Knowledge that Transforms

This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets Kj, j = 1, 2, …, n and a convex resource set K are specified and the feasible region is given by [Formula: see text] where the binary operation + refers to addition of sets. The problem is then to find x̄ ∈ X that maximizes the linear function c · x. When the resource set has a special form, this problem is solved via an auxiliary linear-programming problem and application to inexact linear programming is possible.

This paper considers a class of optimization problems characterized by constraints that themselves contain optimization problems. The problems in the constraints can be linear programs, nonlinear programs, or two-sided optimization problems, including certain types of games. The paper presents theory dealing primarily with properties of the relevant functions that result in convex programming problems, and discusses interpretations of this theory. It gives an application with linear programs in the constraints, and discusses computational methods for solving the problems.

This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-salesman problem. The procedure is based on a general approach to heuristics that is believed to have wide applicability in combinatorial optimization problems. The procedure produces optimum solutions for all problems tested, “classical” problems appearing in the literature, as well as randomly generated test problems, up to 110 cities. Run times grow approximately as n2; in absolute terms, a typical 100-city problem requires less than 25 seconds for one case (GE635), and about three minutes to obtain the optimum with above 95 per cent confidence.

This note shows that the assignment and sequencing of single-operation jobs in a parallel-machine environment can be formulated as an assignment problem. Means for reducing the size and computational difficulty of this problem are identified.