Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming

TLDR

The paper introduces a convex programming formulation that replaces the conventional feasible region definition with an alternative strategy. The authors define the feasible region via set containment of convex activity sets \(K_j\) and a convex resource set \(K\), using set addition, and then solve for \(\bar{x}\in X\) that maximizes the linear objective \(c\cdot x\). For special resource sets, the problem reduces to an auxiliary linear program, enabling applications to inexact linear programming.

Abstract

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, f i (x) ≦ b i , i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {K j , 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.