Thomas W. Reiland

Generalized invexity for nonsmooth vector-valued mappings

We define four types of invexity for Lipschitz vector-valued mappings from Rp to Rq that generalize previous definitions of invexity in the differentiable setting. After establishing relationships between the various definitions, we show the importance of the concept of nonsmooth invexity in the field of optimization. In particular, we obtain conditions sufficient for optimality in unconstrained and cone-constrained nondifferentiable programming that are weaker than previous conditions presented in the literature; we also obtain weak and strong duality results.

Nonsmooth analysis of vector-valued mappings with contributions to nondifferentiable programming

We define order Lipschitz mappings from a Banach space to an order complete vector lattice and present a nonsmooth analysis for such functions. In particular, we establish properties of a generalized directional derivative and gradient and derive results concerning a calculus of generalized gradients (i.e., calculation of the generalized gradient of f when f = f1 + f2, f = f · 2, etc.). We show the relevance of the above analysis to nondifferentiaile programming by deriving optimality conditions for problems of the form min f(x) subject to x [euro] S. For S arbitrary we state the results in terms of cones of displacement of the feasible region at the optimal point; when S ={x ∊ A|g(x) ∊ B}, we obtain Kuhn-Tucker type results.

Nonsmooth analysis and optimization on partially ordered vector spaces

Interval-Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz-type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.

Nonsmooth Analysis and Optimization for a Class of Nonconvex Mappings

A class of nonconvex operators from a Banach space to an order complete vector lattice, referred to as order Lipschitz mappings, is introduced. Relations with other Lipschitz-type mappings are mentioned and a derivative in the spirit of Clarke’s generalized gradient is presented. Necessary and sufficient optimality conditions are obtained for constrained programming problems whose data functions are order Lipschitz.

The role of accounting in the death penalty debate

Accounting plays a role in the death penalty debate. Accounting studies have compared the cost of capital punishment with the cost of life imprisonment. The studies have all shown that it is cheaper to lock convicted murderers up than it is to employ capital punishment. A survey was conducted for this paper to determine the attitudes of senior level and graduate accounting majors regarding capital punishment. Participants were asked, yes or no, whether they favoured capital punishment. Then they were given information about the accounting studies that point to the greater costs associated with the death penalty. The purpose was to determine if accounting applications and dollar cost, factors in classical, rational and economic choice structure, had an influence on those with an academic investment in the accounting profession. The data collected show that accounting students were statistically consistent with the population in general in favour of the death penalty.

Invexity: an updated survey and new results for nonsmooth functions

The purpose of this paper is to introduce two recent developments in optimization theory which have the potential for significant impact when applied to the many optimization problems in mathematical statistics. The needs of optimization theory have served as the catalyst for the introduction and recent development of two important concepts: invexity and generalized differentiability. The former is a significant generalization of the simple yet powerful concept of convexity, while the latter extends the differentiabe calculus to functions that are not differentiabe in any traditional two-sided sense. For the most part these concepts have been developed independently in the literature. In this paper we provide an updated review of many of the results concerning smooth invex functions and merge the two concepts by introducing invexity for nonsmooth functions. For both real-valued and vector-valued nonsmooth functions we present recent results on optimality conditions, duality, and converse duality. No previ...