A Framework for Generalized Benders' Decomposition and Its Application to Multilevel Optimization

We describe a framework for reformulating and solving optimization problems that generalizes the well-known framework originally introduced by Benders. We discuss details of the application of the procedures to several classes of optimization problems that fall under the umbrella of multilevel/multistage mixed integer linear optimization problems. The application of this abstract framework to this broad class of problems provides new insights and a broader interpretation of the core ideas, especially as they relate to duality and the value functions of optimization problems that arise in this context

On the Complexity of Inverse Mixed Integer Linear Optimization

Inverse optimization is the problem of determining the values of missing input parameters for an associated forward problem that are closest to given estimates and that will make a given target vector optimal. This study is concerned with the relationship of a particular inverse mixed integer linear optimization problem (MILP) to both the forward problem and the separation problem associated with its feasible region. We show that a decision version of the inverse MILP in which a primal bound is verified is coNP-complete, whereas primal bound verification for the associated forward problem is NP-complete, and that the optimal value verification problems for both the inverse problem and the associated forward problem are complete for the complexity class D^P. We also describe a cutting-plane algorithm for solving inverse MILPs that illustrates the close relationship between the separation problem for the convex hull of solutions to a given MILP and the associated inverse problem. The inverse problem is shown to be equivalent to the separation problem for the radial cone defined by all inequalities that are both valid for the convex hull of solutions to the forward problem and binding at the target vector. Thus, the inverse, forward, and separation problems can be said to be equivalent

On the Relationship Between the Value Function and the Efficient Frontier of a Mixed Integer Linear Optimization Problem

In this paper, we investigate the connection between the efficient frontier (EF) of a general multiobjective mixed integer linear optimization problem (MILP) and the so-called restricted value function (RVF) of a closely related single-objective MILP. We demonstrate that the EF of the multiobjective MILP is comprised of points on the boundary of the epigraph of the RVF so that any description of the EF suffices to describe the RVF and vice versa. In the first part of the paper, we describe the mathematical structure of the RVF, including characterizing the set of points at which it is differentiable, the gradients at such points, and the subdifferential at all nondifferentiable points. Because of the close relationship of the RVF to the EF, we observe that methods for constructing so-called value functions and methods for constructing the EF of a multiobjective optimization problem, each of which have been developed in separate communities, are effectively interchangeable. By exploiting this relationship, we propose a generalized cutting plane algorithm for constructing the EF of a multiobjective MILP based on a generalization of an existing algorithm for constructing the classical value function. We prove that the algorithm is finite under a standard boundedness assumption and comes with a performance guarantee if terminated early

Bilevel Programming and Maximally Violated Valid Inequalities

In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MIPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MIPs are a critical element of branch-and-cut. Thi

Bilevel Programming and the Separation Problem

In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is that of generating a valid inequality violated by a given real vector, usually arising as the solution to a relaxation of the original problem. We show that the problem of generating a maximally violated valid inequality often has a natural interpretation as a bilevel program. In some cases, this bilevel program can be easily reformulated as a simple single-level mathematical program, yielding a standard mathematical programming formulation for the separation problem. In other cases, no such polynomial-size single-level reformulation exists unless the polynomial hierarchy collapses to its first level (an event considered extremely unlikely in computational complexity theory). We illustrate our insights by considering the separation problem for two well-known classes of valid inequalities.

An Improved Algorithm for Biobjective Integer Programming and Its Application to Network Routing Problems

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including a Pareto set approximation scheme and an interactive version that provides access to all Pareto outcomes. In addition, an application is presented in which the tradeoff between the fixed and variable costs associated with solutions to a class of network routing problems closely related to the fixed-charge network flow problem is examined using the algorithm

Bilevel Programming and the Separation Problem Bilevel Programming and the Separation Problem

Abstract In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is that of generating a valid inequality violated by a given real vector, usually arising as the solution to a relaxation of the original problem. We show that the problem of generating a maximally violated valid inequality often has a natural interpretation as a bilevel program. In some cases, this bilevel program can be easily reformulated as a simple single-level mathematical program, yielding a standard mathematical programming formulation for the separation problem. In other cases, no such polynomial-size single-level reformulation exists unless the polynomial hierarchy collapses to its first level (an event considered extremely unlikely in computational complexity theory). We illustrate our insights by considering the separation problem for two well-known classes of valid inequalities

SYMPHONY 3.0.1 User's Manual

Contents 1 How to Use This Manual 1 2 A Brief History 1 3 Related Work 2 4 Introduction to Branch, Cut, and Price 2 4.1 Branch and Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Branch, Cut, and Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 Design of SYMPHONY 5 5.1 An Object-oriented Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.2 Data Structures and Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.2.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2.3 Search Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.3 Modular Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.3.1 The Master Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3