B. M. Glover

Decreasing Functions with Applications to Penalization

The theory of increasing positively homogeneous functions defined on the positive orthant is applied to the class of decreasing functions. A multiplicative version of the inf-convolution operation is studied for decreasing functions. Modified penalty functions for some constrained optimization problems are introduced that are in general nonlinear with respect to the objective function of the original problem. As the perturbation function of a constrained optimization problem is decreasing, the theory of decreasing functions is subsequently applied to the study of modified penalty functions, the zero duality gap property, and the exact penalization.

CUTTING ANGLE METHODS IN GLOBAL OPTIMIZATION

Abstract A generalization of the cutting plane method from convex minimization is proposed applicable to a very broad class of nonconvex global optimization problems. Convergence results are described along with details of the initial numerical implementation of the algorithms. In particular, we study minimization problems in which the objective function is increasing and convex-along-rays.

Extended Lagrange And Penalty Functions in Continuous Optimization

We consider problems of continuous constrainedoptimization in finite dimensional space and studygeneralized nonlinear Lagrangian and penaltyfunctions which are formed by increasing convolutionfunctions

Complete Characterizations of Global Optimality for Problems Involving the Pointwise Minimum of Sublinear Functions

Necessary and sufficient global optimality conditions are presented for certain non-convex minimization problems subject to inequality constraints that are expressed as the pointwise minimum of sublinear (MSL) functions. A generalized Farkas lemma for inequality systems with MSL functions plays a crucial role in presenting the conditions in dual forms. Applications to certain multiplicative sublinear programming problems and fractional programming problems are also given.

A Farkas lemma for difference sublinear systems and quasidifferentiable programming

A new generalized Farkas theorem of the alternative is presented for systems involving functions which can be expressed as the difference of sublinear functions. Various other forms of theorems of the alternative are also given using quasidifferential calculus. Comprehensive optimality conditions are then developed for broad classes of infinite dimensional quasidifferentiable programming problems. Applications to difference convex programming and infinitely constrained concave minimization problems are also discussed.

Dual conditions characterizing optimality for convex multi-objective programs

Asymptotic necessary and sufficient conditions for a point to be a Pareto minimum, and weak minimum (proper minimum) for a convex multi-objective program are given without a regularity condition. It is further shown that, in the cases of weak minimum and single objective function, the asymptotic dual conditions reduce to nonasymptotic optimality conditions under Slaters constraint qualification. The results are applied to multi-objective quadratic and linar programming problems. Numerical examples are given to illustrate the nature of the conditions.

Extended Lagrange and Penalty Functions in Optimization

We consider nonlinear Lagrange and penalty functions for optimization problems with a single constraint. The convolution of the objective function and the constraint is accomplished by an increasing positively homogeneous of the first degree function. We study necessary and also sufficient conditions for the validity of the zero duality gap property for both Lagrange and penalty functions and for the exact penalization. We also study the so-called regular weak separation functions.

Generalized convexity in nondifferentiable programming

For an abstract mathematical programming problem involving quasidifferentiable cone-constraints we obtain necessary (and sufficient) optimality conditions of the Kuhn-Tucker type without recourse to a constraint qualification. This extends the known results to the non-differentiable setting. To obtain these results we derive several simple conditions connecting various concepts in generalized convexity not requiring differentiability of the functions involved.

Nonlinear Unconstrained Optimization Methods: A Review

In this review paper, we report recent results in the study of nonlinear unconstrained optimization methods, such as nonlinear penalty function method and nonlinear Lagrangian method. One important feature of this line of research is that unconstrained optimization problems may be nonlinear in the cost function of the original constraint problem. Results such as zero duality gaps, exact penalization, and the existence of nonlinear Lagrange multipliers are reviewed.

On Quasidifferentiable Functions and Non-Differentiable Programming

For vector-valued difference sublinear functions a Farkas type solvability result is established in a non-homogeneous form. This is used to establish both necessary and sufficient optimality conditions of the Karush-Kuhn-Tucker type for programming problems involving quasidifferentiable constraint functions, in the sense of Demyanov, and differentiable objective functions