Abstract
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.
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References
Alizadeh F. and Goldfarb D. (2003). Second-order cone programming. Math. Program. 95: 3–51
Andreani, R., Friedlander, A., Mello, M.P., Santos, S.A.: Generalized second-order complementarity problems: theory and numerical experiments. Technical Report, IMECC, Unicamp, January (2005) Available at http://www.ime.unicamp.br/andreani
Andreani R., Friedlander A. and Santos S.A. (2001). On the resolution of generalized nonlinear complementarity problems. SIAM J. Optimiz. 12: 303–321
Andreani R., Martínez J.M. and Schuverdt M.L. (2005). On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J. Optimiz. Theory Appl. 125: 473–485
Bertsekas D.P. (1999). Nonlinear Programming. Athena Scientific, Belmont, Massachusetts
Bielchowsky R.H., Friedlander A., Gomes F.A.M., Martínez J.M. and Raydan M. (1997). An adaptive algorithm for bound constrained quadratic minimization. Invest. Operat. 7: 67–102
Buss M., Hashimoto H. and Moore J.B. (1996). Dextrous hand grasping force optimization. IEEE Trans. Robot. Automat. 12: 406–418
Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Technical Report, July (2004)
Chen, J.-S.: A new merit function and its related properties for the second-order cone complementarity problem. Technical Report, October (2004).
Facchinei F. and Soares J. (1997). A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optimiz. 7: 225–247
Ferris M.C. and Pang J.-S. (1997). Engineering and economic applications for nonlinear complementarity problems. SIAM Rev. 39: 669–713
Fischer A. (1992). A special Newton-type optimization methods. Optimization 24: 269–284
Friedlander A., Martínez J.M. and Santos S.A. (1994). A new trust region algorithm for bound constrained minimization. Appl. Math. Optimiz. 30: 235–266
Geiger C. and Kanzow C. (1996). On the resolution of monotone complementarity problems. Comput. Optimiz. Appl. 5: 155–173
Han L., Trinkle J.C. and Li Z.X. (2000). Grasp analysis as linear matrix inequality problems. IEEE Trans. Robot. Automat. 16(06): 663–674
Hayashi S., Yamashita N. and Fukushima M. (2005). A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optimiz. 15: 593–615
Krejić N., Martínez J.M., Mello M.P. and Pilotta E.A. (2000). Validation of an augmented Lagrangian algorithm with a Gauss–Newton Hessian approximation using a set of hard-spheres problems. Comput. Optimiz. Appl. 16: 247–263
Lobo M., Vandenberghe L., Boyd S. and Lebret H. (1998). Applications of second-order cone programming. Linear Algebra Appl. 284: 193–228
Peng J.-M. and Yuan Y.-X. (1997). Unconstrained methods for generalized complementarity Problems. J. Comput. Math. 15: 253–264
Yamada, K., Yamashita, N., Fukushima,M.: A new derivative-free descent method for the nonlinear complementarity problem. In: Pillo, G.D., Giannessi, F. (eds.) Nonlinear Optimization and Related Topics, pp. 463–489. Kluwer Academic Publishers, Netherlands (2000)
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Supported by FAPESP (01/04597-4), CNPq, PRONEX-Optimization, FAEPEX-Unicamp.
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Andreani, R., Friedlander, A., Mello, M.P. et al. Box-constrained minimization reformulations of complementarity problems in second-order cones. J Glob Optim 40, 505–527 (2008). https://doi.org/10.1007/s10898-006-9109-x
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DOI: https://doi.org/10.1007/s10898-006-9109-x