Abstract
Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Auslender,Optimisation: Méthodes Numériques (Masson, Paris, 1976).
S. Dafermos, “Traffic equilibrium and variational inequalities,”Transportation Science 14 (1980) 42–54.
S. Dafermos, “An iterative scheme for variational inequalities,”Mathematical Programming 26 (1983) 40–47.
J.E. Dennis, Jr. and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).
M. Florian, “Mathematical programming applications in national, regional and urban planning,” in: M. Iri and K. Tanabe, eds.,Mathematical Programming: Recent Developments and Applications (KTK Scientific Publishers, Tokyo, 1989) pp. 57–81.
J.H. Hammond and T.L. Magnanti, “Generalized descent methods for asymmetric systems of equations,”Mathematics of Operations Research 12 (1987) 678–699.
P.T. Harker and J.S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,”Mathematical Programming (Series B) 48 (1990) 161–220.
D.W. Hearn, “The gap function of a convex program,”Operations Research Letters 1 (1982) 67–71.
T. Itoh, M. Fukushima and T. Ibaraki, “An iterative method for variational inequalities with application to traffic equilibrium problems,”Journal of the Operations Research Society of Japan 31 (1988) 82–103.
D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980).
P. Marcotte, “A new algorithm for solving variational inequalities with application to the traffic assignment problem,”Mathematical Programming 33 (1985) 339–351.
P. Marcotte and J.P. Dussault, “A note on a globally convergent Newton method for solving monotone variational inequalities,”Operations Research Letters 6 (1987) 35–42.
A. Nagurney, “Competitive equilibrium problems, variational inequalities and regional science,”Journal of Regional Science 27 (1987) 503–517.
J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313.
W.I. Zangwill,Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, NJ, 1969).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fukushima, M. Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming 53, 99–110 (1992). https://doi.org/10.1007/BF01585696
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585696