On the convergence of conditional ε-subgradient methods for convex programs and convex–concave saddle-point problems

Abstract

The paper provides two contributions. First, we present new convergence results for conditional ε-subgradient algorithms for general convex programs. The results obtained here extend the classical ones by Polyak [Sov. Math. Doklady 8 (1967) 593; USSR Comput. Math. Math. Phys. 9 (1969) 14; Introduction to Optimization, Optimization Software, New York, 1987] as well as the recent ones in [Math. Program. 62 (1993) 261; Eur. J. Oper. Res. 88 (1996) 382; Math. Program. 81 (1998) 23] to a broader framework. Secondly, we establish the application of this technique to solve non-strictly convex–concave saddle point problems, such as primal-dual formulations of linear programs. Contrary to several previous solution algorithms for such problems, a saddle-point is generated by a very simple scheme in which one component is constructed by means of a conditional ε-subgradient algorithm, while the other is constructed by means of a weighted average of the (inexact) subproblem solutions generated within the subgradient method. The convergence result extends those of [Minimization Methods for Non-Differentiable Functions, Springer-Verlag, Berlin, 1985; Oper. Res. Lett. 19 (1996) 105; Math. Program. 86 (1999) 283] for Lagrangian saddle-point problems in linear and convex programming, and of [Int. J. Numer. Meth. Eng. 40 (1997) 1295] for a linear–quadratic saddle-point problem arising in topology optimization in contact mechanics.