A Subgradient Algorithm for Solving K-Convex Inequalities

Abstract

In this paper we discuss an algorithm for numerical solution of a system of K-convex inequalities

$$\begin{array}{*{20}{c}} {f(x){{\mathop < \limits_ = }_K}0} \\ {x \in C,} \end{array}$$

(1)

where f: D ⊂ ℝⁿ → ℝ^m, C is a nonempty closed convex set in ℝⁿ, K is a nonempty closed convex cone in ℝ^m, and where we write y₁ ≦ _K y₂ if y₂ − y₁ ∈ K. Under assumptions of convexity and regularity of f, we show that the algorithm converges, from an arbitrary starting point in C, to a solution of (1) at a rate which is at least linear. The algorithm requires the computation of a subgradient of (1) and the solution of a projection problem (a convex quadratic minimization problem) at each step. With additional differentiability assumptions on f, much faster convergence (e. g., quadratic) can be expected. The algorithm is an extension of a method proposed by Oettli [3], which in turn is related to earlier works such as those of Polyak [4] and Eremin [1], among others.