Convex Programming Methods for Global Optimization

We investigate some approaches to solving nonconvex global optimization problems by convex nonlinear programming methods. We assume that the problem becomes convex when selected variables are fixed. The selected variables must be discrete, or else discretized if they are continuous. We provide a survey of disjunctive programming with convex relaxations, logic-based outer approximation, and logic-based Benders decomposition. We then introduce a branch-and-bound method with convex quasi-relaxations (BBCQ) that can be effective when the discrete variables take a large number of real values. The BBCQ method generalizes work of Bollapragada, Ghattas and Hooker on structural design problems. It applies when the constraint functions are concave in the discrete variables and have a weak homogeneity property in the continuous variables. We address global optimization problems that become convex when selected variables are fixed. If these variables are discrete, the constraints can be reformulated as logical disjunctions of convex constraints. If some of the selected variables are not discrete, we discretize them in order to obtain an approximate global solution. The motivation for this approach is to take advantage of highly developed nonlinear programming methods for convex problems, as well as branch-andbound methods for discrete problems. A branch-and-bound method chooses the appropriate disjunct in each constraint. Nonlinear programming is applied to the convex subproblem that results when the disjuncts are chosen. We present four variations of this general approach. Two of them are most practical when the discrete variables do not take a large number of possible values: (a) disjunctive programming with convex relaxations, and (b) logic-based outer approximation. The disjunctive programming model can also be solved as a mixed integer/nonlinear programming (MINLP) problem. When there are a large number of discrete values, as when some discrete variables represent discretized continuous variables, one can turn to methods that do not require explicit representation of the disjunctions: (c) logic-based Benders decomposition, and (d) branch and bound with convex quasi-relaxations (BBCQ). The convergence rate of the Benders method depends heavily on the problem structure, however. BBCQ is intended for problems in which the discrete variables are real-valued. It does not rely on decomposition but requires that the constraint functions satisfy certain properties. This paper begins with a summary of the first three methods, which are developed elsewhere. It then introduces the BBCQ method as a formalization and generalization of a technique applied by Bollapragada, Ghattas and Hooker to structural design problems [1]. This application is presented at the end of the paper as an illustration of disjunctive programming and BBCQ. 1 General Form of the Problem We solve problems of the form min x0 subject to g(x, yj) ≤ 0, j ∈ J L(y) x ∈ IR, yj ∈ Yj , j ∈ J (1) where g(x, yj) is a vector of functions and L(y) is a logical constraint on possible values of the discrete variables yj . If some of the yj are continuous, we discretize them by converting Yj to a finite set. We assume that when each yj is fixed to some ȳj ∈ Yj we obtain the convex subproblem: min x0 subject to g(x, ȳj) ≤ 0, j ∈ J x ∈ IR (2) It is convex in the sense that each g(x, ȳj) is a vector of convex functions of x. We assume without loss of generality that the objective function is a single variable x0, since x0 can be defined in the constraints. We also suppose that each constraint contains only one discrete variable yj . Many problems naturally occur in this form. Problems that do not can in principle be put into this form by a change of variables. Thus a constraint g(x, y1, . . . , ym) ≤ 0 can be written g(x, y) ≤ 0, where y = (y 1 , . . . , y m) is regarded as a single variable. The variables y can now be related by the logical constraints y = y for all j ∈ J . For instance, the constraints x+y1 +y2 ≥ b and x+y2 +y3 ≥ b can be rewritten x+ y 1 + y 2 ≥ b and x+ y 2 + y 3 ≥ b by adding the constraint y 2 = y 2 . 2 Structural Design Example We use a simple structural design problem to illustrate all four solution methods. The two pillars of Fig. 1 support the two horizontal platforms shown. Pillar 1 bears a weight of 10 and pillar 2 a weight of 20. The weight on pillar 1 causes