Sangho Kum

Penalized complementarity functions on symmetric cones

We show that penalized functions of the Fischer–Burmeister and the natural residual functions defined on symmetric cones are complementarity functions. Boundedness of the solution set of a symmetric cone complementarity problem, based on the penalized natural residual function, is proved under monotonicity and strict feasibility. The proof relies on a trace inequality on Euclidean Jordan algebras.

On the generalized Fischer-Burmeister merit function for the second-order cone complementarity problem

It has been an open question whether the family of merit functions ψp (p > 1), the generalized Fischer-Burmeister (FB) merit function, associated to the second-order cone is smooth or not. In this paper we answer it partly, and show that ψp is smooth for p ∈ (1, 4), and we provide the condition for its coerciveness. Numerical results are reported to illustrate the influence of p on the performance of the merit function method based on ψp.

The penalized Fischer-Burmeister SOC complementarity function

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 R01-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.

A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matrices and . Moreover, an interesting generalization of the geometric mean of and to convex functions was introduced by Atteia and Raissouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raissouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.