Convex Hull Of Feasible Solutions Research Articles

Prescriptive analytics for a maritime routing problem

Port state control (PSC) serves as the final defense against substandard ships in maritime transportation. The port state control officer (PSCO) routing problem involves selecting ships for inspection and determining the inspection sequence for available PSCOs, aiming to identify the highest number of deficiencies. Port authorities face this problem daily, making decisions without prior knowledge of ship conditions. Traditionally, a predict-then-optimize framework is employed, but its machine learning (ML) models' loss function fails to account for the impact of predictions on the downstream optimization problem, potentially resulting in suboptimal decisions. We adopt a decision-focused learning framework, integrating the PSCO routing problem into the ML models' training process. However, as the PSCO routing problem is NP-hard and plugging it into the training process of ML models requires that it be solved numerous times, computational complexity and scalability present significant challenges. To address these issues, we first convert the PSCO routing problem into a compact model using undominated inspection templates, enhancing the model's solution efficiency. Next, we employ a family of surrogate loss functions based on noise-contrastive estimation (NCE) for the ML model, requiring a solution pool treating suboptimal solutions as noise samples. This pool represents a convex hull of feasible solutions, avoiding frequent reoptimizations during the ML model's training process. Through computational experiments, we compare the predictive and prescriptive qualities of both the two-stage framework and the decision-focused learning framework under varying instance sizes. Our findings suggest that accurate predictions do not guarantee good decisions; the decision-focused learning framework's performance may depend on the optimization problem size and the training dataset size; and using a solution pool containing noise samples strikes a balance between training efficiency and decision performance.

The polytope of binary sequences with bounded variation

We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length n with bounded variation, where the latter is defined as the number of pairs of consecutive entries with different value. This problem arises naturally in many applications, e.g., in unit commitment problems or when discretizing binary optimal control problems subject to a bounded total variation. We study two variants of the problem. In the first one, the variation of the binary vector is penalized in the objective function, while in the second one, it is bounded by a hard constraint. We show that the first variant is easy to deal with while the second variant turns out to be more complex, but still tractable. For the latter case, we present a complete polyhedral description of the convex hull of feasible solutions by facet-inducing inequalities and devise an exact linear-time separation algorithm. The proof of completeness also yields a new exact primal algorithm with a running time of O(nlogn), which is significantly faster than the straightforward dynamic programming approach. Finally, we devise a compact extended formulation.A preliminary version of this article has been published in the Proceedings of the 7th International Symposium on Combinatorial Optimization (ISCO 2022) (Buchheim and Hügging, 2022).

Integer Programming Formulations for Minimum Spanning Tree Interdiction

We consider a two-player interdiction problem staged over a graph where the attacker’s objective is to minimize the cost of removing edges from the graph so that the defender’s objective, that is, the weight of a minimum spanning tree in the residual graph, is increased up to a predefined level r. Standard approaches for graph interdiction frame this type of problems as bilevel formulations, which are commonly solved by replacing the inner problem by its dual to produce a single-level reformulation. In this paper, we study an alternative integer program derived directly from the attacker’s solution space and show that this formulation yields a stronger linear relaxation than the bilevel counterpart. Furthermore, we analyze the convex hull of the feasible solutions of the problem and identify several families of facet-defining inequalities that can be used to strengthen this integer program. We then proceed by introducing a different formulation defined by a set of so-called supervalid inequalities that may exclude feasible solutions, albeit solutions whose objective value is not better than that of an edge cut of minimum cost. We discuss several computational aspects required for an efficient implementation of the proposed approaches. Finally, we perform an extensive set of computational experiments to test the quality of these formulations, analyzing and comparing the benefits of each model, as well as identifying further enhancements.Summary of Contribution: Network interdiction has received significant attention over the last couple of decades, with a notable peak of interest in recent years. This paper provides an interesting balance between the theoretical and computational aspects of solving a challenging network interdiction problem via integer programming. We present several technical developments, including a detailed study of the problem's solution space, multiple formulations, and a polyhedral analysis of the convex hull of feasible solutions. We then analyze the results of an extensive set of computational experiments that were used to validate the effectiveness of the different methods we developed in this paper.

