Infinite Dimensional Optimization

Abstract

The classical proximal point methodfor optimization was analyzed in details by Rockafellar [106]. It is devised to minimize a proper, lower semicontinuous, convex function g: H→ (−∞,+∞] defined on a Hilbert space H. The iteration is of the form

$$ {{x}^{{k + 1}}} = \arg \min \{ g(x) + ({{\omega }_{k}}/2)||x - {{x}^{k}}|{{|}^{2}}\} $$

where {ωκ}_κ∈ℕ is a bounded sequence of positive real numbers. If the function g is differentiable, then x^κ+1 is the unique solution of

$$ g'(x) + {{\omega }_{k}}x = {{\omega }_{k}}{{x}^{k}} $$

where g′(x) denotes the derivative of g at x. If we have a constrained optimization problem, that is, if we have to minimize the function g over a closed, convex, nonempty subset C of H, then we have to replace in (3.1) the function g by the function h:= g +I _c , where I _c is the indicator function of the set C, i.e., I _c (x) = 0 if x ∈ C and I _c (x) = +∞, otherwise. In this case the equation (3.2) becomes

$$ g'(x) + {{N}_{C}}(x) + {{\omega }_{k}}x \mathrel\backepsilon {{\omega }_{k}}{{x}^{k}} $$

where N _c (x) is the normal cone to C at x.