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

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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:Axb}. We prove that under an appropriate choice of the sequences λ k , ε k and with some control on the residual νk, 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.

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Received: May 2000 / Accepted: November 2001¶Published online June 25, 2002

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Alvarez, F., Cominetti, R. Primal and dual convergence of a proximal point exponential penalty method for linear programming. Math. Program. 93, 87–96 (2002). https://doi.org/10.1007/s10107-002-0295-0

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  • DOI: https://doi.org/10.1007/s10107-002-0295-0

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