Mixed-integer Sets Research Articles

On the convex hull of convex quadratic optimization problems with indicators

We consider the convex quadratic optimization problem in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{n}$$\\end{document} with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of an (n+1)×(n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+1) \ imes (n+1)$$\\end{document} positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are “finitely generated.” In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.

Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints

We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of $$f(a^\top x)$$ , where f is a univariate concave function, a is a non-negative vector, and x is a binary vector of appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when a has two distinct values. We show how to use these inequalities to obtain valid inequalities for general a that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when a contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework.

A Column Generation Algorithm for the Resource-Constrained Order Acceptance and Scheduling on Unrelated Parallel Machines

In this paper, we investigate the resource-constrained order acceptance and scheduling on unrelated parallel machines that arise in make-to-order systems. The objective of this problem is to simultaneously select a subset of orders to be processed and schedule the accepted orders on unrelated machines in such a way that the resources are not overutilized at any time. We first propose two formulations for the problem: mixed integer linear programming formulation and set partitioning. In view of the complexity of the problem, we then develop a column generation approach based on the set partitioning formulation. In the proposed column generation approach, a differential evolution algorithm is designed to solve subproblems efficiently. Extensive numerical experiments on different-sized instances are conducted, and the results demonstrate that the proposed column generation algorithm reports optimal or near-optimal solutions that are evidently better than the solutions obtained by solving the mixed integer linear programming formulation.

On a generalization of the Chvátal–Gomory closure

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvátal closure for polytopes in the unit cube, http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf, 2012) considered a strengthened version of Chvátal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also extend this polyhedrality result to mixed-integer sets defined by linear constraints.

A robust optimization model for distribution network design under a mixed integer set of scenarios

The design of a distribution network under uncertainty is addressed by using the minmax regret paradigm. Economic or technical uncertain factors may have a strong influence on the behavior of a distribution system and should be taken into account in the design process. These factors can affect to the construction and operating costs, origin or destination of the supplied commodities, required demands or arc capacities. In this paper, some of these uncertain parameters are supposed to be continuous while others are clearly discrete in nature, giving rise to a mixed integer set of scenarios under which the system may be assessed. We explore how to design a robust distribution network when the uncertainty in the continuous parameters is modeled through polyhedral sets combined with other sources of uncertainty for the discrete parameters. The corresponding optimization problem is solved using a Benders decomposition framework. Finally, a computational experiment with specialized solvers is carried out in order to test the efficiency of our algorithms.

Modeling Defender-Attacker Problems as Robust Linear Programs with Mixed-Integer Uncertainty Sets

We study a class of sequential defender-attacker optimization problems where the defender’s objective is uncertain and depends on the operations of the attacker, which are represented by a mixed-integer uncertainty set. The defender seeks to hedge against the worst possible data realization, resulting in a robust optimization problem with a mixed-integer uncertainty set, which requires the solution of a challenging mixed-integer problem, which can be seen as a saddle-point problem over a nonconvex domain. We study two exact solution algorithms and present two feature applications for which the uncertainty is naturally modeled as a mixed-integer set. Our computational experiments show that the considered algorithms greatly outperform standard algorithms both in terms of computational time and solution quality. Moreover, our results show that modeling uncertainty with mixed-integer sets, instead of approximating the data using convex sets, results in less conservative solutions, which translates to a lower cost for the defender to protect from uncertainty. Summary of Contribution: We consider a class of defender-attacker problems where the defender has to make operational decisions that depend on uncertain actions from an adversarial attacker. Due to the type of information available to the defender, neither probabilistic modeling, nor robust optimization methods with convex uncertainty sets, are well suited to address the defender’s decision-making problem. Consequently, we frame the defender’s problem as a class of robust optimization problems with a mixed-integer uncertainty sets, and devise two exact algorithms that solve this class of problems. A comprehensive computational study shows that for the considered applications, our algorithms improves the performance of existing robust optimization approaches that can be adapted to solve this class of problems. Moreover, we show how mixed-integer uncertainty sets can reduce the level of over-conservatism that is a known issue of robust optimization approaches.

Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs

In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1–20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of \(l_p\)-norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. \(p=1\) and \(p =2\), respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts.

Strong formulations for conic quadratic optimization with indicator variables

We study the convex hull of the mixed-integer set given by a conic quadratic inequality and indicator variables. Conic quadratic terms are often used to encode uncertainties, while the indicator variables are used to model fixed costs or enforce sparsity in the solutions. We provide the convex hull description of the set under consideration when the continuous variables are unbounded. We propose valid nonlinear inequalities for the bounded case, and show that they describe the convex hull for the two-variable case. All the proposed inequalities are described in the original space of variables, but extended SOCP-representable formulations are also given. We present computational experiments demonstrating the strength of the proposed formulations.

