\{i,j\}\succeq i,j\), Problem P–3.2.7 is analogous to the inputonly Problem P–3.1.5, with input flows \(x_i\) replaced by direct flows \(z_i\) and fixed pool concentration p replaced by a product quality limit P(i, j) (either lower or upper). Thus, the flow and dominance results for pairs in Sect. 3.1 are mirrored via Corollaries 3.2.5–3.2.6. Moreover, any pair viable as the dominant direct active set needs to first dominate both its individual nodes, so only such pairs and their solutions are of interest.
Corollary 3.2.5
Corollary 3.2.6
(Domination condition between active direct pairs)
Proof
Analogous to Proposition 3.1.4, but for Problem P–3.2.7 rather than Problem P–3.1.5. \(\square \)
Theorem 3.2.7
(Directsonly optimal solution and dominant direct active set)
 (a)
If \(\{i,j\}\succeq i,j\) then \(\mathcal {A}_D=\{i,j\}\) with the flows in Eq. (13) and \(f^*=D\cdot (d\gamma _{ij})\).
 (b)
Else, \(\mathcal {A}_D= \alpha = {{\mathrm{arg\,min}}}_{\alpha \in \{i,j\}} \gamma _\alpha \) with \(z_{\alpha }=D\) and \(f^*=D\cdot (d\gamma _{\alpha })\).
Proof
 (a)
If \(\{i,j\}\succeq i,j\), Lemma 3.2.3 implies one of the nodes, w.l.o.g j, is infeasible, so w.l.o.g \(C_j>P^U,C_i<P^U\) and \(\gamma _j<\gamma _i\). If \(\exists k\in T_Z\setminus \{i\}\) s.t. \( C_k<P^U,\ \gamma _k<\gamma _i\), then \(\{k,j\}\succeq \{i,j\}\), contradiction. Therefore (feasible) i dominates any other alternative feasible direct, and by transitivity, \(\{i,j\}\succeq i\succeq k\ \forall k\in T_Z\setminus \{i\}, C_k\in [P^L,P^U]\) . The latter and Remark 3.2.1 imply that \(\mathcal {A}_D=\{i,j\}\).
 (b)
Assume \(i\succeq \{i,j\}\), so i is feasible with \(P^L \le C_i \le P^U\). If \(\gamma _j<\gamma _i\) then \(\{i,j\}\) as a pair with a linearly weighted cost would dominate i, contradiction, and therefore \(\gamma _j>\gamma _i\). If more restrictively, \(P^L< C_i < P^U\), since by transitivity \(i\succeq \{i,j\}\succeq \{i,k\}\ \forall k\in T_Z\setminus \{i,j\}\) (\(\{i,k\}\) feasible due to \(P^L< C_i < P^U\)), therefore \(\gamma _k>\gamma _i \ \forall k\in T_Z\setminus \{i\}\) and \(\mathcal {A}_D=i\). Now the complementary restriction of \(C_i=P^U\) is assumed (\(C_i=P^L\) is analogous). If \(\exists k\in T_Z\setminus \{j\},\ C_k<P^U,\ \gamma _k<\gamma _i\) then \(\{i,k\}\succeq i\succeq \{i,j\}\), contradiction, therefore \(\forall k\in T_Z\setminus \{j\},\ C_k<P^U \) we have \( \gamma _k>\gamma _i\). Additionally, since \(\{i,j\}\) feasible implies \(C_j<P^U\), if \(\exists k\in T_Z\) s.t. \(C_k>P^U, \gamma _k<\gamma _i\) then \(\{k,j\}\succeq \{i,j\}\), contradiction, thus \(\forall k\in T_Z,\ C_k>P^U,\ \gamma _k>\gamma _i\). Therefore, all subcases after assuming \(i\succeq \{i,j\}\) result in \(\mathcal {A}_D=i\).\(\square \)
3.3 Inputs and directs analysis (I\({+}\mathrm{H}{}1{}1\) subclass)
This subsection considers the original pparametric Problem P–3.4, allowing mixed active sets of both input and direct nodes. Theorem 3.3.1 uses the interplay of earlier results for both input (Sect. 3.1) and direct (Sect. 3.2) active sets to pinpoint mixed active sets that can be dominant (overall) active sets as triples of two inputs and one direct. This section focuses on mixed triples not dominated by the dominant input active set \(\mathcal {A}_I\). Definition 3.3.2 first extends the feasibility conditions from Sect. 3.2 for mixed triples viewed as direct pairs to pintervals by partitioning any pinterval \({\varPhi }\) around \(\{P^L,P^U\}\) if necessary and building \(Q({\varPhi })\), the set of directs making the mixed active set feasible. Definition 3.3.2e also extends the directs domination result in Lemma 3.2.3 to pintervals. Lastly, Definition 3.3.2f splits pintervals \({\varPhi }\) into subintervals \({\varPhi }_I\) and \({\varPhi }_M\) based on whether the dominant mixed active triple is dominated or not by \(\mathcal {A}_I\) over the subintervals, respectively. Using the breakpoints between mixed and input active sets identified in Lemma 3.3.3, Lemma 3.3.4 then implements the \({\varPhi }_I/{\varPhi }_M\) split of any interval \({\varPhi }\).
