# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Two new self-adaptive projection methods for variational inequality problems. (English) Zbl 1012.65064
The usual variational inequality Find $u^* \in K$ such that $$F(u^*)^T (v-u^*) \geq 0 \quad\text{for any }v \in K,$$ where $K$ is a nonempty closed convex subset of $R^n$, is considered. The function $F$ is continuous and satisfies only some generalized monotonicity assumptions. The new methods use only function evaluations and projections onto the set $K$, together with a line search strategy. Numerical tests are reported.

##### MSC:
 65K10 Optimization techniques (numerical methods) 49J40 Variational methods including variational inequalities 49M15 Newton-type methods in calculus of variations
Full Text:
##### References:
 [1] Bertsekas, D. P.; Gafni, E. M.: Projection method for variational inequalities with applications to the traffic assignment problem. Mathematical programming study 17, 139-159 (1982) · Zbl 0478.90071 [2] Dafermos, S.: Traffic equilibrium and variational inequalities. Transportation science 14, 42-54 (1980) [3] Harker, P. T.; Pang, J. S.: Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Mathematical programming 48, 161-220 (1990) · Zbl 0734.90098 [4] Nagurney, A.; Ramanujam, P.: Transportation network policy modeling with goal targets and generalized penalty functions. Transportation science 30, 3-13 (1996) · Zbl 0849.90055 [5] Sibony, M.: Méthodes itératives pour LES équations et inéquations aux dérivées partiells nonlinéares de type monotone. Calcolo 7, 65-183 (1970) [6] He, B. S.: A projection and contraction method for a class of linear complementary problems and its applications to in convex quadratic programming. Applied mathematics and optimization 25, 247-262 (1992) · Zbl 0767.90086 [7] He, B. S.: On a class of iterative projection and contraction methods for linear programming. Journal of optimization theory and application 78, 247-266 (1993) · Zbl 0792.90042 [8] He, B. S.: Further developments in an iterative projection and contraction method for linear programming. Journal of computational mathematics 11, 350-364 (1993) · Zbl 0795.65032 [9] Korpelevich, G. M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747-756 (1976) · Zbl 0342.90044 [10] Sun, D. F.: A new step-size skill for solving a class of nonlinear projection equations. Journal of computational mathematics 13, 357-368 (1995) · Zbl 0854.65048 [11] Iusem, A. N.; Svaiter, B. F.: A variant of korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309-321 (1997) · Zbl 0891.90135 [12] Solodov, M. V.; Svaiter, B. F.: A new projection method for variational inequality problems. SIAM journal on control and optimization 37, 765-776 (1999) · Zbl 0959.49007 [13] Pini, R.; Singh, C.: A survey of recent (1985--1995) advances in generalized convexity with applications to duality theory and optimality conditions. Optimization 39, 311-360 (1997) · Zbl 0872.90074 [14] He, B. S.: A class of projection and contraction methods for monotone variational inequalities. Applied mathematics and optimization 35, 69-76 (1997) · Zbl 0865.90119 [15] Solodov, M. V.; Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM journal of control and optimization 34, 1814-1830 (1996) · Zbl 0866.49018 [16] A. Chen, H.K. Lo and H. Yang, A self-adaptive projection and contraction algorithm for traffic equilibrium problem with path-specific costs, Europe Journal of Operation Research (to appear). · Zbl 1077.90516 [17] Eaves, B. C.: On the basic theorem of complementarity. Mathematical programming 1, 68-75 (1971) · Zbl 0227.90044 [18] He, B. S.: Some predict-correct projection methods for monotone variational inequalities. Reports of the faculty of technical mathematics and informatics, 95-98 (1995) [19] Kojima, M.; Shindo, S.: Extensions of Newton and quasi-Newton methods to systems of PC1 equations. Journal of operations research society of Japan 29, 352-374 (1986) · Zbl 0611.65032