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

We consider the diagonal inexact proximal point iteration ((u^k − u^k−1)/(λk)) ∈ − ∂ϵk f(u^k,rk) + v^k where f(x,r) = c^Tx + rΣexp[(Aix − bi)/r] is the exponential penalty approximation of the linear program min{c^Tx : Ax ≤ b}. We prove that under an appropriate choice of the sequences λk, ϵk and with some control on the residual v^k, for every rk → 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^ki = exp[(Aiu^k − bi)/rk] towards a dual optimal solution.