Based on the latter results, Proposition 3.3.5 finds the dominant mixed active set for fixed p and all mixed dominance breakpoints, while Proposition 3.3.6 finds all dominance breakpoints. Moreover, Proposition 3.3.7 finds the pparametric optimal objective function to be monotone convex/concave for mixed active sets. Consequently, Theorem 3.3.8 summarizes all cases of optimal objective monotonicity. Finally, Theorem 3.3.9 uses all objective monotonicity results to find the optimal solution at a breakpoint dominance point in stronglypolynomial time.
Theorem 3.3.1
Proof
Finally, the flow solutions in Eq. (16) are found by combining Eq. (18) with Eq. (20a). \(\square \)
Definition 3.3.2
 (a)Let subintervals of the partition \(\{P^L,P^U\}\) of inputs/directs concentrations be denoted by:$$\begin{aligned} \begin{aligned} I^X_{L}&= \big [\min \limits _{i\in T_X}C_i, P^L\big ),\\ I^Z_{L}&= \big [\min \limits _{i\in T_Z}C_i, P^L\big ), \end{aligned}\quad I_{LU}=\big [P^L,P^U\big ],\quad \begin{aligned} I^X_{U}&= \big (P^U, \max \limits _{i\in T_X}C_i \big ],\\ I^Z_{U}&= \big (P^U, \max \limits _{i\in T_Z}C_i \big ]. \end{aligned} \end{aligned}$$
 (b)Let \({\varPhi }\subseteq [C_i,C_j]\) denote a closed pinterval between two consecutive input dominance breakpoints, i.e. \(C_i,C_j\in \mathcal {B}_I,\ \mathcal {B}_I\cap (C_i,C_j)=\emptyset \), with \((\forall p\in {\varPhi })\ \mathcal {A}_I(p)=ij\). Let \({\varPhi }\in \{[C_i,C_j]\cap I_L^X,\ [C_i,C_j]\cap I_{LU},\ [C_i,C_j]\cap I_U^X \}\) such that:$$\begin{aligned} \mathbb {1}_{{\varPhi }\subseteq I^X_L}+\mathbb {1}_{{\varPhi }\subseteq I^X_{LU}}+\mathbb {1}_{{\varPhi }\subseteq I^X_{U}}=1. \end{aligned}$$
 (c)Let \(Q({\varPhi })\) be the set of directs with concentration outside \({\varPhi }\)’s partition around \(\{P^L,P^U\}\), i.e.:$$\begin{aligned} \begin{aligned} Q({\varPhi })&= \big \{q\in T_Z \big \ C_q\in [\min \limits _{i\in T_Z}C_i,\max \limits _{i\in T_Z}C_i]\setminus I^Z_\beta , \text { where } {\varPhi }\subseteq I^X_\beta ,\ \beta \in \{L,LU,U\} \big \}\\&= \big \{q\in T_Z \big \ \text { mixed active set }\{ij,q\} \text { is feasible, where}\ (\forall p\in {\varPhi })\ \mathcal {A}_I(p)=ij \big \}. \end{aligned} \end{aligned}$$
 (d)Let \(R({\varPhi })\subseteq Q({\varPhi })\) denote a subset s.t. \((\forall q\in R({\varPhi }))(\forall p\in {\varPhi })\ \gamma _q < \gamma _{ij}(p)\). Since \(\gamma _{ij}(p)\), as defined in Eq. (8), is a linear function of p with extremes at \({\varPhi }\) endpoints,$$\begin{aligned} \begin{aligned} R({\varPhi })=\big \{q\in Q({\varPhi }) \big \ \gamma _q<\max \{\gamma _{ij}(b_l),\gamma _{ij}(b_u)\}, \text { where } {\varPhi }=[b_l,b_u]\big \}. \end{aligned} \end{aligned}$$
 (e)
Let \(\begin{aligned} {\varTheta }({\varPhi }) = {\left\{ \begin{array}{ll} R({\varPhi })\text { if } R({\varPhi })\not =\emptyset \text { or }\ \ {\varPhi }\subseteq I_{LU},\\ Q({\varPhi })\text { if } R({\varPhi }) =\emptyset \text { and } {\varPhi }\not \subseteq I_{LU}. \end{array}\right. } \end{aligned}\)
 (f)Let \({\varPhi }_I,{\varPhi }_M\) be partition subintervals of \({\varPhi }\), where \(\mathcal {A}_I, \mathcal {A}_M\) dominate, respectively. Assuming \((\forall p\in {\varPhi })\ \mathcal {A}_I(p)=ij\),$$\begin{aligned} \begin{aligned}&{\varPhi }_I = \left\{ p\in {\varPhi }\ (\forall q\in {\varTheta }({\varPhi }))\ \gamma _{ij}(p)\le \gamma _{q} \Rightarrow \text {if }{\varPhi }\subseteq I_{LU}\text { then } (\forall q)\ ij\succeq _p \{ij,q\}, \mathcal {A}_I(p)\succeq _p \mathcal {A}_M(p) \right\} ,\\&{\varPhi }_M=\{ p\in {\varPhi }\ (\exists q\in {\varTheta }({\varPhi }))\ \gamma _{ij}(p)\ge \gamma _{q} \Leftrightarrow (\exists q\in {\varTheta }({\varPhi }))\ ij\preceq _p \{ij,q\} \Leftrightarrow \mathcal {A}_I(p)\preceq _p \mathcal {A}_M(p) \}.\\ \end{aligned} \end{aligned}$$(21)
Lemma 3.3.3
(Domination/breakpoints between dominant input and mixed active sets)
Proof
Proof in “Appendix A”. \(\square \)
Lemma 3.3.4
 (i)
\((\forall p\in {\varPhi })\ (\mathcal {A}_M(p)=\{ij,q\} \Rightarrow q\in {\varTheta }({\varPhi }))\).
