Convergence of an adaptive penalty scheme for finding constrained equilibria. (English) Zbl 0773.90092

Summary: This note is devoted to establishing the convergence of a certain penalty scheme for solving monotonic constrained equilibrium problems via a sequence of penalized, but less constrained, subproblems. The penalty parameter, \(t\), is steered adaptively and tends to a limiting value \(t_ *\) \((\neq 0,\;\neq +\infty)\), which prevents the subproblems from becoming unstable in the limit. The method is similar to the one of the first author [U.S.S.R. Comput. Math. Math. Phys. 26, No. 6, 117-122 (1986; Zbl 0635.90073)] and the authors [Numer. Funct. Anal. Optimization 10, No. 9/10, 1003-1017 (1989; Zbl 0703.49011)], but is augmented by an interpolatory step. This permits the avoidance of an unpleasant lower semicontinuity requirement for the solution set, \(S(t)\), of the penalized subproblems, and moreover permits the estimation of the rate of convergence in the convex case.
Equilibrium problems in the sense used here are considered, under various headings, for instance by J. Gwinner, G. Minty, U. Mosco, the second author and S. Simons, and, without monotonicity by H. Brézis, L. Nirenberg and G. Stampacchia, Ky Fan and J. Gwinner. Adaptive penalty methods of the type considered here do not seem to have been discussed for general equilibrium problems.


90C48 Programming in abstract spaces
49M30 Other numerical methods in calculus of variations (MSC2010)
49J45 Methods involving semicontinuity and convergence; relaxation
49J40 Variational inequalities
91B52 Special types of economic equilibria
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[1] Brézis, H.; Nirenberg, L.; Stampacchia, G., A remark on Ky Fan’s minimax principle, Boll. Unione Mat. Ital., 6, 293-300 (1972) · Zbl 0264.49013
[2] Céa, J., Optimization—Théorie et Algorithmes (1971), Dunod: Dunod Paris · Zbl 0211.17402
[3] Ky, Fan, A minimax inequality and applications, (Shisha, O., Inequalities III (1972), Academic Press: Academic Press New York), 103-113 · Zbl 0302.49019
[4] Gwinner, J., On the regularization of monotone variational inequalities, Ops Res. Verf., 28, 374-386 (1978) · Zbl 0387.90085
[5] Gwinner, J., On the penalty method for constrained variational inequalities, (Hiriart-Urruty, J.-B.; Oettli, W.; Stoer, J., Optimization—Theory and Algorithms (1981), Marcel Dekker: Marcel Dekker New York), 197-211 · Zbl 0519.49021
[6] Gwinner, J., On fixed points and variational inequalities—a circular tour, Nonlinear Analysis, 5, 565-583 (1981) · Zbl 0461.47037
[7] Minty, G. J., On variational inequalities for monotone operators, Adv. Math., 30, 1-7 (1978) · Zbl 0407.49008
[8] Mosco, U., Implicit variational problems and quasi variational inequalities, Lecture Notes in Mathematics, Vol. 543, 83-156 (1976) · Zbl 0346.49003
[9] Muu, L. D., An augmented penalty function method for solving a class of variational inequalities, U.S.S.R. Comput. Math. Math. Phys., 26, 117-122 (1986) · Zbl 0635.90073
[10] Muu, L. D.; Oettli, W., A Lagrangian penalty function method for monotone variational inequalities, Numer. Funct. Analysis Optimiz., 10, 1003-1017 (1989) · Zbl 0703.49011
[11] Oettli, W., Monoton-konvexe Funktionen—Eine Bemerkung zum Satz von Browder-Minty, (Guddat, J., Advances in Mathematical Optimization, Vol. 45 (1988), Akademie: Akademie Berlin), 130-136, Mathematical Research · Zbl 0668.47038
[12] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 0229.90020
[13] Rockafellar, R. T., Monotone operators and augmented Lagrangian methods in nonlinear programming, (Mangasarian, O. L.; Meyer, R. R.; Robinson, S. M., Nonlinear Programming 3 (1978), Academic Press: Academic Press New York), 1-25 · Zbl 0458.90050
[14] Rockafellar, R. T., Lagrange multipliers and variational inequalities, (Cottle, R. W.; Giannessi, F.; Lions, J.-L., Variational Inequalities and Complementarity Problems (1980), Wiley: Wiley Chichester), 303-322 · Zbl 0491.49007
[15] Simons, S., Variational inequalities via the Hahn-Banach theorem, Arch. Math., 31, 482-490 (1978) · Zbl 0388.46001
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