Primal and dual convergence of a proximal point exponential penalty method for linear programming

We consider the diagonal inexact proximal point iteration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{u^k-u^{k-1}}{\lambda_k}\in-\partial_{\varepsilon_k} f( u^k,r_k) + \nu^k$$\end{document} where f(x,r)=cTx+r∑exp[(Aix-bi)/r] is the exponential penalty approximation of the linear program min{cTx:Ax≤b}. We prove that under an appropriate choice of the sequences λk, εk and with some control on the residual νk, for every rk→0+ the sequence uk converges towards an optimal point u∞ of the linear program. We also study the convergence of the associated dual sequence μik=exp[(Aiuk-bi)/rk] towards a dual optimal solution.