Convex Optimization Problems

Abstract

If the set M of feasible solutions of an optimization problem is a convex subset of a linear space X and the objective function f : X → ℝ is convex, then one speaks of a convex optimization problem. We shall investigate problems of the form

$$ {\text{Minimize f(x) on}}\,M\,: = \{ x \in {\text{ X : x }} \in {\text{ C, g(x) }} \in - K\} $$

((P))

and later generally assume that f : X → ℝ is convex, C ⊂ X is convex and g : X → Y is a map which is convex with respect to a cone K contained in the linear space Y. One easily convinces oneself that under these conditions (P) is a convex optimization problem.