Convex two-level optimization problem

Abstract

A two-level optimization problem is considered in which the objective functional of the second-level problem is minimized on the solution set of the first-level problem. Convergence of the modified penalty method is established. The main results include a continuous two-level optimization method based on the regularized extremal shifting principle [3, 5, 6]. For a linearly convex problem, two-sided bounds on the approximation by the first-level functional are established in addition to convergence. For a linearly quadratic problem, two-sided bounds on the approximation by the second-level functional are derived. For a linearly quadratic problem with interval constraints, an explicit form of differential inclusions is presented for the implementation of the method.