Inexact proximal penalty alternating linearization decomposition scheme of nonsmooth convex constrained optimization problems

Abstract

In this paper, the convex constrained optimization problems are studied via the alternating linearization approach. The objective function f is assumed to be complex, and its exact oracle information (function values and subgradients) is not easy to obtain, while the constraint function h is expected to be “simple” relatively. With the help of the exact penalty function, we present an alternating linearization method with inexact information. In this method, the penalty problem is replaced by two relatively simple linear subproblems with regularized form which are needed to be solved successively in each iteration. An approximate solution is utilized instead of an exact form to solve each of the two subproblems. Moreover, it is proved that the generated sequence converges to some solution of the original problem. The dual form of this approach is discussed and described. Finally, some preliminary numerical test results are reported. Numerical experiences provided show that the inexact scheme has good performance, certificate and reliability.