Two classes of merit functions for the second-order cone complementarity problem (original) (raw)
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A new merit function and its related properties for the second-order cone complementarity problem
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Recently, J.-S. Chen and P. Tseng extended two merit functions for the nonlinear complementarity problem (NCP) and the semidefinite complementarity problem (SDCP) to the second-order cone commplementarity problem (SOCCP) and showed several favorable properties. In this paper, we extend a merit function for the NCP studied by Yamada, Yamashita, and Fukushima to the SOCCP and show that the SOCCP is equivalent to an unconstrained smooth minimization via this new merit function. Furthermore, we study conditions under which the new merit function provides a global error bound which plays an important role in analyzing the convergence rate of iterative methods for solving the SOCCP; and conditions under which the new merit function has bounded level sets which ensures that the sequence generated by a descent method has at least one accumulation point.
A one-parametric class of merit functions for the symmetric cone complementarity problem
Journal of Mathematical Analysis and Applications, 2009
In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [18] to the SCCP. By exploiting the Cartesian P -properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R 02 -property. All of these results generalize some recent important works in ] under a unified framework.
A one-parametric class of merit functions for the second-order cone complementarity problem
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We investigate a one-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) which is closely related to the popular Fischer-Burmeister (FB) merit function and natural residual merit function. In fact, it will reduce to the FB merit function if the involved parameter τ equals 2, whereas as τ tends to zero, its limit will become a multiple of the natural residual merit function. In this paper, we show that this class of merit functions enjoys several favorable properties as the FB merit function holds, for example, the smoothness. These properties play an important role in the reformulation method of an unconstrained minimization or a nonsmooth system of equations for the SOCCP. Numerical results are reported for some convex second-order cone programs (SOCPs) by solving the unconstrained minimization reformulation of the KKT optimality conditions, which indicate that the FB merit function is not the best. For the sparse linear SOCPs, the merit function corresponding to τ = 2.5 or 3 works better than the FB merit function, whereas for the dense convex SOCPs, the merit function with τ = 0.1, 0.5 or 1.0 seems to have better numerical performance.
Merit Functions and Nonsmooth Functions for the Second-order Cone Complementarity Problem
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There are three popular approaches, merit functions approach, nonsmooth functions approach, and smoothing methods approach, for the second-order cone complementarity problem (SOCCP). In this article, we survey recent results on the most popular approach, merit functions approach. In particular, we investigate and present several merit functions for SOCCP. We also cdot\cdotcdot propose some open questions for future study.
Construction of merit functions for ellipsoidal cone complementarity problem
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Nonsymmetric cone program and its corresponding complementarity problem have long been mysterious to optimization researchers because of no unified analysis technique to handle these cones. Nonetheless, merit function approach is a popular method to deal with general conic complementarity problem, for which the key lies on constructing appropriate merit functions. In this paper, we focus on a special class of nonsymmetric cone complementarity problem, that is, the ellipsoidal cone complementarity problem (ECCP). We not only show the readers how to construct merit functions for solving the ellipsoidal cone complementarity problem, but also we study the conditions under which the level sets of the corresponding merit functions are bounded. In addition, we assert that these merit functions provide an error bound for the ellipsoidal cone complementarity problem. All these results build up a theoretical basis for the merit function method for solving ellipsoidal cone complementarity prob...
Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems
Journal of Optimization Theory and Applications, 2009
In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P -property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P -property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in .
2006
] where an NCP-function and a descent method were proposed for the nonlinear complementarity problem. An unconstrained reformulation was formulated due to a merit function based on the proposed NCP-function. We continue to explore properties of the merit function in this paper. In particular, we show that the gradient of the merit function is globally Lipschitz continuous which is important from computational aspect. Moreover, we show that the merit function is SC 1 function which means it is continuously differentiable and its gradient is semismooth. On the other hand, we provide an alternative proof, which uses the new properties of the merit function, for the convergence result of the descent method considered in [
Two Classes of Merit Functions for Infinite-Dimensional Second Order Complimentary Problems
Numerical Functional Analysis and Optimization, 2010
In this article, we extend two classes of merit functions for the second-order complementarity problem (SOCP) to infinite-dimensional SOCP. These two classes of merit functions include several popular merit functions, which are used in nonlinear complementarity problem, (NCP)/(SDCP) semidefinite complementarity problem, and SOCP, as special cases. We give conditions under which the infinite-dimensional SOCP has a unique solution and show that all these merit functions provide an error bound for infinite-dimensional SOCP and have bounded level sets. These results are very useful for designing solution methods for infinite-dimensional SOCP.
An unconstrained smooth minimization reformulation of the second-order cone complementarity problem
Mathematical Programming, 2005
In honor of Terry Rockafellar on his 70th birthday Abstract. A popular approach to solving the nonlinear complementarity problem (NCP) is to reformulate it as the global minimization of a certain merit function over IR n . A popular choice of the merit function is the squared norm of the Fischer-Burmeister function, shown to be smooth over IR n and, for monotone NCP, each stationary point is a solution of the NCP. This merit function and its analysis were subsequently extended to the semidefinite complementarity problem (SDCP), although only differentiability, not continuous differentiability, was established. In this paper, we extend this merit function and its analysis, including continuous differentiability, to the second-order cone complementarity problem (SOCCP). Although SOCCP is reducible to a SDCP, the reduction does not allow for easy translation of the analysis from SDCP to SOCCP. Instead, our analysis exploits properties of the Jordan product and spectral factorization associated with the second-order cone. We also report preliminary numerical experience with solving DIMACS second-order cone programs using a limited-memory BFGS method to minimize the merit function.