×

Equilibrium programming using proximal-like algorithms. (English) Zbl 0890.90150

Summary: We compute constrained equilibria satisfying an optimality condition. Important examples include convex programming, saddle problems, noncooperative games, and variational inequalities. Under a monotonicity hypothesis, we show that equilibrium solutions can be found via iterative convex minimization. In the main algorithm each stage of computation requires two proximal steps, possibly using Bregman functions. One step serves to predict the next point; the other helps to correct the new prediction. To enhance practical applicability we tolerate numerical errors.

MSC:

90C25 Convex programming
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A.S. Antipin, Controlled proximal differential systems for saddle problems,Differential Equations 28 (1992) 1498–1510. · Zbl 0821.49017
[2] A.S. Antipin, Feed-back controlled saddle gradient processes,Automat. Remote Control 55 (1994) 311–320. · Zbl 0847.93050
[3] A.S. Antipin, Convergence and estimates of the rate of convergence of proximal methods to fixed points of extremal mappings,Zh. Vychisl. Mat. Fiz. 35 (1995) 688–704. · Zbl 0852.65046
[4] A. Auslender, J.P. Crouzeix and P. Fedit, Penalty-proximal methods in convex programming,Journal of Optimization Theory and Applications 55 (1987) 1–21. · Zbl 0622.90065
[5] A. Auslender and M. Haddou, An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities,Mathematical Programming 71 (1995) 77–100. · Zbl 0855.90095
[6] D.P. Bertsekas and P. Tseng, Partial proximal minimization algorithms for convex programming,SIAM Journal of Optimization 4 (1994) 551–572. · Zbl 0819.90069
[7] L.M. Bregman, The relaxation method of finding the common point of convex sets and its applications to the solution of problems in convex programming,USSR Computational Mathematics and Mathematical Physics 7 (1967) 200–217. · Zbl 0186.23807
[8] A. Brønsted and R.T. Rockafellar, On the subdifferentiability of convex functions,Proceedings of the American Mathematical Society 16 (1965) 605–611. · Zbl 0141.11801
[9] Y. Censor and S. Zenios, The proximal minimization algorithm with D-functions,Journal of Optimization Theory and Applications 73 (1992) 451–464. · Zbl 0794.90058
[10] G. Chen and M. Teboulle, Convergence analysis of a proximal-like minimization algorithm using Bregman functions,SIAM Journal of Optimization 3 (1993) 538–543. · Zbl 0808.90103
[11] A. Cournot,Recherches sur les Principes Mathématiques de la Théorie des Richesses (Paris, 1838); English translation:Researches into the Mathematical Principles of the Theory of Wealth (N. Bacon, ed.) (Macmillan, New York, 1897).
[12] J. Eckstein, Nonlinear proximal point algorithms using Bregman functions with applications to convex programming,Mathematics of Operations Research 18 (1) (1993) 202–226. · Zbl 0807.47036
[13] P.P.B. Eggermont, Multiplicative iterative algorithms for convex programming,Linear Algebra and its Applications 130 (1990) 25–42. · Zbl 0715.65037
[14] Yu.M. Ermol’ev and S.P. Uryas’ev, Nash equilibrium inn-person games,Kibernetika 3 (1982) 85–88 (in Russian).
[15] S.D. Flåm, Paths to constrained Nash equilibria,Applied Mathematics and Optimization 27 (1993) 275–289. · Zbl 0805.90122
[16] S.D. Flåm and A. Ruszczynski, Noncooperative games: Computing equilibrium by partial regularization, Working Paper IIASA 42, 1994.
[17] S.D. Flåm, Approaches to economic equilibrium,Journal of Economic Dynamics and Control 20 (1996) 1505–1522.
[18] N. Hadjisavvas and S. Schaible, On strong monotonicity and (semi)strict quasimonotonicty,Journal of Optimization Theory and Applications 79 (1993) 139–155. · Zbl 0792.90068
[19] P.T. Harker and J.-S. Pang, Finite-dimensional variational inequalities and nonlinear complementarity problems: A survey of theory, algorithms and applications,Mathematical Programming 48 (1990) 161–220. · Zbl 0734.90098
[20] J.-B. Hiriart-Urruty and C. Lemaréchal,Convex Analysis and Minimization Algorithms (Springer, Berlin, 1993).
[21] A.N. Iusem, B.F. Svaiter and M. Teboulle, Entopy-like proximal methods in convex programming,Mathematics of Operations Research 19 (1994) 790–814. · Zbl 0821.90092
[22] A. N. Iusem, Some properties of generalized proximal point methods for quadratic and linear programming,Journal of Optimization Theory and Applications 85 (1995) 593–612. · Zbl 0831.90092
[23] G.M. Korpelevich, The extragradient method for finding saddle points and other problems,Ekon. i Mat. Metody 12 (1976) 747–756. · Zbl 0342.90044
[24] B. Martinet, Perturbations des methodes d’optimisation,RAIRO Anal. Numer. 12 (1978) 153–171. · Zbl 0379.90088
[25] M.J. Osborne and A. Rubinstein,A Course in Game Theory (MIT Press, Cambridge, MA, 1994). · Zbl 1194.91003
[26] H. Robbins and D. Siegmund, A convergence theorem for nonnegative almost surmartingales and some applications, in: J. Rustagi, ed.,Optimizing Methods in Statistics (Academic Press, New York, 1971) 235–257.
[27] R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970). · Zbl 0193.18401
[28] R.T. Rockafellar, Monotone operators and the proximal point algorithm,SIAM Journal of Control and Optimization 14 (1976) 877–898. · Zbl 0358.90053
[29] J.B. Rosen, Existence and uniqueness of equilibrium points for concaveN-person games,Econometrica 33 (1965) 520–534. · Zbl 0142.17603
[30] A. Ruszczynski, A partial regularization method for saddle point seeking, Working Paper IIASA 20, 1994.
[31] M. Teboulle, Entropic proximal mappings with applications to nonlinear programming,Mathematics of Operations Research 17 (1992) 670–690. · Zbl 0766.90071
[32] J. Tirole,The Theory of Industrial Organization (MIT Press, Cambridge, MA, 1988). · Zbl 0664.90023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.