 (ii)\((\forall {\varPhi })\) Subintervals \({\varPhi }_I,{\varPhi }_M\) can be found explicitly as:where:$$\begin{aligned} \left\{ \begin{aligned}&(a)\ {\varPhi }_I={\varPhi },\ {\varPhi }_M=\emptyset , \text { if } {\varTheta }({\varPhi })=\emptyset ,\\&(b)\ {\varPhi }_I=\{ p\in {\varPhi }\ b_l \le p \le \min (S) \},\ {\varPhi }_M=\{ p\in {\varPhi }\ \min (S) \le p \le b_u \}, \text { else if } \frac{\gamma _i\gamma _j}{C_iC_j}>0,\\&(c)\ {\varPhi }_I=\{ p\in {\varPhi } \max (S) \le p \le b_u \},\ {\varPhi }_M=\{ p\in {\varPhi }\ b_l \le p \le \max (S) \}, \text { else if } \frac{\gamma _i\gamma _j}{C_iC_j}<0, \end{aligned}\right. \end{aligned}$$(23)$$\begin{aligned} \begin{aligned} S&=\{p\ \{i,j\}\asymp _p\{i,j,q\},\ \forall q\in {\varTheta }({\varPhi }) \} \\&= \left\{ \frac{C_i(\gamma _q\gamma _j)C_j(\gamma _q\gamma _i)}{\gamma _i\gamma _j}\bigg \ \forall q\in {\varTheta }({\varPhi }) \right\} . \end{aligned} \end{aligned}$$(24)
 (iii)\((\forall q\in {\varTheta }({\varPhi })) (\forall p\in \hat{{\varPhi }}\in \{{\varPhi }_I,{\varPhi }_M\} )\) if \(\mathcal {A}_M(p)=\{ij,q\}\) then P(ij, q) reduces to:independent of specific \(p\in \hat{{\varPhi }}\).$$\begin{aligned} P(ij,q)={\left\{ \begin{array}{ll} P^L,\text {if } ((C_q<b_u)\oplus (\hat{{\varPhi }}={\varPhi }_M))\wedge (P^LC_q)(P^Lb_u)<0,\\ P^U,\text {if } ((C_q>b_u)\oplus (\hat{{\varPhi }}={\varPhi }_M))\wedge (P^UC_q)(P^Ub_u)<0, \end{array}\right. } \end{aligned}$$(25)
Proof
(i) The restriction \(Q({\varPhi })\in T_Z\), and its subset \(R({\varPhi })\in T_Z\), enforces Lemma 3.2.2 feasibility for \(\{ij,q\},\ \forall p\in {\varPhi }\) viewed as a direct active set. Set \({\varTheta }({\varPhi })\) also enforces the Lemma 3.2.3 domination condition \(\{ij,q\}\succeq _p ij \Leftrightarrow \forall p\in {\varPhi }\ \gamma _q\le \gamma _{ij}(p)\) via \({\varTheta }({\varPhi })=R({\varPhi })\). When \(R({\varPhi })=\emptyset \) and \({\varPhi }\subseteq I_{LU}\) then ij feasible and \((\forall q\in Q({\varPhi }))(\forall p\in {\varPhi })\ \gamma _q\ge \gamma _{ij}(p) \Leftrightarrow (\forall p\in {\varPhi })\ ij\succeq \mathcal {A}_M(p)\), in which case \({\varTheta }({\varPhi })=R({\varPhi })=\emptyset \) since \((\forall p\in {\varPhi })\ \mathcal {A}_M(p)\not =\mathcal {A}^*(p)\). When \(R({\varPhi })=\emptyset \) and \({\varPhi }\not \subseteq I_{LU}\) then ij infeasible, and since a direct needs to be active for feasibility, in this case \({\varTheta }({\varPhi })=Q({\varPhi })\) and \((\forall q\in Q({\varPhi }))(\forall p\in {\varPhi })\ \gamma _q\ge \gamma _{ij}(p)\) but \(\{ij,q\}\succeq _p ij,q\). Therefore in this case, \({\varTheta }({\varPhi })\) is extended from \(R({\varPhi })\) to \(Q({\varPhi })\).
(ii) Case (a) results by construction of \({\varPhi }_I,{\varPhi }_M, {\varTheta }({\varPhi })\). For Case (b)(c), S is built as the set of all breakpoints from Lemma 3.3.3 where \(\gamma _{ij}(p)=\gamma _{q},\forall p\in S, q\in {\varTheta }({\varPhi })\). From Eq. (8), \(\partial \gamma _{ij}(p)/\partial p = (\gamma _i\gamma _j)/(C_iC_j)\), which implies if \(\partial \gamma _{ij}(p)/\partial p>0\) then \(\gamma _{ij}(p)\) is increasing with p. Therefore, given \(\forall b\in S,\gamma _{ij}(b)=\gamma _q\), if \(\partial \gamma _{ij}(p)/\partial p>0\) then \(\forall p\le b\ \gamma _{ij}(p)\le \gamma _q\) and viceversa. Consequently, if \(\partial \gamma _{ij}(p)/\partial p>0\) then \({\varPhi }_I\) is the restriction of \({\varPhi }\) up till \(\min (S)\), and otherwise \({\varPhi }_I\) is the restriction of \({\varPhi }\) from \(\max (S)\).