Convex hull results for generalizations of the constant capacity single node flow set

For single node flow sets with fixed costs and constant capacities on the inflow and outflow arcs, a family of constant capacity flow covers are known to provide the convex hull in different special cases and are conjectured to provide it in the general case. Here we study more general mixed integer sets for which such single node flow cover inequalities suffice to give the convex hull. In particular we consider the case of a path in which each node has one (or several) incoming and outgoing arcs with constant capacities and fixed costs. This can be seen as a lot-sizing set with production and sales decisions driven by costs and prices and by the lower and upper bounds on stocks instead of being driven by demands as in the standard lot-sizing model. The approach we take is classical: we characterize the extreme points, derive tight extended formulations and project out the additional variables. Specifically we show that Fourier–Motzkin elimination, though far from elegant, can be used to carry out the non-trivial projections. The validity of the conjecture for the single node flow set follows from our results.

Lattice closures of polyhedra

Given \(P\subset {\mathbb {R}}^n\), a mixed-integer set \(P^I=P\cap ({\mathbb {Z}}^{t}\times {\mathbb {R}}^{n-t}\)), and a k-tuple of n-dimensional integral vectors \((\pi _1, \ldots , \pi _k)\) where the last \(n-t\) entries of each vector is zero, we consider the relaxation of \(P^I\) obtained by taking the convex hull of points x in P for which \( \pi _1^Tx,\ldots ,\pi ^T_kx\) are integral. We then define the k-dimensional lattice closure of \(P^I\) to be the intersection of all such relaxations obtained from k-tuples of n-dimensional vectors. When P is a rational polyhedron, we show that given any collection of such k-tuples, there is a finite subcollection that gives the same closure; more generally, we show that any k-tuple is dominated by another k-tuple coming from the finite subcollection. The k-dimensional lattice closure contains the convex hull of \(P^I\) and is equal to the split closure when \(k=1\). Therefore, a result of Cook et al. (Math Program 47:155–174, 1990) implies that when P is a rational polyhedron, the k-dimensional lattice closure is a polyhedron for \(k=1\) and our finiteness result extends this to all \(k\ge 2\). We also construct a polyhedral mixed-integer set with n integer variables and one continuous variable such that for any \(k < n\), finitely many iterations of the k-dimensional lattice closure do not give the convex hull of the set. Our result implies that t-branch split cuts cannot give the convex hull of the set, nor can valid inequalities from unbounded, full-dimensional, convex lattice-free sets.

Binary extended formulations of polyhedral mixed-integer sets

We analyze different ways of constructing binary extended formulations of polyhedral mixed-integer sets with bounded integer variables and compare their relative strength with respect to split cuts. We show that among all binary extended formulations where each bounded integer variable is represented by a distinct collection of binary variables, what we call “unimodular” extended formulations are the strongest. We also compare the strength of some binary extended formulations from the literature. Finally, we study the behavior of branch-and-bound on such extended formulations and show that branching on the new binary variables leads to significantly smaller enumeration trees in some cases.

Valid inequalities for two-period relaxations of big-bucket lot-sizing problems: Zero setup case

In this paper, we investigate two-period subproblems for big-bucket lot-sizing problems, which have shown a great potential for obtaining strong bounds. In particular, we investigate the special case of zero setup times and identify two important mixed integer sets representing relaxations of these subproblems. We analyze the polyhedral structure of these sets, deriving several families of valid inequalities and presenting their facet-defining conditions. We then extend these inequalities in a novel fashion to the original space of two-period subproblems, and also propose a new family of valid inequalities in the original space. In order to investigate the true strength of the proposed inequalities, we propose and implement exact separation algorithms, which are computationally tested over a broad range of test problems. In addition, we develop a heuristic framework for separation, in order to extend computational tests to larger instances. These computational experiments indicate the proposed inequalities can be indeed very effective improving lower bounds substantially.

Generalized coefficient strengthening cuts for mixed integer programming

Cutting plane methods are an important component in solving the mixed integer programming (MIP). By carefully studying the coefficient strengthening method, which is originally a presolving method, we are able to generalize this method to generate a family of valid inequalities called generalized coefficient strengthening (GCS) inequalities. The invariant property of the GCS inequalities is established under bound substitutions. Furthermore, we develop a separation algorithm for finding the violated GCS inequalities for a general mixed integer set. The separation algorithm is proved to have the polynomial time complexity. Extensive numerical experiments are made on standard MIP test sets, which demonstrate the usefulness of the resulting GCS separator.

