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 Sturm and Zhang [J.F. Sturm and S. Zhang, A dual and interior point approach to solve convex min-max problems, in: D.-Z. Du and P.M. Pardalos eds., Minimax and Applications, (1995) 69-78, Kluwer]. 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.

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Frenk, J.B.G, Sturm, J.F, & Zhang, S. (1996). An interior point subgradient method for linearly constrained nondifferentiable convex programming (No. EI 9612-/A). Econometric Institute Research Papers. Retrieved from http://hdl.handle.net/1765/1376