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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 ${t}_{*}$ $\left(\ne 0,\phantom{\rule{4pt}{0ex}}\ne +\infty \right)$, 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, $S\left(t\right)$, 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.

##### MSC:
 90C48 Programming in abstract spaces 49M30 Other numerical methods in calculus of variations 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 49J40 Variational methods including variational inequalities 91B52 Special types of equilibria in economics