Imperial College London

ProfessorRuthMisener

Faculty of EngineeringDepartment of Computing

Professor in Computational Optimisation
 
 
 
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Contact

 

+44 (0)20 7594 8315r.misener Website CV

 
 
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Location

 

379Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

107 results found

Wiebe J, Misener R, 2021, ROmodel: Modeling robust optimization problems in Pyomo, Publisher: arXiv

This paper introduces ROmodel, an open source Python package extending themodeling capabilities of the algebraic modeling language Pyomo to robustoptimization problems. ROmodel helps practitioners transition fromdeterministic to robust optimization through modeling objects which allowformulating robust models in close analogy to their mathematical formulation.ROmodel contains a library of commonly used uncertainty sets which can begenerated using their matrix representations, but it also allows users todefine custom uncertainty sets using Pyomo constraints. ROmodel supportsadjustable variables via linear decision rules. The resulting models can besolved using ROmodels solvers which implement both the robust reformulation andcutting plane approach. ROmodel is a platform to implement and compare customuncertainty sets and reformulations. We demonstrate ROmodel's capabilities byapplying it to six case studies. We implement custom uncertainty sets based on(warped) Gaussian processes to show how ROmodel can integrate data-drivenmodels with optimization.

Working paper

Ceccon F, Misener R, 2021, Solving the pooling problem at scale with extensible solver GALINI, Publisher: arXiv

This paper presents a Python library to model pooling problems, a class ofnetwork flow problems with many engineering applications. The libraryautomatically generates a mixed-integer quadratically-constrained quadraticoptimization problem from a given network structure. The library additionallyuses the network structure to build 1) a convex linear relaxation of thenon-convex quadratic program and 2) a mixed-integer linear restriction of theproblem. We integrate the pooling network library with galini, an open-sourceextensible global solver for quadratic optimization. We demonstrate galini'sextensible characteristics by using the pooling library to develop two galiniplug-ins: 1) a cut generator plug-in that adds valid inequalities in the galinicut loop and 2) a primal heuristic plug-in that uses the mixed-integer linearrestriction. We test galini on large scale pooling problems and show that,thanks to the good upper bound provided by the mixed-integer linear restrictionand the good lower bounds provided by the convex relaxation, we obtainoptimality gaps that are competitive with Gurobi 9.1 on the largest probleminstances.

Working paper

Pistikopoulos EN, Barbosa-Povoa A, Lee JH, Misener R, Mitsos A, Reklaitis G, Venkatasubramanian V, You F, Gani Ret al., 2021, Process systems engineering - The generation next?, COMPUTERS & CHEMICAL ENGINEERING, Vol: 147, ISSN: 0098-1354

Journal article

Letsios D, Bradley JT, Suraj G, Misener R, Page Net al., 2021, Approximate and robust bounded job start scheduling for Royal Mail delivery offices, Journal of Scheduling, Vol: 24, Pages: 237-258, ISSN: 1094-6136

Motivated by mail delivery scheduling problems arising in Royal Mail, we study a generalization of the fundamental makespan scheduling P||Cmax problem which we call the bounded job start scheduling problem. Given a set of jobs, each specified by an integer processing time pj, that have to be executed non-preemptively by a set of m parallel identical machines, the objective is to compute a minimum makespan schedule subject to an upper bound g≤m on the number of jobs that may simultaneously begin per unit of time. With perfect input knowledge, we show that Longest Processing Time First (LPT) algorithm is tightly 2-approximate. After proving that the problem is strongly NP-hard even when g=1, we elaborate on improving the 2-approximation ratio for this case. We distinguish the classes of long and short instances satisfying pj≥m and pj<m, respectively, for each job j. We show that LPT is 5/3-approximate for the former and optimal for the latter. Then, we explore the idea of scheduling long jobs in parallel with short jobs to obtain tightly satisfied packing and bounded job start constraints. For a broad family of instances excluding degenerate instances with many very long jobs, we derive a 1.985-approximation ratio. For general instances, we require machine augmentation to obtain better than 2-approximate schedules. In the presence of uncertain job processing times, we exploit machine augmentation and lexicographic optimization, which is useful for P||Cmax under uncertainty, to propose a two-stage robust optimization approach for bounded job start scheduling under uncertainty aiming in a low number of used machines. Given a collection of schedules of makespan ≤D, this approach allows distinguishing which are the more robust. We substantiate both the heuristics and our recovery approach numerically using Royal Mail data. We show that for the Royal Mail application, machine augmentation, i.e., short-term van rental, is especially relevant.

