Maximum Representation of Some Convex Marginal Functions Without Constant Rank Regularity

Summary

We consider a convex optimization problem of two groups of variables x and y, where the objective function and the constraints are jointly convex and continuously differentiable. Supposing strong second order sufficient optimality condition (SOC), constant rank constraint qualification (CR) and using a decomposition approach with the variables x at the lower level, it is known that the marginal function ϕ(y) local can be represented as the maximum of a finite number of convex functions f _i (y). Hereby the (locally defined) functions f _i depend on the point y, and no such representation of ϕ is known in the neighborhood of points y where (CR) is violated.

But in the particular case under consideration better properties can be obtained without supposing (CR). For this, we use an extension technique and construct a special convex differentiate intermediate function with values between two arbitrary convex functions. In this way, the functions f _i (y) can be rearranged and partially redefined such that a global maximum representation holds where all functions f _i are defined and differentiable on int dom(ϕ).