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Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. (English) Zbl 0756.90081
The variational inequality problem is (1): Find $$x^*\in S\subset\mathbb{R}^ n$$ such that $$\langle F(x^*),x-x^*\rangle\geq 0$$, $$\forall x\in\mathbb{R}^ n$$ where $$S\neq\emptyset$$ is closed and convex and $$F: \mathbb{R}^ n\to\mathbb{R}^ n$$. Now with $$G$$ being any $$n\times n$$ positive definite matrix consider the program (2): $$\min\{\phi(y): y\in S\}$$ where $$\phi(y)=\langle F(x),(y-x)\rangle+{1\over 2}\langle(y-x),G(y-x)\rangle$$, and let $$-f(x): \mathbb{R}^ n\to\mathbb{R}$$ be the optimal objective value of $$\phi(y)$$ in (2).
The main result is now: (i) $$f(x)\geq 0$$, $$\forall x\in S$$, and (ii) $$x^*$$ solves (1) if and only if it solves the program (3): $$\min\{f(x): x\in S\}$$ and that happens if and only if $$f(x^*)=0$$, $$x^*\in S$$. Moreover $$f$$ is continuously differentiable (continuous) if $$F$$ is continuously differentiable (continuous). In the first case descent methods are presented to solve the program (3) by an iteration process.
A list of sixteen references closes the paper.

##### MSC:
 90C30 Nonlinear programming 49J40 Variational inequalities 90-08 Computational methods for problems pertaining to operations research and mathematical programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C99 Mathematical programming
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##### 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. · Zbl 0506.65026  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. · Zbl 0679.90087  J.H. Hammond and T.L. Magnanti, ”Generalized descent methods for asymmetric systems of equations,”Mathematics of Operations Research 12 (1987) 678–699. · Zbl 0642.65035  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. · Zbl 0734.90098  D.W. Hearn, ”The gap function of a convex program,”Operations Research Letters 1 (1982) 67–71. · Zbl 0486.90070  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. · Zbl 0637.90037  D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980). · Zbl 0457.35001  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. · Zbl 0623.65073  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). · Zbl 0241.65046  J.S. Pang and D. Chan, ”Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313. · Zbl 0499.90074  W.I. Zangwill,Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, NJ, 1969).
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