Distributionally robust mixed integer linear programs: Persistency models with applications

In this paper, we review recent advances in the distributional analysis of mixed integer linear programs with random objective coefficients. Suppose that the probability distribution of the objective coefficients is incompletely specified and characterized through partial moment information. Conic programming methods have been recently used to find distributionally robust bounds for the expected optimal value of mixed integer linear programs over the set of all distributions with the given moment information. These methods also provide additional information on the probability that a binary variable attains a value of 1 in the optimal solution for 0–1 integer linear programs. This probability is defined as the persistency of a binary variable. In this paper, we provide an overview of the complexity results for these models, conic programming formulations that are readily implementable with standard solvers and important applications of persistency models. The main message that we hope to convey through this review is that tools of conic programming provide important insights in the probabilistic analysis of discrete optimization problems. These tools lead to distributionally robust bounds with applications in activity networks, vertex packing, discrete choice models, random walks and sequencing problems, and newsvendor problems.     

Description of 2-integer continuous knapsack polyhedra

In this paper we discuss the polyhedral structure of several mixed integer sets involving two integer variables. We show that the number of the corresponding facet-defining inequalities is polynomial on the size of the input data and their coefficients can also be computed in polynomial time using a known algorithm [D. Hirschberg, C. Wong, A polynomial-time algorithm for the knapsack problem with two variables, Journal of the Association for Computing Machinery 23 (1) (1976) 147–154] for the two integer knapsack problem. These mixed integer sets may arise as substructures of more complex mixed integer sets that model the feasible solutions of real application problems.     

A note on the MIR closure

In 1988, Nemhauser and Wolsey introduced the concept of MIR inequality for mixed integer linear programs. In 1998, Wolsey gave another definition of MIR inequalities. This note points out that the natural concepts of MIR closures derived from these two definitions are distinct. Dash, Günlük and Lodi made the same observation independently.     

Lifting,superadditivity, mixed integer rounding and single node flow sets revisited

In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs. The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0–1 single node flow set. This paper appeared in 4OR, 1, 173–208 (2003). The first author is supported by the FNRS as a chercheur qualifié. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.     

混合整数二次规划问题的全局最优性条件

本文给出了混合整数二次规划问题的全局最优性条件,包括全局最优充分性条件和全局最优必要性条件.我们还给出了一个数值实例用以说明如何利用本文所给出的全局最优性条件来判定一个给定点是否是全局最优解.     

Design of the federal express large package sort facility

In this paper, two mixed integer programming models, the Aggregate Bin Assignment Model (ABAM) and Aggregate Rack Assignment Model (ARAM), are developed for the analysis of possible large package sort facility designs for Federal Express Corporation. Applying the ABAM and RAM algorithm on the current topological design of the sort facility reduces the number of forklifts and total forklift travel time for accomplishing the sort by about 20%. Tests on 16 other configurations proposed by Federal Express indicated that savings of 33% with respect to the number of forklifts required and over 50% in the total forklift travel time when compared to the existing operations can be achieved.     

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.     

Regulation of Overlaps in Technology Development Activities

In this paper, we present an algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems involving (i) 0-1 integer variables, and, (ii) more than one parameter, bounded between lower and upper bounds, present on the right hand side (RHS) of constraints. The solution is approached by decomposing the mp-MILP into two subproblems and then iterating between them. The first subproblem is obtained by fixing integer variables, resulting in a multiparametric linear programming (mp-LP) problem, whereas the second subproblem is formulated as a mixed integer linear programming (MILP) problem by relaxing the parameters as variables.     

Projected Chvátal–Gomory cuts for mixed integer linear programs

Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure integer linear program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the Chvátal–Gomory (CG) separation problem as a mixed integer linear program (MILP) which is then solved by a general- purpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory mixed integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive Chvátal–Gomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, which can typically be solved in a reasonable amount of computing time. Gérard Cornuéjols was supported in part by NSF grant DMI-0352885, ONR grant N00014-03-1-0188, and ANR grant BLAN 06-1-138894. Matteo Fischetti was supported in part by the EU projects ADONET (contract n. MRTN-CT-2003-504438) and ARRIVAL (contract n. FP6-021235-2). Andrea Lodi was supported in part by the EU projects ADONET (contract n. MRTN-CT-2003-504438) and ARRIVAL (contract n. FP6-021235-2).     

