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.

1. Motivation and basic concepts In this note, we consider two signomial functions whose convexity play important roles in some recent papers [4, 6, 7, 8, 9] dealing with geometric programming problems. However, the verifications therein contain some certain flaws and those incorrect arguments are repeatedly appeared and cited. From point of scientific researchs view, we hereby provide correct proofs for them.

We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the FischerBurmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function _FB as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan <i>P</i>-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (<i>An unconstrained smooth minimization reformulation of the second-order cone complementarity problem</i>, Math. Program. 104(2005), pp

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.

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x H, a spectral decomposition is introduced, and for any function f: R R, we define a corresponding vector-valued function f H (x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.