The Jacobian consistency of a smoothed Fischer-Burmeister function associated with second-order cones. (English) Zbl 1278.90406
Summary: This paper deals with the second-order cone complementarity problem (SOCCP), which is an important class of problems containing various optimization problems. The SOCCP can be reformulated as a system of nonsmooth equations. For solving this system of nonsmooth equations, smoothing Newton methods are widely used. The Jacobian consistency property plays an important role for achieving a rapid convergence of the methods. In this paper, we show the Jacobian consistency of a smoothed Fischer-Burmeister function. Moreover, we estimate the distance between the subgradient of the Fischer-Burmeister function and the gradient of its smoothing function.
MSC:
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
| 90C53 | Methods of quasi-Newton type |
Keywords:
second-order cone complementarity problem; smoothed Fischer-Burmeister function; Jacobian consistencyReferences:
| [1] | Facchinei, F.; Pang, J.-S., Finite-Dimensional Variational Inequalities and Complementarity Problems (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1062.90001 |
| [2] | Chen, X. D.; Sun, D.; Sun, J., Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems, Comput. Optim. Appl., 25, 39-56 (2003) · Zbl 1038.90084 |
| [3] | Hayashi, S.; Yamashita, N.; Fukushima, M., A combined smoothing and regularization method for monotone second-order cone complementarity problems, SIAM J. Optim., 15, 593-615 (2005) · Zbl 1114.90139 |
| [4] | Chen, J. S.; Tseng, P., An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program., 104, 293-327 (2005) · Zbl 1093.90063 |
| [5] | Fukushima, M.; Luo, Z. Q.; Tseng, P., Smoothing functions for second-order-cone complementarity problems, SIAM J. Optim., 12, 436-460 (2001) · Zbl 0995.90094 |
| [6] | Chen, J. S.; Chen, X.; Tseng, P., Analysis of nonsmooth vector-valued functions associated with second-order cones, Math. Program., 101, 95-117 (2004) · Zbl 1065.49013 |
| [7] | Han, D., On the coerciveness of some merit functions for complementarity problems over symmetric cones, J. Math. Anal. Appl., 336, 727-737 (2007) · Zbl 1124.49027 |
| [8] | Narushima, Y.; Sagara, N.; Ogasawara, H., A smoothing Newton method with Fischer-Burmeister function for second-order cone complementarity problems, J. Optim. Theory Appl., 149, 79-101 (2011) · Zbl 1221.90085 |
| [9] | Pan, S.; Chen, J. S., A damped Gauss-Newton method for the second-order cone complementarity problem, Appl. Math. Optim., 59, 293-318 (2009) · Zbl 1169.49031 |
| [10] | Sun, D.; Sun, J., Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions, Math. Program., 103, 575-581 (2005) · Zbl 1099.90062 |
| [11] | Chen, J. S.; Pan, S., A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs, Pac. J. Optim., 8, 33-74 (2012) · Zbl 1286.90148 |
| [12] | Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley: Wiley New York, (reprinted by SIAM, Philadelphia, PA, 1990) · Zbl 0727.90045 |
| [13] | Chen, X.; Qi, L.; Sun, D., Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67, 519-540 (1998) · Zbl 0894.90143 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