Journal article

Tsay C, Kronqvist J, Thebelt A, Misener Ret al., 2021, Partition-based formulations for mixed-integer optimization of trained ReLU neural networks, Publisher: arXiv

This paper introduces a class of mixed-integer formulations for trained ReLUneural networks. The approach balances model size and tightness by partitioningnode inputs into a number of groups and forming the convex hull over thepartitions via disjunctive programming. At one extreme, one partition per inputrecovers the convex hull of a node, i.e., the tightest possible formulation foreach node. For fewer partitions, we develop smaller relaxations thatapproximate the convex hull, and show that they outperform existingformulations. Specifically, we propose strategies for partitioning variablesbased on theoretical motivations and validate these strategies using extensivecomputational experiments. Furthermore, the proposed scheme complements knownalgorithmic approaches, e.g., optimization-based bound tightening capturesdependencies within a partition.

Working paper

Olofsson S, Schultz ES, Mhamdi A, Mitsos A, Deisenroth MP, Misener Ret al., 2021, Design of dynamic experiments for black-box model discrimination, Publisher: arXiv

Diverse domains of science and engineering require and use mechanisticmathematical models, e.g. systems of differential algebraic equations. Suchmodels often contain uncertain parameters to be estimated from data. Consider adynamic model discrimination setting where we wish to chose: (i) what is thebest mechanistic, time-varying model and (ii) what are the best model parameterestimates. These tasks are often termed modeldiscrimination/selection/validation/verification. Typically, several rivalmechanistic models can explain data, so we incorporate available data and alsorun new experiments to gather more data. Design of dynamic experiments formodel discrimination helps optimally collect data. For rival mechanistic modelswhere we have access to gradient information, we extend existing methods toincorporate a wider range of problem uncertainty and show that our proposedapproach is equivalent to historical approaches when limiting the types ofconsidered uncertainty. We also consider rival mechanistic models as dynamicblack boxes that we can evaluate, e.g. by running legacy code, but wheregradient or other advanced information is unavailable. We replace theseblack-box models with Gaussian process surrogate models and thereby extend themodel discrimination setting to additionally incorporate rival black-box model.We also explore the consequences of using Gaussian process surrogates toapproximate gradient-based methods.

Working paper

Kronqvist J, Misener R, Tsay C, 2021, Between steps: Intermediate relaxations between big-M and convex hull formulations, Publisher: arXiv

This work develops a class of relaxations in between the big-M and convexhull formulations of disjunctions, drawing advantages from both. The proposed"P-split" formulations split convex additively separable constraints into Ppartitions and form the convex hull of the partitioned disjuncts. Parameter Prepresents the trade-off of model size vs. relaxation strength. We examine thenovel formulations and prove that, under certain assumptions, the relaxationsform a hierarchy starting from a big-M equivalent and converging to the convexhull. We computationally compare the proposed formulations to big-M and convexhull formulations on a test set including: K-means clustering, P_ball problems,and ReLU neural networks. The computational results show that the intermediateP-split formulations can form strong outer approximations of the convex hullwith fewer variables and constraints than the extended convex hullformulations, giving significant computational advantages over both the big-Mand convex hull.

Conference paper

Wiebe J, Misener R, 2021, ROmodel: A Python Robust Optimization Modeling Toolbox, Computer Aided Chemical Engineering, Pages: 683-688

We introduce ROmodel, a Python package that extends the modeling capabilities of the popular modeling language Pyomo to robust optimization problems. ROmodel contains a library of commonly used uncertainty sets which can be generated using their matrix representations, but it also allows the definition of custom uncertainty sets using Pyomo constraints. The resulting models can be solved using ROmodels solvers which implement both the robust reformulation and cutting plane approach. We apply the problem to a number of instances of three case studies and show some results.

