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AbstractMany optimization problems occur in both theory and practice when one has to optimize an objective function while an infinite number of constraints must be satisfied. The aim of this paper in to describe methods of handling such problems numerically in an effective manner. We also indicate a number of applications.

AbstractWe first present a survey on the theory of semi‐infinite programming as a generalization of linear programming and convex duality theory. By the pairing of a finite dimensional vector space over an arbitrarily ordered field with a generalized finite sequence space, the major theorems of linear programming are generalized. When applied to Euclidean spaces, semi‐infinite programming theory yields a dual theorem associating as dual problems minimization of an arbitrary convex function over an arbitrary convex set in n‐space with maximization of a linear function in non‐negative variables of a generalized finite sequence space subject to a finite system of linear equations.We then present a new generalization of the Kuhn‐Tucker saddle‐point equivalence theorem for arbitrary convex functions in n‐space where differentiability is no longer assumed.

The problem of assigning cell probabilities to maximize a multinomial likelihood with order restrictions on the probabilies and/or restrictions on the local odds ratios is modeled as a posynomial geometric program (GP), a class of nonlinear optimization problems with a well-developed duality theory and collection of algorithms. (Local odds ratios provide a measure of association between categorical random variables.) A constrained multinomial MLE example from the literature is solved, and the quality of the solution is compared with that obtained by the iterative method of El Barmi and Dykstra, which is based upon Fenchel duality. Exploiting the proximity of the GP model of MLE problems to linear programming (LP) problems, we also describe as an alternative, in the absence of special-purpose GP software, an easily implemented successive LP approximation method for solving this class of MLE problems using one of the readily available LP solvers.

AbstractIn this paper marginal investment costs are assumed known for two kinds of equipment stocks employed to supply telecommunications services: trunks and switching facilities. A network hierarchy is defined which includes important cases occurring in the field and also appearing in the literature. A different use of the classical concept of the marginal capacity of an additional trunk at prescribed blocking probability leads to a linear programming supply model which can be used to compute the sizes of all the high usage trunk groups. The sizes of the remaining trunk groups are approximated by the linear programming models, but can be determined more accurately by alternate methods once all high usage group sizes are computed. The approach applies to larger scale networks than previously reported in the literature and permits direct application of the duality theory of linear programming and its sensitivity analyses to the study and design of switched probabilistic communications networks with multiple busy hours during the day. Numerical results are presented for two examples based on field data, one of which having been designed by the multi‐hour engineering method.