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/λ_k ∈ - ∂ _{ε k} f (u^k, r_k) + v^k where f(x, r) = c^Tx + r ∑ exp[(A_ix -b_i)/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 λ_k, ε_k and with some control on the residual v^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_iu^k - b_i) / r_k] towards a dual optimal solution.