A study of general and security Stackelberg game formulations

In this paper, we analyze different mathematical formulations for general Stackelberg games (GSGs) and Stackelberg security games (SSGs). We consider GSGs in which a single leader commits to a utility maximizing strategy knowing that p possible followers optimize their own utility taking the leader’s strategy into account. SSGs are a type of GSG that arise in security applications where the strategies of the leader consist of protecting a subset of targets and the strategies of the p followers consist of attacking a single target. We compare existing mixed integer linear programming (MILP) formulations for GSGs, ranking them according to the tightness of their linear programming (LP) relaxations. We show that SSG formulations are projections of GSG formulations and exploit this link to derive a new SSG MILP formulation that (i) has the tightest LP relaxation known among SSG MILP formulations and (ii) has an LP relaxation that coincides with the convex hull of feasible solutions in the case of a single follower. We present computational experiments empirically comparing the difficulty of solving the formulations in the general and security settings. The new SSG MILP formulation remains computationally efficient as problem size increases.

Exact algorithms for the minimum cost vertex blocker clique problem

We study the minimum cost vertex blocker clique problem (CVCP), which is defined as follows. Given a simple undirected graph with weights and costs on its vertices and r > 0, detect a minimum cost subset of vertices to be removed such that the weight of any clique in the remaining graph is at most r. We propose a new characterization of the set of feasible solutions that results in new linear 0–1 programming formulations solved by lazy-fashioned branch-and-cut algorithms. A new set of facet-inducing inequalities for the convex hull of feasible solutions to CVCP are also identified. We also propose new combinatorial bounding schemes and employ them to develop the first combinatorial branch-and-bound algorithm for this problem. The computational performance of these exact algorithms is studied on a test-bed of randomly generated graphs, and real-life instances.

A mixed-integer bilevel programming approach for a competitive prioritized set covering problem

The competitive set covering problem is a two-player Stackelberg (leader–follower) game involving a set of items and clauses. The leader acts first to select a set of items, and with knowledge of the leader’s action, the follower then selects another subset of items. There exists a set of clauses, where each clause is a prioritized set of items. A clause is satisfied by the selected item having the highest priority, resulting in a reward for the player that introduced the highest-priority selected item. We examine a mixed-integer bilevel programming (MIBLP) formulation for a competitive set covering problem, assuming that both players seek to maximize their profit. This class of problems arises in several fields, including non-cooperative product introduction and facility location games. We develop an MIBLP model for this problem in which binary decision variables appear in both stages of the model. Our contribution regards a cutting-plane algorithm, based on inequalities that support the convex hull of feasible solutions and induce faces of non-zero dimension in many cases. Furthermore, we investigate alternative verification problems to equip the algorithm with cutting planes that induce higher-dimensional faces, and demonstrate that the algorithm significantly improves upon existing general solution method for MIBLPs.

Generalized multiple depot traveling salesmen problem—Polyhedral study and exact algorithm

The generalized multiple depot traveling salesmen problem (GMDTSP) is a variant of the multiple depot traveling salesmen problem (MDTSP), where each salesman starts at a distinct depot, the targets are partitioned into clusters and at least one target in each cluster is visited by some salesman. The GMDTSP is an NP-hard problem as it generalizes the MDTSP and has practical applications in design of ring networks, vehicle routing, flexible manufacturing, scheduling and postal routing. We present an integer programming formulation for the GMDTSP and valid inequalities to strengthen the linear programming relaxation. Furthermore, we present a polyhedral analysis of the convex hull of feasible solutions to the GMDTSP and derive facet-defining inequalities that strengthen the linear programming relaxation of the GMDTSP. All these results are then used to develop a branch-and-cut algorithm to obtain optimal solutions to the problem. The performance of the algorithm is evaluated through extensive computational experiments on several benchmark instances.

A Two-Stage Stochastic Integer Programming Approach to Integrated Staffing and Scheduling with Application to Nurse Management

We study the problem of integrated staffing and scheduling under demand uncertainty. This problem is formulated as a two-stage stochastic integer program with mixed-integer recourse. The here-and-now decision is to find initial staffing levels and schedules. The wait-and-see decision is to adjust these schedules at a time closer to the actual date of demand realization. We show that the mixed-integer rounding inequalities for the second-stage problem convexify the recourse function. As a result, we present a tight formulation that describes the convex hull of feasible solutions in the second stage. We develop a modified multicut approach in an integer L-shaped algorithm with a prioritized branching strategy. We generate 20 instances (each with more than 1.3 million integer and 4 billion continuous variables) of the staffing and scheduling problem using 3.5 years of patient volume data from Northwestern Memorial Hospital. Computational results show that the efficiency gained from the convexification of the recourse function is further enhanced by our modifications to the L-shaped method. The results also show that compared with a deterministic model, the two-stage stochastic model leads to a significant cost savings. The cost savings increase with mean absolute percentage errors in the patient volume forecast.