A Polyhedral Study of Binary Polynomial Programs

We study the polyhedral convex hull of a mixed-integer set 𝒮 defined by a collection of multilinear equations over the unit hypercube. Such sets appear frequently in the factorable reformulation of mixed-integer nonlinear optimization problems. In particular, the set 𝒮 represents the feasible region of a linearized unconstrained binary polynomial optimization problem. We define an equivalent hypergraph representation of the mixed-integer set 𝒮, which enables us to derive several families of facet-defining inequalities, structural properties, and lifting operations for its convex hull in the space of the original variables. Our theoretical developments extend several well-known results from the Boolean quadric polytope and the cut polytope literature, paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub-expressions.

A new lift-and-project operator

In this paper, we analyze the strength of split cuts in a lift-and-project framework. We first observe that the Lovász–Schrijver and Sherali–Adams lift-and-project operator hierarchies can be viewed as applying specific 0–1 split cuts to an appropriate extended formulation and demonstrate how to strengthen these hierarchies using additional split cuts. More precisely, we define a new operator that adds all 0–1 split cuts to the extended formulation. For 0–1 mixed-integer sets with k binary variables, this new operator is guaranteed to obtain the integer hull in ⌈k/2⌉ steps compared to k steps for the Lovász–Schrijver or the Sherali–Adams operator. We also present computational results on the stable set problem with our new operator.

Valid inequalities for a single constrained 0-1 MIP set intersected with a conflict graph

In this paper a mixed integer set resulting from the intersection of a single constrained mixed 0–1 set with the vertex packing set is investigated. This set arises as a subproblem of more general mixed integer problems such as inventory routing and facility location problems. Families of strong valid inequalities that take into account the structure of the simple mixed integer set and that of the vertex packing set simultaneously are introduced. In particular, the well-known mixed integer rounding inequality is generalized to the case where incompatibilities between binary variables are present. Exact and heuristic algorithms are designed to solve the separation problems associated to the proposed valid inequalities. Preliminary computational experiments show that these inequalities can be useful to reduce the integrality gaps and to solve integer programming problems.

Cutting planes from extended LP formulations

Given a mixed-integer set defined by linear inequalities and integrality requirements on some of the variables, we consider extended formulations of its continuous (LP) relaxation and study the effect of adding cutting planes in the extended space. In terms of optimization, extended LP formulations do not lead to better bounds as their projection onto the original space is precisely the original LP relaxation. However, adding cutting planes in the extended space can lead to stronger bounds. In this paper we show that for every 0---1 mixed-integer set with n integer and k continuous variables, there is an extended LP formulation with $$2n+k-1$$2n+k-1 variables whose elementary split closure is integral. The proof is constructive but it requires an inner description of the LP relaxation. We then extend this idea to general mixed-integer sets and construct the best extended LP formulation for such sets with respect to lattice-free cuts. We also present computational results on the two-row continuous group relaxation showing the strength of cutting planes derived from extended LP formulations.

Using cuts for mixed integer knapsack sets to generate cuts for mixed integer polyhedral conic sets

Based on a bijective mapping between two mixed integer sets, we introduce a new perspective on developing cuts for the mixed integer polyhedral conic (MIPC) set by establishing a one-to-one correspondence between the cuts for this set and those for a related mixed integer knapsack (MIK) set. The face/facet-defining properties of the corresponding cuts are identical for their respective sets. We further show that the cut generation approach for the MIPC set resulting from this new perspective always produces cuts that dominate those generated based on any of the two individual MIK constraints corresponding to the MIPC constraint. Our computational results show this dominance can be quite significant. As a special case of this new perspective, the conic MIR inequality of Atamturk and Narayanan for the MIPC set and its properties can be directly derived from the MIR inequality for the MIK set and its properties. We also generalize these cuts to the n-step conic MIR inequalities, which are directly derived form the n-step MIR inequalities for the MIK set.

Lifted Euclidean inequalities for the integer single node flow set with upper bounds

In this paper we discuss the polyhedral structure of the integer single node flow set with two possible values for the upper bounds on the arc flows. Such mixed integer sets arise as substructures in complex mixed integer programs for real application problems.This work builds on results for the integer single node flow polytope with two arcs given by Agra and Constantino, 2006a. Valid inequalities are extended to a new family, the lifted Euclidean inequalities, and a complete description of the convex hull is given. All the coefficients of the facet-defining inequalities can be computed in polynomial time.We report on some computational experimentations for three problems: an inventory distribution problem, a facility location problem and a multi-item production planning model.

Relaxations of mixed integer sets from lattice-free polyhedra

This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitely.