Abstract
Motivated by network design problems, we study a special class of mathematical programs with equilibrium constraints to which the so-called implicit programming approach cannot be applied directly. By using the tools of nonsmooth analysis, however, also these problems may be converted to Lipschitz programs, solvable by existing methods of non-differentiable optimization.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Anandalingam and T. Friesz (eds.), “Hierarchical optimization”, Annals Oper. Res. 34 (1992).
J.-P. Aubin and H. Frankowska, Set—valued Analysis, (J. Wiley & Sons, New York, 1990).
F. Bonnans, “Local analysis of Newton—type methods for variational inequalities and nonlinear programming”, Appl. Math. Optimization 29 (1994) 161–186.
F. H. Clarke, Optimization and Nonsmooth Analysis, (J. Wiley & Sons, New York, 1983).
S. Dafermos, “Traffic equilibrium and variational inequalities”, Transportation Science 14 (1980) 42–54.
S. Dempe, “On generalized differentiability of optimal solutions and its application to an algorithm for solving bilevel optimization problems”, in: D.-Z. Du, L. Qi and R. S. Womersley, eds., Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1995) 36–56.
P. T. Harker and S. C. Choi, “A penalty function approach for mathematical programs with variational inequality constraints”, WP 87–09–08 (Decision Sciences Dep., Univ. of Pennsylvania, Philadelphia, PA, 1987).
P. T. Harker and J.-S. Pang, “Finite—dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications”, Math. Programming 48 (1990) 161–220.
N. H. Josephy, “Newton method for generalized equations”, Technical Report # 1965 (Mathematics Research Center, Univ. of Wisconsin, Madison, WI, 1979).
M. Kocvara and J. V. Outrata, “On the solution of optimum design problems with variational inequalities”, in: D.-Z. Du, L. Qi and R. S. Womersley, eds., Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1995) 172–192.
B. Kummer, “Newton’s method based on generalized derivatives for nonsmooth functions: Convergence analysis”, in: W. Oettli and D. Pallaschke, eds., Advances in Optimization (Lect. Notes in Econ. and Math. Systems 382, Springer—Verlag, Berlin, 1992) 171–194.
J. Kyparisis, “Solution differentiability for variational inequalities”, Math. Programming 48 (1990) 285–301.
Z. Q. Luo, J.-S. Pang and D. Ralph, “Mathematical Programs with Equilibrium Constraints”, (Cambridge Univ. Press, Cambridge, 1996).
P. Marcotte, “ Design optimal d’un réseau de transport en présence d’effets de congestion”, Ph. D. Thesis (Univ. de Montréal, Montréal, Canada, 1981).
P. Marcotte, “Network design problems with congestion effects: A case of bilevel programming”, Math. Programming 34 (1986) 142–162.
J. M. Ortega and W. G. Rheinboldt, “ Iterative Solution of Nonlinear Equations in Several Variables”, (Academic Press, New York, 1970).
J. V. Outrata, “On optimization problems with variational inequality constraints”, SIAM J. Optimization 4 (1994) 340–357.
J. V. Outrata and J. Zowe, “A numerical approach to optimization problems with variational inequality constraints”, Math. Programming 68 (1995) 105–130.
Y. Qiu and T. L. Magnanti, “Sensitivity analysis for variational inequalities defined on polyhedral sets”, Math. Oper. Res. 14 (1989) 410–432.
S. M. Robinson, “Strongly regular generalized equations”, Math. Oper. Res. 5 (1980) 43–62.
H. Schramm and J. Zowe, “A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results”, SIAM J. Optimization 2 (1992) 121–152.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Outrata, J.V. (1997). On a Special Class of Mathematical Programs with Equilibrium Constraints. In: Gritzmann, P., Horst, R., Sachs, E., Tichatschke, R. (eds) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59073-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-59073-3_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63022-7
Online ISBN: 978-3-642-59073-3
eBook Packages: Springer Book Archive