Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.

DC programming: overview

The purpose of this chapter is to present the essential elements of the theory, applications, and solution algorithms of concave minimization. Concave minimization problems seek to globally minimize real-valued concave functions over closed convex sets. As in other global optimization problems, in concave minimization problems there generally exist many local optima which are not global. Concave minimization problems are NP-hard. Even seemingly-simple cases can possess an exponential number of local minima. However, in spite of these difficulties, concave minimization problems are more tractable than general global optimization problems. This is because concave functions and minimizations display some special mathematical properties. Concave minimization problems also have a surprisingly-diverse range of direct and indirect applications. The special mathematical properties and diverse range of applications of concave minimization have motivated the construction of a rich and varied set of approaches for their solution. Three fundamental classes of solution approaches can be distinguished. These are the enumerative, successive approximation, and successive partitioning (branch and bound) approaches. After giving a brief introduction, the chapter describes and discusses some of the most important special mathematical properties of concave functions and of concave minimization problems. Following this, it describes several of the most important categories of direct and indirect applications of concave minimization. The chapter then describes each of the three fundamental solution approaches for concave minimization. For each approach, the essential elements are explained in a unified framework. Following this, for each approach, most of the well-known individual concave minimization algorithms that use the approach are explained and compared. The scope of the presentation of the solution approaches and algorithms is limited to those that use deterministic (rather than stochastic) methods. The chapter concludes with some suggested topics for further research.

Concave Minimization: Theory, Applications and Algorithms

An algorithm for solving a linear multiplicative programming problem (referred to as LMP) is proposed. LMP minimizes the product of two linear functions subject to general linear constraints. The product of two linear functions is a typical non-convex function, so that it can have multiple local minima. It is shown, however, that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure. The computational results indicate that the amount of computation is not much different from that of solving linear programs of the same size. In addition, the method proposed for LMP can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.

Linear multiplicative programming

This chapter presents recent advances in the development and application of global optimization algorithms for solving mixed-integer nonlinear programs (MINLPs). It is demonstrated that practically relevant nonlinear programs can be solved to global optimality in a completely automated fashion when carefully chosen relaxation schemes, branching strategies, and domain reduction techniques are are used in conjunction with branch and bound to enhance its performance. In particular, this chapter presents a) applications of the convex extensions theory for constructing tight relaxations, b) unifying ideas behind domain reduction schemes, c) linear outer-approximation schemes with proven convergence guarantees, and d) branching schemes for factorable nonlinear programs. The chapter concludes with computational results on some benchmark mixed-integer nonlinear problems. New solutions are reported for four of these problems.

Exact Algorithms for Global Optimization of Mixed-Integer Nonlinear Programs

On finding new vertices and redundant constraints in cutting plane algorithms for global optimization

This paper deals with outer approximation methods for solving possibly multiextremal global optimization problems. A general theorem on convergence is presented and new classes of outer approximation methods using polyhedral convex sets are derived. The underlying theory is then related to the cut map-separator theory of Eaves and Zangwill. Two constraint dropping strategies are deduced.

Outer Approximation by Polyhedral Convex Sets

A simple outer approximation concept for global optimization problems is presented that simplifies and generalizes several previous approaches. Unbounded feasible regions and non-affine cuts are admitted. Constraint dropping strategies and a large class of cutting plane methods are derived.

On an outer approximation concept in global optimization

For the problem of minimizing a concave function over a convex polyedral-set an algorithm is given, which is based on the extension principle developed by Schoch. This algorithm yields after a finite number of steps an exact optimal solution of the problem. On the other hand one can find out throughout the algorithm an approximate optimal solution with any given precision.

Ng.V. Thoai

Papers

Outer Approximation by Polyhedral Convex Sets

On an outer approximation concept in global optimization

Aanwendung des erweiterungsprinzips zur lösung konkayer optimierungsaufgaben