Book chapter

Mistry M, Letsios D, Krennrich G, Lee R, Misener Ret al., 2020, Mixed-integer convex nonlinear optimization with gradient-boosted trees embedded, Informs Journal on Computing, ISSN: 1091-9856

Decision trees usefully represent sparse, high-dimensional, and noisy data. Having learned a function from these data, we may want to thereafter integrate the function into a larger decision-making problem, for example, for picking the best chemical process catalyst. We study a large-scale, industrially relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pretrained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models or they may wish to optimize a discrete model that accurately represents a data set. We develop several heuristic methods to find feasible solutions and an exact branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on a concrete mixture design instance and a chemical catalysis industrial instance.

Journal article

Kronqvist J, Misener R, 2020, A disjunctive cut strengthening technique for convex MINLP, OPTIMIZATION AND ENGINEERING, ISSN: 1389-4420

Journal article

Wiebe J, Cecílio I, Dunlop J, Misener Ret al., 2020, A robust approach to warped Gaussian process-constrained optimization, Publisher: arXiv

Optimization problems with uncertain black-box constraints, modeled by warpedGaussian processes, have recently been considered in the Bayesian optimizationsetting. This work introduces a new class of constraints in which the sameblack-box function occurs multiple times evaluated at different domain points.Such constraints are important in applications where, e.g., safety-criticalmeasures are aggregated over multiple time periods. Our approach, which usesrobust optimization, reformulates these uncertain constraints intodeterministic constraints guaranteed to be satisfied with a specifiedprobability, i.e., deterministic approximations to a chance constraint. Thisapproach extends robust optimization methods from parametric uncertainty touncertain functions modeled by warped Gaussian processes. We analyze convexityconditions and propose a custom global optimization strategy for non-convexcases. A case study derived from production planning and an industriallyrelevant example from oil well drilling show that the approach effectivelymitigates uncertainty in the learned curves. For the drill scheduling example,we develop a custom strategy for globally optimizing integer decisions.

Working paper

Thebelt A, Kronqvist J, Mistry M, Lee RM, Sudermann-Merx N, Misener Ret al., 2020, ENTMOOT: A framework for optimization over ensemble tree models, Publisher: arXiv

Gradient boosted trees and other regression tree models perform well in awide range of real-world, industrial applications. These tree models (i) offerinsight into important prediction features, (ii) effectively manage sparsedata, and (iii) have excellent prediction capabilities. Despite theiradvantages, they are generally unpopular for decision-making tasks andblack-box optimization, which is due to their difficult-to-optimize structureand the lack of a reliable uncertainty measure. ENTMOOT is our new frameworkfor integrating (already trained) tree models into larger optimizationproblems. The contributions of ENTMOOT include: (i) explicitly introducing areliable uncertainty measure that is compatible with tree models, (ii) solvingthe larger optimization problems that incorporate these uncertainty aware treemodels, (iii) proving that the solutions are globally optimal, i.e. no bettersolution exists. In particular, we show how the ENTMOOT approach allows asimple integration of tree models into decision-making and black-boxoptimization, where it proves as a strong competitor to commonly-usedframeworks.

Working paper

Letsios D, Baltean-Lugojan R, Ceccon F, Mistry M, Wiebe J, Misener Ret al., 2020, Approximation algorithms for process systems engineering, COMPUTERS & CHEMICAL ENGINEERING, Vol: 132, ISSN: 0098-1354

Journal article

Thebelt A, Kronqvist J, Lee RM, Sudermann-Merx N, Misener Ret al., 2020, Global Optimization with Ensemble Machine Learning Models, Editors: Pierucci, Manenti, Bozzano, Manca, Publisher: ELSEVIER SCIENCE BV, Pages: 1981-1986

