Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Abstract

Let f: X→Y be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f ^′(x) is Lipschitz in B(a,r) with Lipschitz constant L and f ^′(a) is a surjection: f ^′(a)X=Y; this implies the existence of ν>0 such that ‖f ^′(a)^* y‖≥ν‖y‖, ∀y∈Y. Then, if ε<min r,ν/(2L), the image F=f(B(a,ε)) of the ball B(a,ε) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint ‖x−a‖≤ε. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for ‘small power control’ is convex. This leads to various results in optimal control.