The single-item lot-sizing polytope with continuous start-up costs and uniform production capacity

In this work we consider the uniform capacitated single-item single-machine lot-sizing problem with continuous start-up costs. A continuous start-up cost is generated in a period whenever there is a nonzero production in the period and the production capacity in the previous period is not saturated. This concept of start-up does not correspond to the standard (discrete) start-up considered in previous models, thus motivating a polyhedral study of this problem. In this work we explore a natural integer programming formulation for this problem. We consider the polytope obtained as convex hull of the feasible points in this problem. We state some general properties, study whether the model constraints define facets, and present an exponentially-sized family of valid inequalities for it. We analyze the structure of the extreme points of this convex hull, their adjacency and bounds for the polytope diameter. Finally, we study the particular case when the demands are high enough in order to require production in all the periods. We provide a complete description of the convex hull of feasible solutions in this case and show that all the inequalities in this description are separable in polynomial time, thus proving its polynomial time solvability.

Minimum edge blocker dominating set problem

This paper introduces and studies the minimum edge blocker dominating set problem (EBDP), which is formulated as follows. Given a vertex-weighted undirected graph and r > 0, remove a minimum number of edges so that the weight of any dominating set in the remaining graph is at least r. Dominating sets are used in a wide variety of graph-based applications such as the analysis of wireless and social networks. We show that the decision version of EBDP is NP-hard for any fixed r > 0. We present an analytical lower bound for the value of an optimal solution to EBDP and formulate this problem as a linear 0–1 program with a large number of constraints. We also study the convex hull of feasible solutions to EBDP and identify facet-inducing inequalities for this polytope. Furthermore, we develop the first exact algorithm for solving EBDP, which solves the proposed formulation by a branch-and-cut approach where nontrivial constraints are applied in a lazy fashion. Finally, we also provide the computational results obtained by using our approach on a test-bed of randomly generated instances and real-life power-law graphs.

ON THE GRAPH COLORING POLYTOPE

The graph coloring problem consists in assigning colors to the vertices of a given graph G such that no two adjacent vertices receive the same color and the number of used colors is as small as possible. In this paper, we investigate the graph coloring polytope P(G) defined as the convex hull of feasible solutions to the binary programming formulation of the problem. We remark that P(G) coincides with the stable set polytope of a graph constructed from the complement G of G. We derive facet-defining inequalities for P(G) from independent sets, odd holes, odd anti-holes and odd wheels in G.

Minimum vertex blocker clique problem

We study the minimum vertex blocker clique problem (VBCP),1 which is to remove a subset of vertices of minimum cardinality in a weighted undirected graph, such that the maximum weight of a clique in the remaining graph is bounded above by a given integer r ≥ 1. Cliques are among the most popular concepts used to model cohesive clusters in different graph‐based applications, such as social, biological, and communication networks. The general case of VBCP on weighted graphs is known to be NP‐hard, and we show that the special case on unweighted graphs is also NP‐hard for any fixed integer r ≥ 1. We present an analytical lower bound on the cardinality of an optimal solution to VBCP, as well as formulate VBCP as a linear 0–1 program with an exponential number of constraints. Facet‐inducing inequalities for the convex hull of feasible solutions to VBCP are also identified. Furthermore, we develop the first exact algorithm for solving VBCP, which solves the proposed formulation by using a row generation approach. Computational results obtained by utilizing this algorithm on a test‐bed of randomly generated instances are also provided. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(1), 48–64 2014

An FPTAS for minimizing a class of low-rank quasi-concave functions over a convex set

We consider the problem of minimizing a class of quasi-concave functions over a convex set. Quasi-concave functions are generalizations of concave functions and NP-hard to minimize in general. We present a simple fully polynomial time approximation scheme (FPTAS) for minimizing a class of low-rank quasi-concave functions. Our algorithm solves a polynomial number of linear minimization problems and computes an extreme point near-optimal solution. Therefore, it applies directly to combinatorial 0-1 problems where the convex hull of feasible solutions is known.

