We investigate a one-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) which is closely related to the popular FischerBurmeister (FB) merit function and natural residual merit function. In fact, it will reduce to the FB merit function if the involved parameter <i></i> equals 2, whereas as <i></i> 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

No abstract uploaded!

In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone complementarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian <i>R</i> ₀₁-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.

In last decades, there has been much effort on the solution and the analysis of the nonlinear complementarity problem (NCP) by reformulating NCP as an unconstrained minimization involving an NCP function. In this paper, we propose a family of new NCP functions, which include the Fischer-Burmeister function as a special case, based on a <i>p</i>-norm with <i>p</i> being any fixed real number in the interval (1,+), and show several favorable properties of the proposed functions. In addition, we also propose a descent algorithm that is indeed derivative-free for solving the unconstrained minimization based on the merit functions from the proposed NCP functions. Numerical results for the test problems from MCPLIB indicate that the descent algorithm has better performance when the parameter <i>p</i> decreases in (1,+). This implies that the merit functions associated with <i>p</i>(1,2), for example <i>p</i>=1.5, are more effective

Necessary conditions of optimality are derived for multiobjective optimal control problems with free end-time, in which the dynamics constraint is modeled as a nonconvex differential inclusion. The obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems.