Descent Methods for Nonsmooth Convex Constrained Minimization

Abstract

We are concerned with methods for solving the problem

$$minimize\,f(x)\,over\,all\,x \in \,{R^N}$$

(1. 1a)

$$satisfying\,F(x)\, \leqslant 0,$$

(1.1b)

$${h_i}(x)\, \leqslant \,0\,for\,each\,i \in r,$$

(1.1c)

where the (possibly nonsmooth) functions f and F are realvalued and convex on R^N, h_i are affine and |I| < ∞. We assume that the feasible set S=S_h ∩ S_F is nonempty, where S_h = {x: h_i(x) ≤ 0 , i ∈ I} and S_F={x: F(x)≤ 0}, and that F( $\tilde x$) < 0 for some $\tilde x$ in S_h(the Slater condition). We suppose that for each x ∈ S_h one can compute f(x), F(x) and two arbitrary subgradients g_f(x) ∈ ∂f(x) and g_F(x) ∈ ∂f(x); these evaluations are not required for x ∉ S_h.