Box-constrained minimization reformulations of complementarity problems in second-order cones

Reformulations of a generalization of a second-order cone complementarity problem (GSOCCP) as optimization problems are introduced, which preserve differentiability. Equivalence results are proved in the sense that the global minimizers of the reformulations with zero objective value are solutions to the GSOCCP and vice versa. Since the optimization problems involved include only simple constraints, a whole range of minimization algorithms may be used to solve the equivalent problems. Taking into account that optimization algorithms usually seek stationary points, a theoretical result is established that ensures equivalence between stationary points of the reformulation and solutions to the GSOCCP. Numerical experiments are presented that illustrate the advantages and disadvantages of the reformulations. © 2007 Springer Science+Business Media, Inc.Reformulations of a generalization of a second-order cone complementarity problem (GSOCCP) as optimization problems are introduced, which preserve differentiability. Equivalence results are proved in the sense that the global minimizers of the reformulati404505527CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO01/04597-4sem informaçãoAlizadeh, F., Goldfarb, D., Second-order cone programming (2003) Math. Program., 95, pp. 3-51Andreani, R., Friedlander, A., Mello, M.P., Santos, S.A., (2005) Generalized Second-order Complementarity Problems: Theory and Numerical Experiments, , http://www.ime.unicamp.br/andreani, Technical Report, IMECC, Unicamp, JanuaryAndreani, R., Friedlander, A., Santos, S.A., On the resolution of generalized nonlinear complementarity problems (2001) SIAM J. Optimiz., 12, pp. 303-321Andreani, R., Martínez, J.M., Schuverdt, M.L., On the relation between constant positive linear dependence condition and quasinormality constraint qualification (2005) J. Optimiz. Theory Appl., 125, pp. 473-485Bertsekas, D.P., (1999) Nonlinear Programming, , 2 Athena Scientific Belmont, MassachusettsBielchowsky, R.H., Friedlander, A., Gomes, F.A.M., Martínez, J.M., Raydan, M., An adaptive algorithm for bound constrained quadratic minimization (1997) Invest. Operat., 7, pp. 67-102Buss, M., Hashimoto, H., Moore, J.B., Dextrous hand grasping force optimization (1996) IEEE Trans. Robot. Automat., 12, pp. 406-418Chen, J.-S., Tseng, P., (2004) An Unconstrained Smooth Minimization Reformulation of the Second-order Cone Complementarity Problem, , Technical Report, JulyChen, J.-S., (2004) A New Merit Function and Its Related Properties for the Second-order Cone Complementarity Problem, , Technical Report, OctoberFacchinei, F., Soares, J., A new merit function for nonlinear complementarity problems and a related algorithm (1997) SIAM J. Optimiz., 7, pp. 225-247Ferris, M.C., Pang, J.-S., Engineering and economic applications for nonlinear complementarity problems (1997) SIAM Rev., 39, pp. 669-713Fischer, A., A special Newton-type optimization methods (1992) Optimization, 24, pp. 269-284Friedlander, A., Martínez, J.M., Santos, S.A., A new trust region algorithm for bound constrained minimization (1994) Appl. Math. Optimiz., 30, pp. 235-266Geiger, C., Kanzow, C., On the resolution of monotone complementarity problems (1996) Comput. Optimiz. Appl., 5, pp. 155-173Han, L., Trinkle, J.C., Li, Z.X., Grasp analysis as linear matrix inequality problems (2000) IEEE Trans. Robot. Automat., 16, pp. 663-674. , 06Hayashi, S., Yamashita, N., Fukushima, M., A combined smoothing and regularization method for monotone second-order cone complementarity problems (2005) SIAM J. Optimiz., 15, pp. 593-615Krejić, N., Martínez, J.M., Mello, M.P., Pilotta, E.A., Validation of an augmented Lagrangian algorithm with a Gauss-Newton Hessian approximation using a set of hard-spheres problems (2000) Comput. Optimiz. Appl., 16, pp. 247-263Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H., Applications of second-order cone programming (1998) Linear Algebra Appl., 284, pp. 193-228Peng, J.-M., Yuan, Y.-X., Unconstrained methods for generalized complementarity Problems (1997) J. Comput. Math., 15, pp. 253-264Yamada, K., Yamashita, N., Fukushima, M., Pillo, G.D., Giannessi, F., A new derivative-free descent method for the nonlinear complementarity problem (2000) Nonlinear Optimization and Related Topics, pp. 463-489. , Kluwer Academic Publishers NetherlandsWe thank J.M. Martínez for the fruitful discussions and helpful suggestions. We also thank the anonymous referees for their constructive criticism