Consequently, \(\forall p\in {\varPhi }_I\), if ij is feasible (\({\varPhi }\subseteq I_{LU}\)) then ij dominates any mixed active set \(\{ij,q\}\) and Theorem 3.3.1 then implies \((\forall p\in {\varPhi }_I)\ \mathcal {A}_I(p)\succeq _p \mathcal {A}_M(p)\). Complementary, \(\{ij,q\}\) is feasible by construction (\(q\in {\varTheta }({\varPhi })\)), and therefore \(\forall p\in {\varPhi }_M\ \exists q\in {\varTheta }({\varPhi })\ \gamma _{ij}(p)\ge \gamma _{q} \Leftrightarrow \exists q\in {\varTheta }({\varPhi })\ ij\preceq _p \{ij,q\}\Leftrightarrow \mathcal {A}_I(p)\preceq _p \mathcal {A}_M(p)\).
(iii) Assume first \(\hat{{\varPhi }}={\varPhi }_I\not ={\varPhi }_M\). Then, Eq. (21) implies \((\forall q\in {\varTheta }({\varPhi }))(\forall p \in \hat{{\varPhi }})\ \gamma _{ij}(p)\le \gamma _q\). Consequently, the expression for P(ij, q) according to Eq. (12) reduces to Eq. (25). Now assume \(\hat{{\varPhi }}={\varPhi }_M\not ={\varPhi }_I\). Then, Eq. (21) implies \((\forall p\in \hat{{\varPhi }})\ \mathcal {A}_M(p)=\{ij,q\}\succeq _p \mathcal {A}_I(p)\) and hence \((\forall p\in \hat{{\varPhi }})\ \gamma _{ij}(p)\ge \gamma _{q}\). Thus, again, the expression for P(ij, q) reduces to Eq. (25). For both possible \(\hat{{\varPhi }}\in \{{\varPhi }_I,{\varPhi }_M\}\), due to the construction of \({\varPhi }\) and \({\varTheta }({\varPhi })\) with \(q\in {\varTheta }({\varPhi })\), \((\forall p\in \hat{{\varPhi }})\ (C_q<p)=(C_q<b_u)\), and thus the comparison becomes independent of a specific \(p\in \hat{{\varPhi }}\). Furthermore, to ensure feasibility of \(\{ij,q\}\) according to Lemma 3.2.2.ii, \((P(ij,q)C_q)(P(ij,q)b_u)<0\) is enforced explicitly in Eq. (25). \(\square \)
Proposition 3.3.5
 (i)(The dominant mixed active set at p) For fixed \(p\in \hat{{\varPhi }},\ \mathcal {A}_M(p)=\{ij,q\}\succeq _p \mathcal {A}_I(p)\) with$$\begin{aligned} \ q = \mathop {\mathrm{arg\,min}}\limits _{r\in {\varTheta }({\varPhi })} \left( \gamma _{\{ij,r\}}(p) := {\frac{\gamma _r(P(ij,r)p)+\gamma _{ij}(p)(C_rP(ij,r))}{C_rp}}\right) . \end{aligned}$$(26)
 (ii)(The set of mixed dominance breakpoints over \(\hat{{\varPhi }}\))with \((\forall q\in {\varTheta }({\varPhi }))\ P(ij,q)\) as in Eq. (25) and independent of p; \(b_l,b_u\in \mathcal {B}_M\) as well.$$\begin{aligned} \begin{aligned} \mathcal {B}_M&\supseteq \left\{ p\in {{\mathrm{int}}}{\hat{{\varPhi }}}\ \{ij,q\}\asymp _{p}\{ij,r\} \wedge q,r\in \mathcal {A}_M(p),\ \forall q,r\in {\varTheta }({\varPhi }) \right\} \\&= \left\{ p\in {{\mathrm{int}}}{\hat{{\varPhi }}}\ p\in \text {SqrRoots}\left( {\varGamma }\big (p,P(ij,q),P(ij,r)\big )\right) \wedge q,r\in \mathcal {A}_M(p),\ \forall q,r\in {\varTheta }({\varPhi }) \right\} , \end{aligned} \end{aligned}$$(27)
Proof
(i) If \(\hat{{\varPhi }}\not \subseteq I_{LU}\), then \(\mathcal {A}_I(p)=ij\) is infeasible and by default \(\mathcal {A}_M(p)\succeq _p \mathcal {A}_I(p)\). If \(\hat{{\varPhi }}={\varPhi }_M\), then, from the construction of \({\varPhi }_M\) in Eq. (21), \(\mathcal {A}_M(p)\succeq _p \mathcal {A}_I(p)\). Thus Theorem 3.3.1 asserts via Eq. (20a) that \(\mathcal {A}_D^p =\{ij,q\}=\mathcal {A}_M(p)\) for a \(q\in {\varTheta }({\varPhi })\), as required by Lemma 3.3.4.i. By viewing \(\{ij,q\}\) as a dominant direct pair with ij active in order to find \(\mathcal {A}_D^p\), the condition Eq. (15) in Theorem 3.2.7 becomes Eq. (26). Since p is fixed, Eq. (26) can be solved using the original Eq. (12) for \(P(ij,q)\ \forall q\in {\varTheta }({\varPhi })\).
