Abstract
Optimization problems with variational inequality constraints are converted to constrained minimization of a local Lipschitz function. To this minimization a non-differentiable optimization method is used; the required subgradients of the objective are computed by means of a special adjoint equation. Besides tests with some academic examples, the approach is applied to the computation of the Stackelberg—Cournot—Nash equilibria and to the numerical solution of a class of quasi-variational inequalities.
Similar content being viewed by others
References
W. Alt and I. Kolumbán, “An implicit function theorem for a class of monotone generalized equations,”Kybernetika 29 (1993) 210–221.
J.-P. Aubin and I. Ekeland,Applied Nonlinear Analysis (Wiley, New York, 1984).
D. Chan and J.S. Pang, “The generalized quasi-variational inequality problem,”Mathematics of Operations Research 7 (1982) 211–222.
F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
A.H. DeSilva, “Sensitivity formulas for nonlinear factorable programming and their application to the solution of an implicitly defined optimization model of US crude oil production,” Dissertation, George Washington University (Washington, D.C., 1978).
B.C. Eaves, “On the basic theorem of complementarity,”Mathematical Programming 1 (1971) 68–75.
S.D. Flåm, “Paths to constrained Nash equilibria,” Research Report UTM 345, University of Trento (Trento, 1991).
T.L. Friesz, R.L. Tobin, H.-J. Cho and N.J. Mehta, “Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints,”Mathematical Programming 48 (1990) 265–284.
T.L. Friesz, H.-J. Cho, N.J. Mehta, R.L. Tobin and G. Anandalingam, “A simulated annealing approach to the network design problem with variational inequality constraints,”Transportation Science 26 (1992) 18–26.
P.T. Harker and S.C. Choi, “A penalty function approach for mathematical programs with variational inequality constraints,” WP 87-08-08, University of Pennsylvania (Pennsylvania, 1987).
P.T. Harker and J.-S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,”Mathematical Programming 48 (1990) 161–220.
P.T. Harker, “Generalized Nash games and quasi-variational inequalities,”European Journal of Operational Research 54 (1991) 81–94.
J. Haslinger and P. Neittaanmäki,Finite Element Approximation for Optimal Shape Design: Theory and Application (Wiley, Chichester, 1988).
J.M. Henderson and R.E. Quandt,Microeconomic Theory, 3rd ed. (McGraw-Hill, New York, 1980).
Y. Ishizuka and E. Aiyoshi, “Double penalty method for bilevel optimization problems,” in: G. Anandalingam and T. Friesz, eds.,Hierarchical Optimization, Annals of Operations Research 34 (1992) 73–88.
M. Kočvara and J.V. Outrata, “A nondifferentiable approach to the solution of optimum design problems with variational inequalities,” in: P. Kall, ed.,System Modelling and Optimization, Lecture Notes in Control and Information Sciences 180 (Springer, Berlin, 1992) 364–373.
M. Kočvara and J.V. Outrata, “Shape optimization of elasto-plastic bodies governed by variational inequalities,” in: J.-P. Zolesio, ed.,Boundary Control and Variation, Lecture Notes in Pure and Applied Mathematics 163 (Marcel Dekker, New York, 1994) 261–272.
J. Kyparisis, “Solution differentiability for variational inequalities,”Mathematical Programming 48 (1990) 285–301.
C. Lemaréchal, J.-J. Strodiot and A. Bihain, “On a bundle algorithm for nonsmooth optimization,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 4 (Academic Press, New York, 1991).
O.L. Mangasarian,Nonlinear Programming (McGraw-Hill, New York, 1969).
P. Marcotte, “Network design problem with congestion effects: A case of bilevel programming,”Mathematical Programming 34 (1986) 142–162.
U. Mosco, “Implicit variational problems and quasi-variational inequalities,” in: A. Dold and B. Eckmann, eds.,Nonlinear Operators and the Calculus of Variations, Lecture Notes in Mathematics, 543 (Springer, Berlin, 1976).
F.H. Murphy, H.D. Sherali and A.L. Soyster, “A mathematical programming approach for determining oligopolistic market equilibrium,”Mathematical Programming 24 (1982) 92–106.
J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1980).
J.V. Outrata, “On the numerical solution of a class of Stackelberg problems,”Zeitschrift für Operations Research 4 (1990) 255–278.
J.V. Outrata, “On necessary optimality conditions for Stackelberg problems,”Journal of Optimization Theory and Applications 76 (1993) 305–320.
J.S. Pang, “Two characterization theorems in complementarity theory,”Operations Research Letters 7 (1988) 27–31.
J.S. Pang, S.P. Han and N. Rangaraj, “Minimization of locally Lipschitz functions,”SIAM Journal on Optimization 1 (1991) 57–82.
Y. Qiu and T.L. Magnanti, “Sensitivity analysis for variational inequalities defined on polyhedral sets,”Mathematics of Operations Research 14 (1989) 410–432.
S.M. Robinson, “Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62.
S.M. Robinson, “An implicit-function theorem for a class of nonsmooth functions,”Mathematics of Operations Research 16 (1991) 282–309.
K. Schittkowski, “NLPQL: A Fortran subroutine solving constrained nonlinear programming problems,”Annals of Operations Research 5 (1985) 485–500.
H. Schramm, “Bundle Trust methods: Fortran codes for nondifferentiable optimization. User's Guide,” DFG Research Report No. 269, University of Bayreuth (Bayreuth, 1991).
H. Schramm and J. Zowe, “A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results,”SIAM Journal on Optimization 2 (1992) 121–152.
M.J. Smith, “A decent algorithm for solving monotone variational inequalities and monotone complementarity problems,”Journal of Optimization Theory and Applications 44 (1984) 485–496.
R.L. Tobin, “Uniqueness results and algorithms for Stackelberg—Cournot—Nash equilibria,” in: G. Anandalingam and T. Friesz, eds.,Hierarchical Optimization, Annals of Operations Research 34 (1992) 21–36.
Author information
Authors and Affiliations
Additional information
Corresponding author.
Rights and permissions
About this article
Cite this article
Outrata, J., Zowe, J. A numerical approach to optimization problems with variational inequality constraints. Mathematical Programming 68, 105–130 (1995). https://doi.org/10.1007/BF01585759
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585759