Book chapter

Botoeva E, Kouvaros P, Kronqvist J, Lomuscio A, Misener Ret al., 2020, Efficient Verification of ReLU-Based Neural Networks via Dependency Analysis, 34th AAAI Conference on Artificial Intelligence / 32nd Innovative Applications of Artificial Intelligence Conference / 10th AAAI Symposium on Educational Advances in Artificial Intelligence, Publisher: ASSOC ADVANCEMENT ARTIFICIAL INTELLIGENCE, Pages: 3291-3299, ISSN: 2159-5399

Conference paper

Čyras K, Karamlou A, Lee M, Letsios D, Misener R, Toni Fet al., 2020, AI-assisted schedule explainer for nurse rostering, Pages: 2101-2103, ISSN: 1548-8403

We present an argumentation-supported explanation generating system, called Schedule Explainer, that assists with makespan scheduling. Our stand-alone generic tool explains to a lay user why a resource allocation schedule is good or not, and offers actions to improve the schedule given the user's constraints. Schedule Explainer provides actionable textual explanations via an interactive graphical interface. We illustrate our system with a proof-of-concept application tool in a nurse rostering scenario whereby a shift-lead nurse aims to account for unexpected events by rescheduling some patient procedures to nurses and is aided by the system to do so.

Conference paper

Sedgwick R, Goertz J, Stevens M, Misener R, Wilk MVDet al., 2020, Design of Experiments for Verifying Biomolecular Networks

Working paper

Wiebe J, Cecílio I, Misener R, 2019, Data-driven optimization of processes with degrading equipment, Industrial & Engineering Chemistry Research, Vol: 57, Pages: 17177-17191, ISSN: 0888-5885

In chemical and manufacturing processes, unit failures due to equipment degradation can lead to process downtime and significant costs. In this context, finding an optimal maintenance strategy to ensure good unit health while avoiding excessive expensive maintenance activities is highly relevant. We propose a practical approach for the integrated optimization of production and maintenance capable of incorporating uncertain sensor data regarding equipment degradation. To this end, we integrate data-driven stochastic degradation models from Condition-based Maintenance into a process level mixed-integer optimization problem using Robust Optimization. We reduce computational expense by utilizing both analytical and data-based approximations and optimize the Robust optimization parameters using Bayesian Optimization. We apply our framework to five instances of the State-Task-Network and demonstrate that it can efficiently compromise between equipment availability and cost of maintenance.

Journal article

Letsios D, Bradley JT, Suraj G, Misener R, Page Net al., 2019, Approximating bounded job start scheduling with application in Royal Mail deliveries under uncertainty, Publisher: arXiv

Motivated by mail delivery scheduling problems arising in Royal Mail, westudy a generalization of the fundamental makespan scheduling P||Cmax problemwhich we call the "bounded job start scheduling problem". Given a set of jobs,each specified by an integer processing time p_j, that have to be executednon-preemptively by a set of m parallel identical machines, the objective is tocompute a minimum makespan schedule subject to an upper bound g<=m on thenumber of jobs that may simultaneously begin per unit of time. We show thatLongest Processing Time First (LPT) algorithm is tightly 2-approximate. Afterproving that the problem is strongly NP-hard even when g=1, we elaborate onimproving the 2-approximation ratio for this case. We distinguish the classesof long and short instances satisfying p_j>=m and p_j<m, respectively, for eachjob j. We show that LPT is 5/3-approximate for the former and optimal for thelatter. Then, we explore the idea of scheduling long jobs in parallel withshort jobs to obtain tightly satisfied packing and bounded job startconstraints. For a broad family of instances excluding degenerate instanceswith many very long jobs, we derive a 1.985-approximation ratio. For generalinstances, we require machine augmentation to obtain better than 2-approximateschedules. Finally, we exploit machine augmentation and lexicographicoptimization, which is useful for P||Cmax under uncertainty, to propose atwo-stage robust optimization approach for bounded job start scheduling underuncertainty aiming in good trade-offs in terms of makespan and number of usedmachines. We substantiate this approach numerically using Royal Mail data.

