Generalized derivatives of optimal-value functions with parameterized convex programs embedded

Abstract

This article proposes new practical methods for furnishing generalized derivative information of optimal-value functions with embedded parameterized convex programs, with potential applications in nonsmooth equation-solving and optimization. We consider three cases of parameterized convex programs: (1) partial convexity—functions in the convex programs are convex with respect to decision variables for fixed values of parameters, (2) joint convexity—the functions are convex with respect to both decision variables and parameters, and (3) linear programs where the parameters appear in the objective function. These new methods calculate an LD-derivative, which is a recently established useful generalized derivative concept, by constructing and solving a sequence of auxiliary linear programs. In the general partial convexity case, our new method requires that the strong Slater conditions are satisfied for the embedded convex program’s decision space, and requires that the convex program has a unique optimal solution. It is shown that these conditions are essentially less stringent than the regularity conditions required by certain established methods, and our new method is at the same time computationally preferable over these methods. In the joint convexity case, the uniqueness requirement of an optimal solution is further relaxed, and to our knowledge, there is no established method for computing generalized derivatives prior to this work. In the linear program case, both the Slater conditions and the uniqueness of an optimal solution are not required by our new method.