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Mathematical Programming

, Volume 62, Issue 1–3, pp 277–297 | Cite as

Nonlinear complementarity as unconstrained and constrained minimization

  • O. L. Mangasarian
  • M. V. Solodov
Article

Abstract

The nonlinear complementarity problem is cast as an unconstrained minimization problem that is obtained from an augmented Lagrangian formulation. The dimensionality of the unconstrained problem is the same as that of the original problem, and the penalty parameter need only be greater than one. Another feature of the unconstrained problem is that it has global minima of zero at precisely all the solution points of the complementarity problem without any monotonicity assumption. If the mapping of the complementarity problem is differentiable, then so is the objective of the unconstrained problem, and its gradient vanishes at all solution points of the complementarity problem. Under assumptions of nondegeneracy and linear independence of gradients of active constraints at a complementarity problem solution, the corresponding global unconstrained minimum point is locally unique. A Wolfe dual to a standard constrained optimization problem associated with the nonlinear complementarity problem is also formulated under a monotonicity and differentiability assumption. Most of the standard duality results are established even though the underlying constrained optimization problem may be nonconvex. Preliminary numerical tests on two small nonmonotone problems from the published literature converged to degenerate or nondegenerate solutions from all attempted starting points in 7 to 28 steps of a BFGS quasi-Newton method for unconstrained optimization.

Keywords

Complementarity Problem Constrain Optimization Problem Unconstrained Optimization Solution Point Nonlinear Complementarity Problem 
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.

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References

  1. [1]
    G. Auchmuty, “Variational principles for variational inequalities,”Numerical Functional Analysis and Optimization 10 (1989) 863–874.Google Scholar
  2. [2]
    D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).Google Scholar
  3. [3]
    R.W. Cottle, “Nonlinear programs with positively bounded Jacobians,” ORC64-12(RR) Operations Research Center, University of California (Berkeley, CA, 1964).Google Scholar
  4. [4]
    R.W. Cottle, “Nonlinear programs with positively bounded Jacobians,”Journal of SIAM on Applied Mathematics 14 (1966) 147–158.Google Scholar
  5. [5]
    G.B. Dantzig and R.W. Cottle, “Positive (semi-)definite programming,” in: J. Abadie, ed.,Nonlinear Programming (North-Holland, Amsterdam, 1967) pp. 55–73.Google Scholar
  6. [6]
    J.E. Dennis and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).Google Scholar
  7. [7]
    G. Di Pillo and L. Grippo, “An exact penalty function method with global convergence properties for nonlinear programming problems,”Mathematical Programming 36 (1986) 1–18.Google Scholar
  8. [8]
    G. Di Pillo and L. Grippo, “Exact penalty functions in constrained optimization,”SIAM Journal on Control and Optimization 27 (1989) 1333–1360.Google Scholar
  9. [9]
    A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968).Google Scholar
  10. [10]
    M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,”Mathematical Programming 53 (1992) 99–110.Google Scholar
  11. [11]
    S.-P. Han and O.L. Mangasarian, “Exact penalty functions in nonlinear programming,”Mathematical Programming 17 (1979) 251–269.Google Scholar
  12. [12]
    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.Google Scholar
  13. [13]
    P.T. Harker and B. Xiao, “Newton's method for the nonlinear complementarity problem: A B-differentiable equation approach,”Mathematical Programming (Series B) 48 (1990) 339–357.Google Scholar
  14. [14]
    N.H. Josephy, “Newton's method for generalized equations,” Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin (Madison, WI, 1979).Google Scholar
  15. [15]
    S. Karamardian, “The nonlinear complementarity problem with applications, Parts 1 and 2,”Journal of Optimization Theory and Applications 4 (1969) 87–98 and 167–181.Google Scholar
  16. [16]
    M. Kojima and S. Shindo, “Extensions of Newton and quasi-Newton methods to systems of PC1 equations,”Journal of Operations Research Society of Japan 29 (1986) 352–374.Google Scholar
  17. [17]
    Z.-Q. Luo and P. Tseng, “Error bound and reduced gradient projection algorithms for convex minimization over polyhedral sets,”SIAM Journal on Optimization 3 (1993) 43–59.Google Scholar
  18. [18]
    Z.-Q. Luo and P. Tseng, “Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem,”SIAM Journal on Optimization 2 (1992) 43–54.Google Scholar
  19. [19]
    O.L. Mangasarian,Nonlinear Programming (McGraw-Hill, New York, 1969).Google Scholar
  20. [20]
    O.L. Mangasarian, “Equivalece of the complementarity problem to a system of nonlinear equations,”SIAM Journal on Applied Mathematics 31 (1976) 89–92.Google Scholar
  21. [21]
    O.L. Mangasarian, “Unconstrained methods in nonlinear programming,” in:Nonlinear Programming. SIAM-AMS Proceedings, Vol. IX (American Mathematical Society, Providence, RI, 1976) pp. 169–184.Google Scholar
  22. [22]
    O.L. Mangasarian, “Global error bounds for monotone affine variational inequality problems,”Linear Algebra and its Applications 174 (1992) 153–163.Google Scholar
  23. [23]
    O.L. Mangasarian and T.-H. Shiau, “Error bounds for monotone linear complementarity problems,”Mathematical Programming 36 (1986) 81–89.Google Scholar
  24. [24]
    O.L. Mangasarian and T.-H. Shiau, “Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems,”SIAM Journal on Control and Optimization 25 (1987) 583–595.Google Scholar
  25. [25]
    B. Murtagh and M. Saunders, “MINOS 5.0 user's guide,” Technical Report SOL83.20, Systems Optimization Laboratory, Stanford University (Stanford, CA, 1983).Google Scholar
  26. [26]
    J.M. Ortega,Numerical Analysis: A Second Course (Academic Press, New York, 1972).Google Scholar
  27. [27]
    J.-S. Pang, “Newton's method for B-differentiable equations,”Mathematics of Operations Research 15 (1990) 311–341.Google Scholar
  28. [28]
    J.-S. Pang, “A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems,”Mathematical Programming 51 (1991) 101–131.Google Scholar
  29. [29]
    J.-S. Pang and S.A. Gabriel, “NE/SQP: A robust algorithm for the nonlinear complementarity problem,”Mathematical Programming 60 (1993) 295–337.Google Scholar
  30. [30]
    R.T. Rockafellar, “The multiplier method of Hestenes and Powell applied to convex programming,”Journal of Optimization Theory and Applications 12 (1973) 555–562.Google Scholar
  31. [31]
    P.K. Subramanian, “Gauss—Newton methods for the complementarity problem,” to appear in:Journal of Optimization Theory and Applications. Google Scholar
  32. [32]
    P. Wolfe, “A duality theorem for nonlinear programming,”Quarterly of Applied Mathematics 19 (1961) 239–244.Google Scholar
  33. [33]
    Z.Q. Luo, O.L. Mangasarian, J. Ren and M.V. Solodov, “New error bounds for the linear complementarity problem,” to appear in:Mathematics of Operations Research. Google Scholar
  34. [34]
    J. Ren, “Computable error bounds in mathematical programming,” Ph.D. Dissertation, Computer Sciences Department, University of Wisconsin (Madison, WI, July 1993).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • O. L. Mangasarian
    • 1
  • M. V. Solodov
    • 1
  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA

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