Abstract
For a solvable monotone complementarity problem we show that each feasible point which is not a solution of the problem provides simple numerical bounds for some or all components of all solution vectors. Consequently for a solvable differentiable convex program each primal-dual feasible point which is not optimal provides simple bounds for some or all components of all primal-dual solution vectors. We also give an existence result and simple bounds for solutions of monotone compementarity problems satisfying a new, distributed constraint qualification. This result carries over to a simple existence and boundedness result for differentiable convex programs satisfying a similar constraint qualification.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L.E.J. Brouwer, “Über Abbildung von Mannigfaltigkeiten”,Mathematische Annalen 71 (1910) 97–115.
R.W. Cottle, “Nonlinear programs with positively bounded Jacobians”,SIAM Journal of Applied Mathematics 14 (1966) 147–158.
G.B. Dantzig, E. Eisenberg and R.W. Cottle, “Symmetric dual nonlinear programs”,Pacific Journal of Mathematics 15 (1965) 809–812.
B.C. Eaves, “On the basic theorem of complementarity”,Mathematical Programming 1 (1971) 68–75.
S. Karamardian, “The nonlinear complementarity problem with applications, part 2”,Journal of Optimization Theory and Applications 4 (1969) 167–181.
O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969).
O.L. Mangasarian, “Simple computable bounds for solutions of linear complementarity problems and linear programs”, Technical Report #519, Computer Sciences Department, University of Wisconsin-Madison, October 1983, to appear inMathematical Programming Study.
L. McLinden, “The complementarity problem for maximal monotone multifunctions”, Chapter 17, 251–270 in: R. W. Cottle, F. Giannessi and J.-L. Lions, eds.,Variational inequalities and complementarity problems, (Wiley New York, 1980).
L. McLinden, “Stable monotone variational inequalities”, Technical Summary Report #2734, Mathematics Research Center, University of Wisconsin-Madison, August 1984.
N. Megiddo, “A monotone complementarity problem with feasible solutions but no complementary solutions”,Mathematical Programming 12 (1977) 131–132.
J.J. Moré, “Coercivity conditions in nonlinear complementarity problems”,SIAM Review 16 (1974) 1–16.
J.J. Moré, “Classes of functions and feasibility conditions in nonlinear complementarity problems”,Mathematical Programming 6 (1974) 327–338.
R.T. Rockafellar, “A general correspondence between dual minimax problems and convex programs”,Pacific Journal of Mathematics 25 (1968) 597–611.
D.R. Smart,Fixed point theorems (Cambridge University Press, Cambridge, 1974).
Additional information
Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grants MCS-8200632 and MCS-8102684.
Rights and permissions
About this article
Cite this article
Mangasarian, O.L., McLinden, L. Simple bounds for solutions of monotone complementarity problems and convex programs. Mathematical Programming 32, 32–40 (1985). https://doi.org/10.1007/BF01585657
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585657