Abstract
Variational inequality and complementarity have much in common, but there has been little direct contact between the researchers of these two related fields ofmathematical sciences. Several problems arising from fluid mechanics, solid mechanics, structural engineering, mathematical physics, geometry, mathematical programming, and so on have the formulation of a variational inequality or complementarity problem. People working in applied mathematics mostly deal with the infinitedimensional case and they deal with variational inequality whereas people working in operations research mostly deal with the finite-dimensional problem and they use the complementarity problem. Variational inequality is a formulation for solving the problem where we have to optimize a functional. The theory is derived by using the techniques of nonlinear functional analysis such as fixed point theory and the theory of monotone operators, among others.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M. S. Bazaras, J. J. Goode, and M. Z. Nashed, A nonlinear complementarity problem in mathematical programming in Banach space, Proc. Amer. Math. Soc. 35 (1972), 165–170.
F. E. Browder, Nonlinear variational inequalities and maximal monotone mappings in Banach spaces, Math. Annalen, 176 (1968), 89–113.
____, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Natl. Acad. Sci. U. S. 56 (1966a), 1080–1086.
____, On the unification of calculus of variation and the theory of monotone nonlinear operators in Banach spaces, ibid (1966b), 419–425.
____, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965a) 780–758.
____, On a theorem of Beurling and Livingston, Canad. J. Math. 17 (1965b), 367–372.
____, Variational boundary value problems for quasilinear elliptic equations of arbitrary order, Proc. Natl. Acad. Sci., U. S. 50 (1963), 31–37.
B. D. Craven and B. Mond. Complementarity over arbitrary cone, Z. O. R. (1977).
R. Chandrasekharan, A special case of the complementarity pivot problem, Opsearch, 7 (1970) 263–268.
R.W. Cottle and G. B. Dantzig, Complementarity pivot theory of mathematical programming, Linear Algebra Appl. 1 (1968), 108–125.
O. D. Donato and G. Maier, Mathematical-programming methods for the inelastic analysis of reinforced concrete frames allowing for limited rotation capacity, Int. J. Numer. Meth. Eng. 4 (1972) 307–329.
A. T. Dash and S. Nanda, A complementarity problem in mathematical programming in Banach space, J. Math. Anal. Appl. 98 (1984).
B. C. Eaves, On the basic theorem of completemtarity, Math. Program. 1 (1971) 68–75.
M. Edelstein, On nearest points of sets in uniformly convex Banach spaces, J. London Math. Soc. 43 (1968) 375–377.
B. Garcia, Some classes of matrices in linear complementarity problem, Math. Program. 5 (1973), 299–310.
K. M. Ghosh, Fixed-point theorems, Pure Math Manuscript 3 (1984) 51–53.
J. R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1967), 436–46.
M. Gregus, A fixed-point theorem in Banach space, Boll. Un. Mat. Ital. 5(17)—A (1980) 193–198.
G. E. Hardy and T. D. Rogers, A generalization of a fixed-point theorem of Reich, Canad, Math. Bull. 16 (1973) 201–206.
Hartmann and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 125 (1956) 271–310.
G. J. Habetler and A. L. Price, Existence theory for generalized nonlinear complementarity problem, J. Optim. Theory Appl. 7 (1971) 223–239.
G. J. Habetler and M. M. Kostreva, On a direct algorithm for nonlinear complementarity problems SIAM J. Control Optim. 16(3) (1978) 504–511.
M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, Wiley Eastern Ltd. (1985).
S. Karamaradian, The nonlinear complementarity problem with applications I, II, J. Optim. Theory Appl. 4 (1969) 87–98, 167–181.
____, Generalized complementarity problem, ibid 8 (1971) 161–168.
J. L. Lions and G. Stampacchia, Variational inequalilities, Comm. Pure Appl. Math. 20 (1967) 493–519.
C. E. Lemke, Bimatrix equilibrium points and mathematical programming, Manage. Sci. Ser. A, 11 (1965) 168–169.
____, Recent Results on Complementarity Problems, in Nonlinear Programming (J. B. Rosen, O. L. Mangasarian and K. Ritter, Eds.), Academic New York (1970).
C. E. Lemke and J. T. Howson Jr., Equilibrium points of bimatrix games, SIAM J. Appl. Math. 12 (1964), 413–423.
G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961) 29–43.
B. Mond. On the complex complementarity problem, Bull. Austral. Math. Soc. 9 (1973) 249–257.
O. G. Mancino and G. G. Stampacchia, Convex programming and variational inequalities, J. Optim. Theory Appl. 9(1) (1972), 3–23.
U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. Math. 3 (1966), 510–585.
____, A remark on a theorem of F. E. Browder, J. Math. Anal. Appl. 20 (1967a), 90–93.
____, Approximation of the solution of some variational inequalities, Ann Scuolo Normals sup. Pisa 21 (1967b), 373–934; 765.
K. G.Murty, On a characterization of P-matrices, SIAM J. Appl. Math. 20(3) (1971), 378–384.
____, On the number of solutions of the complementarity problem and spanning properties of complementary cones, Linear Algebra Appl. 5 (1972), 65–108.
____, Note on a Bard-type scheme for solving the complementarity problem, Opsearch 11, 2–3 (1974), 123–130.
Sribatsa Nanda and Sudarsan Nanda, A complex nonlinear complementarity problem, Bull. Austral. Math. Soc. 19 (1978) 437–444.
____, On stationary points and the complementarity problem, ibid 20 (1979a) 77–86.
____, A nonlinear complementarity problem in mathematical programming in Hilbert space, ibid 20 (1979b) 233–236.
____, A nonlinear complementarity problem in Banach space, ibid 21 (1980) 351–356.
Sudarsan Nanda, A note on a theorem on a nonlinear complementarity problem, ibid 27 (19783) 161–163.
J. Parida and B. Sahoo, On the complex nonlinear complementarity problem, ibid 14 (1976a), 129–136.
____, Existence theory for the complex nonlinear complementarity problem, ibid 14 (1976b), 417–435.
A. Ravindran, A computer routine for quadratic and linear programming problems [H], Commun. ACM 15(9) (1972) 818–820.
____, A comparison of the primal-simplex and complementary pivot methods for linear programming, Naval Res. Logistics Q. 20(1) (1973) 96–100.
S. Reich, some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971) 121–124, MR 45 # 1145.
B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257–289.
H. Scarf, The approximation of fixed points of a continuous map. SIAM J. Appl. Math. 15 (1967), 328–343.
R. Saigal, On the class of complementary cones and Lemkes’ algorithm, SIAM J. Appl. Math. 23 (1972) 46–60.
____, A note on a special linear complementarity problem, Opsearch 7(3) (1970) 175–183.
S. Simons, Variational inequalities via the Hann-Banach theorem, Arch. Math. 31 (1978) 482–490.
L. T. Watson, Solving nonlinear complementarity problem by homotopy methods, SIAM J. Control Optim. 17(1) (1979) 36.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Nanda, S. (2011). Variational Inequality and Complementarity Problem. In: Mishra, S. (eds) Topics in Nonconvex Optimization. Springer Optimization and Its Applications(), vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9640-4_4
Download citation
DOI: https://doi.org/10.1007/978-1-4419-9640-4_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9639-8
Online ISBN: 978-1-4419-9640-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)