An interior-point based subgradient method for nondifferentiable convex optimization

We propose in this paper an algorithm for solving linearly constrained nondifferentiable convex programming problems. This algorithm combines the ideas of the affine scaling method with the subgradient method. It is a generalization of the dual and interior point method for min-max problems proposed by J.F. Sturm and S. Zhang (1995), A dual and interior point approach to solve convex min-max problems, in D.-Z. Du and P.M. Pardalos (eds.), Minimax and Applications, pp. 69-78, Kluwer Academic Publishers. In the new method, the search direction is obtained by projecting in a scaled space a subgradient of the objective function with a logarithmic barrier term. The stepsize choice is analogous to the stepsize choice in the usual subgradient method. Convergence of the method is established.