(ii) “Appendix A” proves Eq. (27) and introduces function \({\varGamma }\big (p,P(ij,q),P(ij,r))\) which is quadratic in p and linear in P(ij, q), P(ij, r). To solve the function \({\varGamma }\big (p,P(ij,q),P(ij,r)\big )\) as a quadratic of p, \((\forall q\in {\varTheta }({\varPhi }))\ P(ij,q)\) has to be independent of p via the form in Eq. (25). First, Lemma 3.3.4.iii implies \(\forall p\in \hat{{\varPhi }}\) that if \(\{ij,q\}=\mathcal {A}_M(p)\) then P(ij, q) in Eq. (25) is correct; therefore any dominant breakpoint within \({\varPhi }\) between \(\{ij,q\}\) and another dominant mixed set is captured in \(\mathcal {B}_M\). Second, Lemma 3.3.4.iii implies if Eq. (25)\(\not \Leftrightarrow \)Eq. (12) for \(P(ij,q),\ q\in {\varTheta }({\varPhi }),\) then feasible \(\{ij,q\}\not =\mathcal {A}_M(p)\). Consequently, the conjunctive condition \(q,r\in \mathcal {A}_M(p)\) eliminates not only those mixed breakpoints that are not dominant, but also those calculated on potentially incorrect P(ij, q) from Eq. (25). Note that, unlike the inputsonly case of Sect. 3.1, two breakpoints between any \(\{ij,q\}\) and \(\{ij,r\}\) can occur, because \(f_{\{ij,q\}}(p)\) and \(f_{\{ij,r\}}(p)\) are convex or concave functions (see Proposition 3.3.7) which can intersect at two points. The endpoints of \(\hat{{\varPhi }}\) also represent mixed dominance breakpoints, since for \(p\in \{b_l,b_u\}\), given \(\mathcal {A}_M(p)=\{ij,q\}\), either ij, q or P(ij, q) change at p, creating a breakpoint. \(\square \)
Proposition 3.3.6
 (a)\((\forall p \in {\varPhi }_I\subseteq I_{LU})\), ij feasible and \(\mathcal {A}^*(p)\in \{\mathcal {A}_I(p),\mathcal {A}_D\}\) with dominance breakpoint$$\begin{aligned} \left\{ p\in {\varPhi }_I\bigg \ ij\asymp _p \mathcal {A}_D:=\alpha \Leftrightarrow \gamma _{ij}(p)=\gamma _\alpha \Leftrightarrow p= \frac{C_i(\gamma _q\gamma _j)C_j(\gamma _q\gamma _i)}{\gamma _i\gamma _j} \right\} \subseteq \mathcal {B}. \end{aligned}$$(28)
 (b)\((\forall p\in \hat{{\varPhi }})\), for \(\hat{{\varPhi }}\in \{{\varPhi }_M;\ {\varPhi }_I \not \subseteq I_{LU}\}\subseteq {\varPhi }\), \(\mathcal {A}^*(p)\in \{\mathcal {A}_D,\mathcal {A}_M(p)\}\) with dominance breakpoints
Proof
(a) By construction, \((\forall p\in {\varPhi }_I\subseteq I_{LU})\ \mathcal {A}_I(p)\succeq _p \mathcal {A}_M(p)\) and thus \(\mathcal {A}^*(p)\in \{ij,\mathcal {A}_D\}\). The breakpoint in Eq. (28) follows directly from Lemma 3.3.3.
(b) The breakpoint condition in Eq. (29) is similar to the one of Lemma 3.3.3, but with the input active set replaced by \(\mathcal {A}_D = \alpha \). The associated dominance breakpoint \(b_{\{ij,q\},\alpha }\) is also found analogously as in Lemma 3.3.3 (details omitted for brevity). The use of the pindependent expression for P(ij, q) in Eq. (25) for every \(q\in {\varTheta }({\varPhi })\) is justified by the same arguments as in the proof for Proposition 3.3.5.ii (mixed set only on one side of any potential breakpoint). \(\square \)
Proposition 3.3.7

If \(p>C_q\) then \(\mathop {\mathrm {sgn}}({\frac{\partial f_A}{\partial p}})\not = \mathop {\mathrm {sgn}}({\frac{\partial ^{2} f_A}{\partial p^{2}}})\) and \(f_A(p)\) is concave increasing or convex decreasing.

If \(p<C_q\) then \(\mathop {\mathrm {sgn}}({\frac{\partial f_A}{\partial p}}) = \mathop {\mathrm {sgn}}({\frac{\partial ^{2} f_A}{\partial p^{2}}})\) and \(f_A(p)\) is concave decreasing or convex increasing.