Working paper

Kouyialis G, Wang X, Misener R, 2019, Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs, PROCESSES, Vol: 7

Journal article

Campos JS, Misener R, Parpas P, 2019, A multilevel analysis of the Lasserre hierarchy, European Journal of Operational Research, Vol: 277, Pages: 32-41, ISSN: 0377-2217

This paper analyzes the relation between different orders of the Lasserre hierarchy for polynomial optimization (POP). Although for some cases solving the semidefinite programming relaxation corresponding to the first order of the hierarchy is enough to solve the underlying POP, other problems require sequentially solving the second or higher orders until a solution is found. For these cases, and assuming that the lower order semidefinite programming relaxation has been solved, we develop prolongation operators that exploit the solutions already calculated to find initial approximations for the solution of the higher order relaxation. We can prove feasibility in the higher order of the hierarchy of the points obtained using the operators, as well as convergence to the optimal as the relaxation order increases. Furthermore, the operators are simple and inexpensive for problems where the projection over the feasible set is “easy” to calculate (for example integer {0, 1} and {−1,1} POPs). Our numerical experiments show that it is possible to extract useful information for real applications using the prolongation operators. In particular, we illustrate how the operators can be used to increase the efficiency of an infeasible interior point method by using them as an initial point. We use this technique to solve quadratic integer {0, 1} problems, as well as MAX-CUT and integer partition problems.

Journal article

Čyras K, Letsios D, Misener R, Toni Fet al., 2019, Argumentation for explainable scheduling, Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19), Publisher: AAAI, Pages: 2752-2759

Mathematical optimization offers highly-effective tools for finding solutions for problems with well-defined goals, notably scheduling. However, optimization solvers are often unexplainable black boxes whose solutions are inaccessible to users and which users cannot interact with. We define a novel paradigm using argumentation to empower the interaction between optimization solvers and users, supported by tractable explanations which certify or refute solutions. A solution can be from a solver or of interest to a user (in the context of 'what-if' scenarios). Specifically, we define argumentative and natural language explanations for why a schedule is (not) feasible, (not) efficient or (not) satisfying fixed user decisions, based on models of the fundamental makespan scheduling problem in terms of abstract argumentation frameworks (AFs). We define three types of AFs, whose stable extensions are in one-to-one correspondence with schedules that are feasible, efficient and satisfying fixed decisions, respectively. We extract the argumentative explanations from these AFs and the natural language explanations from the argumentative ones.

Conference paper

Wiebe J, Cecílio I, Misener R, 2019, Robust optimization for the pooling problem, Industrial & Engineering Chemistry Research, Vol: 58, Pages: 12712-12722, ISSN: 0888-5885

The pooling problem has applications, e.g., in petrochemical refining, water networks, and supply chains and is widely studied in global optimization. To date, it has largely been treated deterministically, neglecting the influence of parametric uncertainty. This paper applies two robust optimization approaches, reformulation and cutting planes, to the non-linear, non-convex pooling problem. Most applications of robust optimization have been either convex or mixed-integer linear problems. We explore the suitability of robust optimization in the context of global optimization problems which are concave in the uncertain parameters by considering the pooling problem with uncertain inlet concentrations. We compare the computational efficiency of reformulation and cutting plane approaches for three commonly-used uncertainty set geometries on 14 pooling problem instances and demonstrate how accounting for uncertainty changes the optimal solution.

Journal article

Olofsson S, Misener R, 2019, GPdoemd: a python package for design of experiments for model discrimination, Computers & Chemical Engineering

GPdoemd is an open-source python package for design of experiments for model discrimination that uses Gaussian process surrogate models to approximate and maximise the divergence between marginal predictive distributions of rival mechanistic models. GPdoemd uses the divergence prediction to suggest a maximally informative next experiment.

