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

Abstract.

We consider the diagonal inexact proximal point iteration $\frac{u^k-u^{k-1}}{\lambda_k}\in-\partial_{\varepsilon_k} f( u^k,r_k) + \nu^k$ where f(x,r)=c ^T x+r∑exp[(A _i x-b _i )/r] is the exponential penalty approximation of the linear program min{c ^T x:Ax≤b}. We prove that under an appropriate choice of the sequences λ _k , ε _k and with some control on the residual ν^k, for every r _k →0⁺ the sequence u ^k converges towards an optimal point u ^∞ of the linear program. We also study the convergence of the associated dual sequence μ _i ^k=exp[(A _i u ^k-b _i )/r _k ] towards a dual optimal solution.