Conditions for Error Bounds and Bounded Level Sets of Some Merit Functions for the Second-Order Cone Complementarity Problem
- 145 Downloads
Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform P *-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called R 01 or R 02-functions to keep the property of bounded level sets.
KeywordsError bounds Jordan products Level sets Merit functions Second-order cones Spectral factorization
Unable to display preview. Download preview PDF.
- 4.Alizadeh, F., Schmieta, S.: Symmetric cones, potential reduction methods, and word-by-word extensions. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming, pp. 195–233. Kluwer Academic, Boston (2000) Google Scholar
- 17.Fischer, A.: Solution of the monotone complementarity problem with locally Lipschitzian functions. Math. Program. 76, 513–532 (1997) Google Scholar
- 18.Yamashita, N., Fukushima, M.: A new merit function and a descent method for semidefinite complementarity problems. In: Fukushima, M., Qi, L. (eds.) Reformulation—Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 405–420. Kluwer Academic, Boston (1999) Google Scholar
- 19.Luo, Z.-Q., Tseng, P.: A new class of merit functions for the nonlinear complementarity problems. In: Ferris, M.C., Pang, J.-S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 204–225. SIAM, Philadelphia (1997) Google Scholar
- 26.Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York (2003) Google Scholar