Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems. This work was supported by Grant R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization

We present a new duality theory to treat convex optimization problems and we prove that the geometric duality used by Scott and Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions to achieve strong duality are considered and optimality conditions are derived. Next, we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach that we present. Communicated by V. F. Demyanov The first author was supported in part by Gottlieb Daimler and Karl Benz Stiftung 02-48/99. The second author was supported in part by Karl und Ruth Mayer Stiftung.     

Superlinearly Convergent Trust-Region Method without the Assumption of Positive-Definite Hessian

In this paper, we reinvestigate the trust-region method by reformulating its subproblem: the trust-region radius is guided by gradient information at the current iteration and is self-adaptively adjusted. A trust-region algorithm based on the proposed subproblem is proved to be globally convergent. Moreover, the superlinear convergence of the new algorithm is shown without the condition that the Hessian of the objective function at the solution be positive definite. Preliminary numerical results indicate that the performance of the new method is notable. The authors thank the Associate Editor and two anonymous referees for valuable comments and suggestions. This work was supported by the National Science Foundation of China, Grant 70302003. Communicated by T. Rapcsak