Abstract
Given a mappingF from real Euclideann-space into itself, we investigate the connection between various known classes of functions and the nonlinear complementarity problem: Find anx * such thatFx * ⩾ 0 andx * ⩾ 0 are orthogonal. In particular, we study the extent to which the existence of au ⩾ 0 withFu ⩾ 0 (feasible point) implies the existence of a solution to the nonlinear complementarity problem, and extend, to nonlinear mappings, known results in the linear complementarity problem on P-matrices, diagonally dominant matrices with non-negative diagonal elements, matrices with off-diagonal non-positive entries, and positive semidefinite matrices.
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References
R. Chandrasekaran, “A special case of the complementarity pivot problem”,Operations Research 7 (1970) 263–268.
R.W. Cottle, “Note on a fundamental theorem in quadratic programming”,SIAM Journal on Applied Mathematics 12 (1964) 663–665.
R.W. Cottle, “Nonlinear programs with positively bounded Jacobians”,SIAM Journal on Applied Mathematics 14 (1966) 147–157.
R.W. Cottle and G.B. Dantzig, “Positive (semi-) definite programming”, in:Nonlinear programming, Ed. J. Abadie (North-Holland, Amsterdam, 1967).
C.W. Cryer, “The solution of a quadratic programming problem using systematic overrelaxation”,SIAM Journal on Control 9 (1971) 385–392.
C.W. Cryer, “The method of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences”,Mathematics of Computation 25 (1971) 435–443.
B.C. Eaves, “The linear complementarity problem”,Management Science 17 (1971) 68–75.
M. Edelstein, “On fixed and periodic points under contractive mappings”,Journal of the London Mathematical Society 37 (1962) 74–79.
M. Fiedler and V. Pták, “On matrices with non-positive off-diagonal elements and positive principal minors”,Czechoslovak Mathematical Journal 12 (1962) 382–400.
C.B. Garcia, “Some classes of matrices in linear complementarity theory”, Mathematical Programming 5 (1973) 299–310.
A.W. Ingleton, “A problem in linear inequalities”,Proceedings of the London Mathematical Society 16 (1966) 519–536.
S. Karamardian, “Existence of solutions of certain systems of nonlinear inequalities”,Numerische Mathematik 12 (1968) 327–334.
C.E. Lemke, “Recent results on complementarity problems”, in:Nonlinear programming, Eds. J.B. Rosen, O.L. Mangasarian and K. Ritter (Academic Press, New York, 1970).
G.P. McCormick and R.A. Tapia, “The gradient projection method under mild differentiability conditions”,SIAM Journal on Control 10 (1972) 93–98.
G. Minty, “Monotone (nonlinear) operators in Hilbert space”,Duke Mathematical Journal 29 (1962) 341–346.
J.J. Moré, “Nonlinear generalizations of matrix diagonal dominance with application to Gauss—Seidel iterations”,SIAM Journal on Numerical Analysis 9 (1972) 357–378.
J.J. Moré, “Coercivity conditions in nonlinear complementarity problems”,SIAM Review 16 (1974).
J.J. Moré and W.C. Rheinboldt, “On P- and S-functions and related classes ofn-dimensional nonlinear mappings”,Linear Algebra and Its Applications 6 (1973) 45–68.
W.C. Rheinboldt, “On M-functions and their application to nonlinear Gauss—Seidel iterations, and network flows”,Journal of Mathematical Analysis and Applications 32 (1971) 274–307.
A. Tamir, “The complementarity problem of mathematical programming”, Ph. D. thesis, Case Western Reserve University, Cleveland, Ohio (1973).
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This research was supported in part by the National Science Foundation under Grants GJ-28528 and GJ-40903.
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Moré, J.J. Classes of functions and feasibility conditions in nonlinear complementarity problems. Mathematical Programming 6, 327–338 (1974). https://doi.org/10.1007/BF01580248
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DOI: https://doi.org/10.1007/BF01580248