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pp. Price: $21.95. This text on nonlinear programming is radically different from others on the market. It presents a unified optimality theory for both convex and nonconvex programs based on constructive methods, in particular, feasible direction methods. The optimality conditions derived for convex programs are "complete characterizations" (necessary and sufficient). Those given for nonconvex programs are not complete characterizations, but are closer than the neoclassical results. The book presents much original research and makes available for the first time in a text many recent results that have only appeared in technical journals.Part one, covering almost three-quarters of the text, is devoted to convex programs. Chapter one begins with section one on sets of feasible directions and related sets of directions and proves various properties of these sets. Section two gives needed results from convex analysis. The authors state the "the reader should be at least superficially familiar with . . . convex sets and convex functions." This is an understatement. A good familiarity with these topics is essential to understanding the material.Section three is the longest and most important of the book. Here the necessary and sufficient, primal and dual optimality conditions for convex programs are derived in a concise, logical manner. Primal conditions are given in terms of feasible solutions and various sets of directions at these feasible solutions. Using certain separation theorems or theorems of the alternative, dual optimality conditions are derived as well. Then both the primal and dual versions of the Karush-Kuhn-Tucker sufficiency criteria, and the Fritz John necessary condition are derived, as special cases, from the newly proven results. Section four introduces an interesting and useful parametric approach to the optimality conditions of Section three that is particularly suitable for numerical procedure of the previous section. Chapter two concludes with Section eight on contained mality have been derived.Section five reviews the method of feasible directions, and introduces an effective antijamming procedure. A modified direction-finding generator in Section six overcomes numerical difficulties that may occur on the boundary of the feasible region, and is based upon the parametric approach of Section four. Section seven describes and illustrates a parametric feasible direction algorithm for convex programs that incorporates the modified procedure of the previous section. Chapter two concludes with section eight on contained line search problems. Considerable effort is spent examining the MINIROOT problem of finding the smallest roots of a collection of functions in a given interval.Chapter three derives necessary and sufficient conditions for problems in which the Slater condition does not hold, and so in which the Karush-Kuhn-Tucker conditions generally do not hold either. Section nine investigates Pareto optimality, Section ten looks at lexicographic multicriteria programs, and, finally, Section eleven...

This paper is a survey of basic results that characterize optimality in single-and multiobjective mathematical programming models. Many people believe, or want to believe, that the underlying behavioural structure of management, economic, and many other systems, generates basically 'continuous' processes. This belief motivates our definition and study of optimality, termed 'structural' optimality. Roughly speaking, we say that a feasible point of a mathematical programming model is structurally optimal if every improvement of the optimal value function, with respect to parameters, results in discontinuity of the. corresponding feasible set of decision variables. This definition appears to be more suitable for many applications and it is also more general than the usual one: every optimum is a structural optimum but not necessarily vice versa. By characterizing structural optima, we obtain some new, and recover the familiar, optimality conditions in nonlinear programming.The paper is self-contained. Our approach is geometric and inductive: we develop intiution by studying finite-dimensional models before moving on to abstract situations.AMS subject clmsilieatiom (1980). 90C25, 90C48, 90C31, 54C10. 54C60, 49B27, 49B50.

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