On Shor's r-Algorithm for Problems with Constraints

Abstract

Shor's r-algorithm (Shor, Zhurbenko (1971), Shor (1979)) with space stretching in the direction of difference of two adjacent subgradients is a competitive method of nonsmooth optimization. However, the original r-algorithm is designed to minimize convex ravine functions without constraints. The standard technique for solving constraint problems with this algorithm is to use exact nonsmooth penalty functions (Eremin (1967), Zangwill (1967)). At the same time, it is necessary to choose the (sufficiently large) penalty coefficient in this method. In Norkin (2020, 2022) and Galvan et al. (2021), the so-called projective exact penalty function method is proposed, which does not formally require an exact definition of the penalty coefficient. In this paper, a nonsmooth optimization problem with convex constraints is first transformed into a constraintfree problem by the projective penalty function method, and then the r-algorithm is applied to solve the transformed problem. We present the results of testing this approach on problems with linear constraints using a program implemented in Matlab.