Proof
Proof in “Appendix A”. \(\square \)
Theorem 3.3.8
(pParametric structure of the optimal objective function \(f^*(p)\))
 (a)
(direct) \((\forall p\in {\varPhi })\ \mathcal {A}^*(p)=\mathcal {A}_D=\{l,r\} \text { or } r\ \Rightarrow f^*_{{\varPhi }}(p)\) is constant,
 (b)
(input) \((\forall p\in {\varPhi })\ \mathcal {A}^*(p)=\mathcal {A}_I(p)=ij\ \qquad \ \ \Rightarrow f^*_{{\varPhi }}(p)\) is linear,
 (c)
(mixed) \((\forall p\in {\varPhi })\ \mathcal {A}^*(p)=\mathcal {A}_M(p)=\{ij,q\}\ \Rightarrow f^*_{{\varPhi }}(p)\) is monotone convex/concave,
Proof
Case (a) follows from Theorem 3.2.7 and the independence of the results therein w.r.t. p; Case (b) from Lemma 3.1.2 and Proposition 3.1.5; Case 3 from Theorem 3.3.1 and Proposition 3.3.7. \(\square \)
Theorem 3.3.9
(Optimal solution and dominance breakpoints for I\({+}\mathrm{H}{}1{}1\) subclass)
Proof
Figure 6 illustrates the I\({+}\mathrm{H}{}1{}1\) subclass with a numerical example showcasing the implications of Theorem 3.3.8. For a chosen parametrization of five inputs (same inputs as in Fig. 4) and three directs with quality constraints, the pparametric function \(f^*(p)\) reveals additional breakpoints compared to Fig. 4 between mixed and input active sets and between mixed and direct active sets. The structure is still piecewisemonotone, but is extended to piecewise pintervals exhibiting convexity or concavity, e.g. when \(3\le p\le 4\) where mixed active set \(\{in_3,in_4,di_1\}\) is dominant.
Similar to the results in Sect. 3.1, the coupling between the piecewise structure and sparse active sets (up to a mixed node triple) is still present, allowing a full description of the pparametric optimal solution space. Section 4 explicitly uses this full description to find optimal solutions in stronglypolynomial time for a multiple outputs instance.
Remark 3.3.10
(From analytical solutions/sparsity/piecewise structure to nonsparse LP)

Reinstating constraints on feed availability (or analogously pool capacity) will erode solution sparsity proportionately to tightness of the flow bounds on dominant feeds. For fixed p, if the dominant active set reaches its upper flow bounds, the nextinline dominant active set would send flow and so on, recursively. If the bounds on cheaper feeds are very tight, this effect would create a hierarchy of dominant active sets participating in the solution. Therefore, the tighter the flow bounds, the more active feeds, and the less sparsity. The piecewise function \(f^*(p)\) would hence have more breakpoints accounting for dominance relations and optimal balancing between all active sets in the dominance hierarchy, not just the top active set. Moreover, because \(f^*(p)\) would represent an addition over a hierarchy of active sets, \(f^*(p)\) can be piecewise nonmonotone as in Sect. 4. Equally importantly, balancing the flows among the hierarchy of dominant active sets under flow constraints is an inherent LP, i.e. not solvable analytically.

Introducing multiple qualities (\(K>1\)) keeps the problem as an LP, but its polynomial complexity increases in line with K as the dominant active set cardinality becomes \(K{+}1\) (this extension is possible for one pool, one output topologies).

Relaxing the fixed product demand assumption implies the same solution with product demand reaching its upper limit if the problem is (assumed) feasible.
4 Subclass I\(\,+\,\mathrm{H}{}1{}\)J: one pool, multiple outputs
This section extends the analysis in Sect. 3 with Assumption 2.2 to \(I{+}H\) feeds (I inputs, H directs), one pool and multiple J outputs. Again, for simplicity of notation, single index l is dropped via the notation transformations \(T_X \leftarrow \{i:(i,l)\in T_X\}\), \(T_Y \leftarrow \{j:(l,j)\in T_Y\}\). This leads to Problem P–4.10, where for each output only connected to directs surplus variables \(y_j\ \forall j\in \overline{1,J}\setminus T_Y\) are introduced and set to 0 as a surplus condition. Note that eliminating p and \(y_j\ \forall j\in T_Y\) from Problem P–4.10 does not produce a linear program as in Sect. 3, but instead a bilinear problem that can be nonconvex.
To analyze Problem P–4.10, Theorem 4.1 first proves its equivalence to Problem P–4.11. The result allows additively decomposing Problem P–4.11 over outputs into subproblems P–4.11j, which are all pparametric and equivalent to the subclass I+H11 studied in Sect. 3. Definition 4.2 then extends dominance breakpoints and dominant active sets for a multiple outputs problem setting. Proposition 4.3 shows that the composed master Problem P–4.11 can present pparametric nonmonotonicity or nonconvexity on specific pintervals. This hurdle is cleared by Theorem 4.4, which finds in polynomial time all stationary points on nonmonotone breakpoint intervals by solving a univariate rational polynomial. Finally, Corollary 4.5 offers a stronglypolynomial worstcase time complexity for solving Problem P–4.11 to optimality.
Theorem 4.1
(pParametric additive decomposition over multiple outputs)
Proof
Definition 4.2

Let \(\mathcal {B}_J=\bigcup _{j\in \overline{1,J}}\mathcal {B}_j\) be the joint dominance breakpoint set for Problem P–4.11j over all J outputs/subproblems, with \(\mathcal {B}_j\) the set of all dominance breakpoints for the jth subproblem P–4.11j, found as in Sect. 3.

Let \({\varPhi }_J\) denote any closed interval with two consecutive elements in \(\mathcal {B}_J\) as endpoints.

Let \(\mathcal {A}^*_j(p)\) denote the dominant active set at p for the jth subproblem P–4.11j, as found in Theorem 3.3.8; by construction, \(\mathcal {A}^*_j(p)\) remains constant over \({\varPhi }_J\) s.t. \(\forall p\in {\varPhi }_J\ \mathcal {A}^*_j(p)=\mathcal {A}^*_j({\varPhi }_J)\).
