Technical Note—A Duality Theory for Convex Programming with Set-Inclusive Constraints

This paper extends the notion of convex programming with set-inclusive constraints as set forth by Soyster [ Opns. Res. 21, 1154–1157 (1973)] by replacing the objective vector c with a convex set C and formulating a dual problem. The primal problem to be considered is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\sup\Bigl\{\inf_{c\in C} c \cdot x \mid x_{1}K_{1}+x_{2}K_{2}+ \ldots x_{n}K_{n}\subseteq K(b), x_{i}\geq 0\Bigr\}$$\end{document} where the sets { K j } are convex activity sets, K ( b ) is a polyhedral resource set, C is a convex set of objective vectors, and the binary operation + refers to addition of sets. Any feasible solution to the dual problem provides an upper bound to (I) and, at optimality conditions, the value of (I) is equal to the value of the dual. Furthermore, the optimal solution of the dual problem can be used to reduce (I) to an ordinary linear programming problem.