Abstract
Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, form a difficult class of optimization problems. The feasible set of MPECs is described by standard equality and inequality constraints as well as additional complementarity constraints that are used to model equilibrium conditions in different applications. But these complementarity constraints imply that MPECs violate most of the standard constraint qualifications. Therefore, more specialized algorithms are typically applied to MPECs that take into account the particular structure of the complementarity constraints. One popular class of these specialized algorithms are the relaxation (or regularization) methods. They replace the MPEC by a sequence of nonlinear programs NLP(t) depending on a parameter t, then compute a KKT-point of each NLP(t), and try to get a suitable stationary point of the original MPEC in the limit t→0. For most relaxation methods, one can show that a C-stationary point is obtained in this way, a few others even get M-stationary points, which is a stronger property. So far, however, these results have been obtained under the assumption that one is able to compute exact KKT-points of each NLP(t). But this assumption is not implementable, hence a natural question is: What kind of stationarity do we get if we only compute approximate KKT-points? It turns out that most relaxation methods only get a weakly stationary point under this assumption, while in this paper, we show that the smooth relaxation method by Lin and Fukushima (Ann. Oper. Res. 133:63–84, 2005) still yields a C-stationary point, i.e. the inexact version of this relaxation scheme has the same convergence properties as the exact counterpart.
Similar content being viewed by others
References
Anitescu, M.: On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints. SIAM J. Optim. 15, 1203–1236 (2005)
Anitescu, M.: Global convergence of an elastic mode approach for a class of mathematical programs with complementarity constraints. SIAM J. Optim. 16, 120–145 (2005)
Bazaraa, M.S., Shetty, C.M.: Foundations of Optimization. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1976)
Demiguel, A.V., Friedlander, M.P., Nogales, F.J., Scholtes, S.: A two-sided relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 16, 587–609 (2005)
Dempe, S.: Foundations of Bilevel Programming. Nonconvex Optimization and Its Applications, vol. 61. Kluwer Academic, Dordrecht (2002)
Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107–134 (1999)
Flegel, M.L., Kanzow, C.: Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124, 595–614 (2005)
Fletcher, R., Leyffer, S.: Solving mathematical programs with complementarity constraints as nonlinear programs. Optim. Methods Softw. 19, 15–40 (2004)
Hoheisel, T., Kanzow, C., Schwartz, A.: Improved convergence properties of the Lin-Fukushima-regularization method for mathematical programs with complementarity constraints. Numer. Algebra Control Optim. 1, 49–60 (2011)
Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137, 257–288 (2013)
Hu, X.M., Ralph, D.: Convergence of a penalty method for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 123, 365–390 (2004)
Izmailov, A.F., Pogosyan, A.L., Solodov, M.V.: Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints. Comput. Optim. Appl. 51, 199–221 (2012)
Kadrani, A., Dussault, J.-P., Benchakroun, A.: A new regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 20, 78–103 (2009)
Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties. SIAM J. Optim. (to appear)
Kanzow, C., Schwartz, A.: The price of inexactness: convergence properties of relaxation methods for mathematical programs with complementarity constraints revisited. Preprint, Institute of Mathematics, University of Würzburg, Würzburg, Germany, March 2013
Leyffer, S., López-Calva, G., Nocedal, J.: Interior methods for mathematical programs with complementarity constraints. SIAM J. Optim. 17, 52–77 (2007)
Lin, G.H., Fukushima, M.: A modified relaxation scheme for mathematical programs with complementarity constraints. Ann. Oper. Res. 133, 63–84 (2005)
Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969). Reprinted by SIAM, Philadelphia (1994)
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)
Outrata, J.V.: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24, 627–644 (1999)
Outrata, J.V.: A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38, 1623–1638 (2000)
Outrata, J.V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (1998)
Pang, J.-S., Fukushima, M.: Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints. Comput. Optim. Appl. 13, 111–136 (1999)
Raghunathan, A.U., Biegler, L.T.: An interior point method for mathematical programs with complementarity constraints (MPCCs). SIAM J. Optim. 15, 720–750 (2005)
Ralph, D., Wright, S.J.: Some properties of regularization and penalization schemes for MPECs. Optim. Methods Softw. 19, 527–556 (2004)
Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)
Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)
Steffensen, S., Ulbrich, M.: A new regularization scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 20, 2504–2539 (2010)
Stein, O.: Lifting mathematical programs with complementarity constraints. Math. Program. 131, 71–94 (2012)
Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000)
Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)
Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)
Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Masao Fukushima, in great respect, on the occasion of his 65th birthday.
Rights and permissions
About this article
Cite this article
Kanzow, C., Schwartz, A. Convergence properties of the inexact Lin-Fukushima relaxation method for mathematical programs with complementarity constraints. Comput Optim Appl 59, 249–262 (2014). https://doi.org/10.1007/s10589-013-9575-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-013-9575-2