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

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.