Journal article

Furini F, Traversi E, Belotti P, Frangioni A, Gleixner A, Gould N, Liberti L, Lodi A, Misener R, Mittelmann H, Sahinidis N, Vigerske S, Wiegele Aet al., 2019, QPLIB: a library of quadratic programming instances, Mathematical Programming Computation, Vol: 11, Pages: 237-265, ISSN: 1867-2949

This paper describes a new instance library for quadratic programming (QP), i.e., the family of continuous and (mixed)-integer optimization problems where the objective function and/or the constraints are quadratic. QP is a very diverse class of problems, comprising sub-classes ranging from trivial to undecidable. This diversity is reflected in the variety of QP solution methods, ranging from entirely combinatorial approaches to completely continuous algorithms, including many methods for which both aspects are fundamental. Selecting a set of instances of QP that is at the same time not overwhelmingly onerous but sufficiently challenging for the different, interested communities is therefore important. We propose a simple taxonomy for QP instances leading to a systematic problem selection mechanism. We then briefly survey the field of QP, giving an overview of theory, methods and solvers. Finally, we describe how the library was put together, and detail its final contents.

Journal article

Olofsson SCW, Mehrian M, Calandra R, Geris L, Deisenroth MP, Misener Ret al., 2019, Bayesian multi-objective optimisation with mixed analytical and black-box functions: application to tissue engineering, IEEE Transactions on Biomedical Engineering, Vol: 66, Pages: 727-739, ISSN: 0018-9294

Tissue engineering and regenerative medicine looks at improving or restoring biological tissue function in humans and animals. We consider optimising neotissue growth in a three-dimensional scaffold during dynamic perfusion bioreactor culture, in the context of bone tissue engineering. The goal is to choose design variables that optimise two conflicting objectives: (i) maximising neotissue growth and (ii) minimising operating cost. We make novel extensions to Bayesian multi-objective optimisation in the case of one analytical objective function and one black-box, i.e. simulation-based, objective function. The analytical objective represents operating cost while the black-box neotissue growth objective comes from simulating a system of partial differential equations. The resulting multi-objective optimisation method determines the trade-off in the variables between neotissue growth and operating cost. Our method outperforms the most common approach in literature, genetic algorithms, in terms of data efficiency, on both the tissue engineering example and standard test functions. The resulting method is highly applicable to real-world problems combining black-box models with easy-to-quantify objectives like cost.

Journal article

Ceccon F, Siirola J, Misener R, 2019, SUSPECT: MINLP special structure detector for pyomo, Optimization Letters, Vol: 14, Pages: 801-814, ISSN: 1862-4472

We present SUSPECT, an open source toolkit that symbolically analyzesmixed-integer nonlinear optimization problems formulated using the Python alge-braic modeling library Pyomo. We present the data structures and algorithms used toimplement SUSPECT. SUSPECT works on a directed acyclic graph representation ofthe optimization problem to perform: bounds tightening, bound propagation, mono-tonicity detection, and convexity detection. We show how the tree-walking rules inSUSPECT balance the need for lightweight computation with effective special struc-ture detection. SUSPECT can be used as a standalone tool or as a Python library to beintegrated in other tools or solvers. We highlight the easy extensibility of SUSPECTwith several recent convexity detection tricks from the literature. We also report ex-perimental results on the MINLPLib 2 dataset.

Journal article

Wiebe J, Cecilio I, Misener R, 2019, The robust pooling problem, Editors: Kiss, Zondervan, Lakerveld, Ozkan, Publisher: ELSEVIER SCIENCE BV, Pages: 907-912, ISBN: 978-0-12-819939-8

Book chapter

Letsios D, Kouyialis G, Misener R, 2018, Reprint of: Heuristics with performance guarantees for the minimum number of matches problem in heat recovery network design, Computers and Chemical Engineering, Vol: 116, Pages: 422-450, ISSN: 1873-4375

Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an -hard mixed-integer linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods.

Journal article

Baltean-Lugojan, Misener R, 2018, Piecewise parametric structure in the pooling problem - from sparse, strongly-polynomial solutions to NP-hardness, Journal of Global Optimization, Vol: 71, Pages: 655-690, ISSN: 0925-5001

The standard pooling problem is a NP-hard subclass of non-convex quadratically-constrained 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 p-formulation. The structure uncovered in this approach validates Professor Christodoulos A. Floudas’ intuition that pooling problems are rooted in piecewise-defined 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.

Journal article

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