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.     

A probabilistic comparison of the strength of split, triangle, and quadrilateral cuts

We consider mixed integer linear sets defined by two equations involving two integer variables and any number of non-negative continuous variables. We analyze the benefit from adding a non-split inequality on top of the split closure. Applying a probabilistic model, we show that the importance of a type 2 triangle inequality decreases with decreasing lattice width, on average. Our results suggest that this is also true for type 3 triangle and quadrilateral inequalities.     

Strengthening lattice-free cuts using non-negativity

In recent years there has been growing interest in generating valid inequalities for mixed-integer programs using sets with two or more constraints. In particular, Andersen et al. (2007) [2] and Borozan and Cornuéjols (2009) [3] have studied sets defined by equations that contain exactly one integer variable per row. The integer variables are not restricted in sign. Cutting planes based on this approach have already been computationally studied by Espinoza (2008) [8] for general mixed-integer problems, and there is ongoing computational research in this area.In this paper, we extend the model studied in the earlier papers and require the integer variables to be non-negative. We extend the results in [2] and [3] to our case, and show that cuts generated by their approach can be strengthened by using the non-negativity of the integer variables. In particular, it is possible to obtain cuts which have negative coefficients for some variables.     

Valid inequalities for mixed integer linear programs

This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Supported by NSF grant DMI-0352885, ONR grant N00014-03-1-0188 and ANR grant BLAN06-1-138894.     

Using mixed integer programming for solving the capacitated arc routing problem with vehicle/site dependencies with an application to the routing of residential sanitation collection vehicles

We present the Composite Approach for solving the Capacitated Arc Routing Problem with Vehicle/Site Dependencies (CARP-VSD). We also present two mixed integer programs, the Initial Fleet Mix Generator (IFM) and the Mathematical Programming Procedure (MPP), and a multi-criterion function called the Measure of Goodness. The IFM, MPP and Measure of Goodness are critical components of the Composite Approach. A key application area of the CARP-VSD is the routing of residential sanitation vehicles; throughout this paper, we derive parameters specific to this problem. In this paper, we describe the Composite Approach, the IFM, the MPP, the Measure of Goodness and work out several examples in detail.     

Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük  (Math Program 105(1):29–53, 2006) are the first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.      

Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity

A pair of symmetric dual multiobjective variational mixed integer programs for the polars of arbitrary cones are formulated, which some primal and dual variables are constrained to belong to the set of integers. Under the separability with respect to integer variables and partial-invexity assumptions on the functions involved, we prove the weak, strong, converse and self-duality theorems to related minimax efficient solution. These results include some of available results.     

Cutting plane algorithms for the inverse mixed integer linear programming problem

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given feasible solution optimal.     

Models and algorithms to improve earthwork operations in road design using mixed integer linear programming

In road construction, earthwork operations account for about 25% of the construction costs. Existing linear programming models for earthwork optimization are designed to minimize the hauling costs and to balance the earth across the construction site. However, these models do not consider the removal of physical blocks that may influence the earthwork process. As such, current models may result in inaccurate estimates of optimal earthwork costs, leading to poor choices in road design. In this research, we extend the classical linear program model of earthwork operations to a mixed integer linear program model that accounts for blocks. We examine the economic impact of incorporating blocks via mixed integer linear programming, and find significant savings for most road designs in our test-set. However, the resulting model is considerably harder to solve than the original linear program. Based on structural observations, we introduce a set of algorithms that theoretically reduce the solving time of the model. We confirm this reduction in solve time with numerical experiments.     

Two-stage minimax regret robust uncapacitated lot-sizing problems with demand uncertainty

In this paper, we consider the two-stage minimax robust uncapacitated lot-sizing problem with interval uncertain demands. A mixed integer programming formulation is proposed. Even though the robust uncapacitated lot-sizing problem with discrete scenarios is an NP-hard problem, we show that it is polynomial solvable under the interval uncertain demand set.