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

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 as well as recent ones 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 for Lagrangian saddle-point problems in linear and convex programming, and others for a linear–quadratic saddle-point problem arising in topology optimization in contact mechanics.