Proposition 4.3
In Problem P–4.11, for \(\forall p\in {\varPhi }_J\), \(f^*(p)\) can be nonmonotone or nonconvex.
Proof
Theorem 4.4
Proof
Corollary 4.5
(Stronglypolynomial time complexity)
Proof
Fig. 8 illustrates the I+H1J subclass analysis with a numerical example showcasing the implications of Proposition 4.3 and Theorem 4.4. For a chosen parametrization of two inputs, directs and outputs with quality constraints for each output in Fig. 8, the pparametric function \(f^*(p)\) is decomposed additively in the optimal objectives of the one output subproblems. Hence, in the particular example, \(f^*(p)\) has a nonmonotone piecewise section as proven in Proposition 4.3 for \(p\in (1.3,2)\) with a stationary point which can be deterministically found via Theorem 4.4.
Remark 4.6
(A new P / NP boundary point for standard pooling)

Reinstating constraints on feed availability/pool capacity for any particular \(j\in T_Y\) implies the mixed active set term in Eq. (34) gets split into a hierarchy of potential mixed active sets (see Remark 3.3.10) each with total flow an unknown proportion of \(D_j\). This hierarchy of sets leads to a bivariate rational polynomial which is NPhard to solve [25].

Introducing multiple qualities (\(K>1\)) implies variables \(p_k,\ k\in \overline{1,K},\) are not independent  a specific pool concentration in one quality restricts the concentration range in another. Consequently, Eq. (34) becomes an NPhard multivariate polynomial system.

Relaxing the fixed product demand assumption  if any of \(D_j\ \forall j\in \ T_Y\) are not fixed but unknown, then Eq. (34) becomes an NPhard bivariate polynomial system.

Extending to the full topology \(I{+}H{}L{}J\), again Eq. (34) translates to a coupled multivariate polynomial system when two pools send nonzero flow to the same output (as in Sect. 5, Theorem 5.2) and one of the two pools also sends nonzero flow to a different output.
Remark 4.7
(Haverly [33] is stronglypolynomially solvable) The Haverly [33] instances, i.e. the first set of three pooling problems in the literature, are part of the singlequality I+H1J class following Assumption 2.2. We can obtain their exact solutions analytically in stronglypolynomial time!
Remark 4.8
(Connection to queueing theory) Woodside and Tripathi [60] report similar, piecewisemonotone structure in a central processor queueing problem where workstation files are assigned to file servers. The Woodside and Tripathi [60] proofs cannot be directly applied to standard pooling, but the similarity recalls the deep connection between pooling and queueing.
5 Subclass I\(+\mathrm{H}{}\mathrm{L}{}1\): multiple pools, one output
This section extends the analysis in Sect. 3 with Assumption 2.2 to \(I{+}H\) feeds (I inputs, H directs), L pools and one output. Again, for simplicity of notation, single index j is dropped via the notation transformations \(T_Z \leftarrow \{i:(i,j)\in T_Z\}\), \(T_Y \leftarrow \{l:(l,j)\in T_Y\}\). This leads to Problem P–5.12, where eliminating variables \(p_l,y_l\ \forall l\in T_Y\) results in an LP as for Problem P–3.4 in Sect. 3, limited to a maximum cardinality of four in terms of the \(x,\ z\) variables. We further identify this solution analytically, understanding pool sparsity and the parametric structure of the optimal objective in the process. To analyze Problem P–5.12, Definition 5.1 first introduces active pools. Theorem 5.2 then finds a maximum of two active pools contribute to the optimal solution and further shows all cases induced are parametric on pools concentrations. Furthermore, Theorem 5.3 proves all cases involved in Theorem 5.2 in fact reduce to the I+H11 subclass studied in Sect. 3. Finally, Corollary 5.4 offers a stronglypolynomial bound on solving Problem P–5.12 and the section concludes with an illustrative numerical example.
Definition 5.1

Let the \(L=\vert T_Y \vert \) pools in Problem P–5.12 be denoted by i with concentration \(p_i\), \(\forall i\in \overline{1,L}\).

An active pool has incoming and outgoing flows strictly nonzero. A nonactive pool \(l\in T_Y\) has welldefined concentration by assuming only \(y_l=0\) but \(x_{i,l}\not =0,\ \forall i:(i,l)\in T_X\). However, any nonactive pool is disconnected via \(y_l=0\) from the output, does not influence objective function f, and can be removed along with any of its flow connections from Problem P–5.12.