Extended formulations in combinatorial optimization

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called “compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, Minkowski-Weyl’s theorem, Balas’ theorem for the union of polyhedra, Yannakakis’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set, and we present the proof of Fiorini, Massar, Pokutta, Tiwary and de Wolf of an exponential lower bound for the cut polytope. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans’ result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.

The maximum [formula omitted]-colorable subgraph problem and orbitopes

Given an undirected node-weighted graph and a positive integer k , the maximum k -colorable subgraph problem is to select a k -colorable induced subgraph of largest weight. The natural integer programming formulation for this problem exhibits a high degree of symmetry which arises by permuting the color classes. It is well known that such symmetry has negative effects on the performance of branch-and-cut algorithms. Orbitopes are a polyhedral way to handle such symmetry and were introduced in Kaibel and Pfetsch (2008) [2]. The main goal of this paper is to investigate the polyhedral consequences of combining problem-specific structure with orbitope structure. We first show that the LP-bound of the integer programming formulation mentioned above can only be slightly improved by adding a complete orbitope description. We therefore investigate several classes of facet-defining inequalities for the polytope obtained by taking the convex hull of feasible solutions for the maximum k -colorable subgraph problem that are contained in the orbitope. We study conditions under which facet-defining inequalities for the polytope associated with the maximum k -colorable subgraph problem and the orbitope remain facet-defining for the combined polytope or can be modified to yield facets. It turns out that the results depend on both the structure and the labeling of the underlying graph.

Extended formulations in combinatorial optimization

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called “compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, Minkowski–Weyl’s theorem, Balas’ theorem for the union of polyhedra, Yannakakis’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans’ result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.

Integral polyhedra related to integer multicommodity flows on a cycle

The integer multicommodity flow problem on a cycle (IMFC) is to find a feasible integral routing of given demands between κ pairs of nodes on a link-capacitated undirected cycle, which is known to be polynomially solvable. Along with integral polyhedra related to IMFC, this paper shows that there exists a linear program, with a polynomial number of variables and constraints, which solves IMFC. Using the results, we also present a compact polyhedral description of the convex hull of feasible solutions to a certain class of instances of IMFC whose number of variables and constraints is O ( κ ) , which in turn means that there exists a non-trivial special case for which a minimum cost integer multicommodity flow problem can be solved in polynomial time.

Lift-Gas Allocation Under Precedence Constraints: MILP Formulation and Computational Analysis

The distribution of a limited rate of high-pressure gas to gas-lifted wells, while respecting injection bounds and activation precedence constraints, consists of a mixed-integer nonlinear programming problem. This paper proposes a mixed-integer linear formulation obtained by piecewise-linearizing the nonlinear functions, thereby allowing the use of integer programming algorithms. Valid inequalities for the convex hull of feasible solutions are derived from knapsack covers, for which exact and approximate lifting procedures yield stronger inequalities. Numerical results show that these cover-based cuts reduce the number of nodes explored in a branch-and-bound search.

A new formulation for the Traveling Deliveryman Problem

The Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.

Second-order cover inequalities

We introduce a new class of second-order cover inequalities whose members are generally stronger than the classical knapsack cover inequalities that are commonly used to enhance the performance of branch-and-cut methods for 0–1 integer programming problems. These inequalities result by focusing attention on a single knapsack constraint in addition to an inequality that bounds the sum of all variables, or in general, that bounds a linear form containing only the coefficients 0, 1, and –1. We provide an algorithm that generates all non-dominated second-order cover inequalities, making use of theorems on dominance relationships to bypass the examination of many dominated alternatives. Furthermore, we derive conditions under which these non-dominated second-order cover inequalities would be facets of the convex hull of feasible solutions to the parent constraints, and demonstrate how they can be lifted otherwise. Numerical examples of applying the algorithm disclose its ability to generate valid inequalities that are sometimes significantly stronger than those derived from traditional knapsack covers. Our results can also be extended to incorporate multiple choice inequalities that limit sums over disjoint subsets of variables to be at most one.