Rough Stability of Solutions to Nonconvex Optimization Problems

Summary

The optimal solution set M(t) to some parametric optimization problem

$$\begin{array}{*{20}{c}} {minimize f(t,x)} & {subject to x \in D(t)} \\ \end{array}$$

is said to be roughly stable w.r.t. the roughness degree r > 0 at $\bar t \in T$ if for all ∈ > 0 there is a neighborhood $V\left( {\bar t} \right) \subset T$ of $\bar t$ such that $\left( {{ \cup _{t \in V\left( {\bar t} \right)}}M\left( t \right)} \right) < r + \in$ diam This paper states some sufficient conditions for this kind of generalized stability. One of the most important assumptions is that f is strictly roughly convexlike w.r.t. the roughness degree r. The result is applied to some optimal control problems, in particular, to a shipping problem.