Necessary Conditions for Local Optimality in Difference-of-Convex Programming

Abstract

Using ε-subdifferential calculus for difference-of-convex (d.c.) optimization, Dür proposed a condition sufficient for local optimality, and showed that this condition is not necessary in general. Here it is proved that whenever the convex part is strongly convex, this condition is also necessary. Strong convexity can always be ensured by changing the given d.c. decomposition slightly. This approach also allows for a formulation with perturbed ε-subdifferentials which involves only the original d.c. decomposition, even without imposing strong convexity. We relate this result with another inclusion condition on perturbed ε-subdifferentials, which even can serve as a quantitative version of a criterion both necessary and sufficient for local optimality.