Abstract
We solve a general optimization problem, where only approximation sequences are known instead of exact values of the goal function and feasible set. Under these conditions we suggest to utilize a penalty function method. We show that its convergence is attained for rather arbitrary means of approximation via suitable coercivity type conditions.
Similar content being viewed by others
References
Polyak, B. T. Introduction to Optimization (Nauka, Moscow, 1983) [in Russian].
Vasil’ev, F. P. Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].
Pervozvanskii, A. A. and Gaitsgori, V. G. Decomposition, Aggregation, and Approximate Optimization (Nauka, Moscow, 1979) [in Russian].
Levitin, E. S. Perturbation Theory in Mathematical Programming and its Applications (Nauka, Moscow, 1992) [in Russian].
Ermol’yev, Yu.M. and Nurminskii, E. A. “Limit Extremal Problems,” Kibernetika, No. 4, 130–132 (1973).
Antipin, A. S. “Regularization Method in Convex Programming Problems,” Ekon. i Mat. Metody 11, No. 2, 336–342 (1975) [in Russian].
Nurminskii, E. A. “On a Nonstationary Optimization Problem,” Kibernetika, No. 2, 76–77 (1977).
Eremin, I. I. and Mazurov, V. D. Nonstationary Processes of Mathematical Programming (Nauka, Moscow, 1979) [in Russian].
Kaplan, A. A. “On Stability of Solution Methods of Convex Programming and Variational Inequalities,” in Models and Methods of Optimization, Trudy Inst.Mat. 10, 132–159 (1988).
Alart, P. and Lemaire, B. “Penalization in Non-Classical Convex Programming via Variational Convergence,” Math. Program. 51, 307–331 (1991).
Moudafi, A. “Coupling ProximalMethods and Variational Convergence,” Z.Oper. Res. 38, 269–280 (1993).
Bahraoui, M. A. and Lemaire, B. “Convergence of Diagonally Stationary Sequences in Convex Optimization,” Set-Valued Analysis 2, 49–61 (1994).
Cominetti, R. “Coupling the Proximal Point Algorithm with Approximation Methods,” J. Optim. Theory Appl. 95, 581–600 (1997).
Barrientos, O. “A Global Regularization Method for Solving the Finite min-max problem,” Comput. Optim. Appl. 11, 277–295 (1998).
Salmon, G., Nguyen, V. H., and Strodiot, J. J. “Coupling the Auxiliary Problem Principle and Epiconvergence Theory for Solving General Variational Inequalities,” J. Optim. Theory Appl. 104, 629–657 (2000).
Kaplan, A. and Tichatschke, R. “A General View on Proximal Point Methods for Variational Inequalities in Hilbert Spaces,” J. Nonl. Conv. Anal. 2, 305–332 (2001).
Badriev, I. B., Zadvornov, O. A., and Ismagilov, L. N. “On the Methods of Iterative Regularization for the Variational Inequalities of the Second Kind,” Comput.Meth. Appl.Math. 3, 223–234 (2003).
Vasil’ev, F. P. Numerical Methods for Solving Extremal Problems (Nauka, Moscow, 1980) [in Russian].
Podinovskii, V. V. and Nogin, V. D. Pareto Optimal Solutions of Mutli-Criterial Problems (Nauka, Moscow, 1982) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.V. Konnov, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 60–68.
About this article
Cite this article
Konnov, I.V. Application of the penalty method to nonstationary approximation of an optimization problem. Russ Math. 58, 49–55 (2014). https://doi.org/10.3103/S1066369X14080064
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X14080064