Theorem 5.2
(Pool sparsity and poolparametric objective function)
Proof
The case with two pools active (see Fig. 9) is treated separately in Eq. (41) as a two pool parametric restriction of Eq. (42) where no directs are active (therefore associated flow variables \(\{z_i\}\) can be eliminated). The second case in Eq. (41) aggregates the cases with maximum one pool active, and corresponds directly to the class of Problems P–3.4 solved in Sect. 3.3. \(\square \)
Theorem 5.3
(Solution for two active pools at an input dominance breakpoint)
Proof
Suppose concentration \(p_n\) is fixed which implies pool n acts as an additional direct with fixed concentration and cost (optimal). Since \(p_m, p_n\) are independent, parametric optimal objective \(f^*(p_m,p_n)\) behaves like \(f^*(p_m)\) in a single m pool problem where n acts as a direct, not as a pool. Pools m, n both active implies that \(\forall p_m\) the dominant active set for \(f^*(p_m)\) is a mixed active set \(\{i,j,n\}\) with n as a direct and \(\{i,j\}\) the dominant active input set sending flow to m. Since n must be active and therefore part of the dominant active set, a change in the dominant active set can only occur with a change of dominant active input set \(\{i,j\}\), independently of n and therefore fixed value \(p_n\). Consequently, the dominance breakpoints of \(f^*(p_m,p_n)\) w.r.t. \(p_m\) occur independently of the value of \(p_n\) and are always breakpoints between dominant active input sets for pool m. The viceversa independence also holds. Excluding input dominance breakpoint pairs of \(f^*(p_m,p_n)\) w.r.t both \(p_m,p_n\) in \(\mathcal {B}_{I(m)} \times \mathcal {B}_{I(n)}\), function \(f^*(p_m,p_n)\) is linear (Lemma 3.1.2) in at least one parameter (with gradient nonzero). Hence, Eq. (43) follows. \(\square \)
Corollary 5.4
(Stronglypolynomial time complexity)
Proof
According to Theorems 5.3 and 3.1.6, the first case in Eq. (41) for two given active pools involves \(2\cdot O(I^3)\) time for finding the two sets of input dominance breakpoints and \(O(I^2)\) for evaluating the optimal parametric objective function at all breakpoint pairs. Since there are \(\left( {\begin{array}{c}L\\ 2\end{array}}\right) \) possible pairs of active pools, the total time for the first case of Eq. (41) is \(O(I^3\cdot L) + O(I^2\cdot L^2)\).
The second case in Eq. (41) for one possibly active pool is equivalent to solving Problem P–3.4 (Theorem 3.3.9) and there are L active pool choices for a total time \(O((I^3{+} H^3)\cdot L)\).
Adding the two cases of Eq. (41) yields the time complexity in Eq. (44). \(\square \)
Figure 10 illustrates the I+HL1 subclass with a numerical example showcasing the implications of Theorem 5.2 and Theorem 5.3. For a chosen parametrization as in Fig. 10, the parametric function \(f^*(p_1,p_2)\) reveals portions of the domain where only one pool is active (around the edges of the cube) corresponding to the second case in Eq. (41) with piecewisemonotone structure (as shown in Sect. 3.3). Other portions of the domain (in the neighbourhood of \(p_1=2.5,\ p_2=3\)), however, correspond to the first case in Eq. (41), when both pools are active.
Remark 5.5
(Sparsity results extend to a multilayered network) Theorem 5.2 extends the sparsity results from the input layer to the pool layer. These sparsity results would also hold for networks with more layers.
Remark 5.6
(From analytical solutions/sparse piecewise structure to nonsparse LP/NPhardness)
Section 5 finds analytically the optimal solution for a \(\mathrm{I}{+}\mathrm{H}{}\mathrm{L}{}1\) pooling topology with Assumption 2.2. Since the \(\mathrm{I}{+}\mathrm{H}{}\mathrm{L}{}1\) subclass is an LP extension of the \(\mathrm{I}{+}\mathrm{H}{}1{}1\) instance, relaxing any constraint assumption, as described in Remark 3.3.10, leads to intractable analytical solutions and vanishing sparse structure. As explained in Remark 4.6, expanding to full topology \(\mathrm{I}{+}\mathrm{H}{}\mathrm{L}{}\mathrm{J}\) results in NPhardness.
6 Concluding remarks
This paper builds a framework analyzing standard pooling problem subclasses by parametrizing the objective function with respect to pool concentrations. The bottomup analysis develops stronglypolynomial time algorithms for multiple pooling network topological subclasses, all in the presence of a single quality, an unbounded number of feeds to pools and also outputs (direct bypass flows) and certain flow assumptions. Patterns and hierarchies of dominating topologies are used to find active network structure. The sparsity identified in the active network structure at optimality is then linked to a pool parametric piecewise structure of the objective function. The result reveals analytically Professor Floudas’ intuition of piecewise structure in pooling problem instances.
The parametric objective function is then shown to be piecewisemonotone for instances with one output, allowing exact global solutions in stronglypolynomial time as alternatives to blackbox linear programming. The insights are further used for nonlinear instances with multiple outputs and one pool to overcome piecewise nonmonotonicity via stationary points found in stronglypolynomial time. This result introduces a new reference point on the P / NP boundary for standard pooling subclasses, as any relaxation of assumptions or full topology (multiple pools and outputs) are shown to reach NPhardness. The multiple outputs subclass and its assumptions includes the Haverly [33] pooling problems, showing for the first time they have exact, analytical solutions.
The position on the P / NP boundary of the multiple outputs and one pool subclass is thus an ideal starting point for approximating algorithms that cross into NPhardness. Moreover, this paper developed intuition around sparse solutions and the conditions under which sparsity vanishes. This encourages future research in building disjunctive cuts based on the structures identified to partition feasible space in the nonsparse NPhard subclasses, an approach taken by the stateoftheart heuristic developed in [21].
Notes
Acknowledgements
We gratefully acknowledge support from EPSRC DTP to R.B.L., EPSRC EP/P008739/1 and a Royal Academy of Engineering Research Fellowship to R.M.
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