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This paper discusses stationary-point optimality conditions for inequality constrained nonlinear programming problems where the functions involved are continuous but not necessarily differentiable. We obtain generalizations of the well known Fritz John and Kuhn-Tucker necessary conditions. We also discuss the sufficient conditions for optimality where the usual convexity assumption is replaced by a weaker assumption of 'supportability.' [ABSTRACT FROM AUTHOR]

This paper establishes necessary and sufficient conditions for constraint set stability requiring neither convex constraint functions nor convex constraint sets. These conditions then lead to a sufficiency result for the continuity of the optimal objective values as the right-hand side varies. Applications to quasiconvex functions are presented. [ABSTRACT FROM AUTHOR]

Certain types of quadratic programs with linear constraints have the property that an extreme point of the convex set of feasible solutions is an optimal solution. This paper presents a procedure for solving these problems, it involves determining a related linear program having the same constraints, the extreme-point-ranking approach of MURTY is then applied to this linear program to obtain an optimum solution to the quadratic program. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm for the solution of the convex programming problem using the simplex method. The algorithm is computationally very simple, requiring the solution of a single linear programming problem which can be accomplished with only slight modification of existing computer codes for the revised simplex method. [ABSTRACT FROM AUTHOR]

This paper is addressed to the problem of locating the absolute and vertex centers (minimax criterion) of an undirected tree graph. Based upon a convexity property of the criterion function a very simple and efficient algorithm is offered that locates the minimax point by locating first a maximax point. The minimax is at the mid-point of the maximum path from the maximax point. The vertex center is located simultaneously. [ABSTRACT FROM AUTHOR]

This article discusses the convexity of quadratic programs over convex sets. In problems involving the minimization of a quadratic function &b.phiv;(x) = x'Dx +c'x+α over a convex set 2, one hopes that &b.phiv; is a convex function. When this is the case, its local minima are global minima, a property that is true of all convex functions over convex sets. It is intuitively clear that &b.phiv;(x) is convex on S if and only if x'Dx is convex there too. The convexity of quadratic forms is related to positive semidefiniteness. The purpose of this article is to draw attention to this relation and to suggest that it may be helpful in establishing the convexity of a quadratic program. Convexity on the carrying plane V(S) is equivalent to convexity on the linear subspace L(S) parallel to V(S). Finally, the convexity of &b.phiv; on L(S) means non-negativity there, that is, positive semidefiniteness subject to homogeneous linear equality constraints. Therefore if &b.phiv; is convex on S, it is convex on the larger set V(S).

Both the comments by F. Trenery Dolbear and by Talbot Page and John Ferejohn in June,1974, issue of the "Review," reemphasize an important point concerning the achievement of a Pareto optimum via constant per unit taxes. In the author's original paper he intends to bring out two points: to set down a set of sufficient conditions under which the constant per unit taxes or subsidies envisioned in traditional discussions of Pigouvian schemes would correspond to a Pareto optimal outcome for an economy; and to formally introduce "externality technologies" into a model framework. As with any results developed on sufficient conditions, the assumptions about convexity are stronger than necessary to obtain the theoretical results that that the author stated, but they are not irrelevant if one actually contemplates calculation. All algorithms which might be employed in solving a particular applied problem seek a solution by various myopic gropings which are assured of finding a globally optimal solution only under rather stringent convexity conditions.

This article focuses on the class of homothetic isoquant production functions (HIPF). Programming for computers is exposed to subtle nuances because it remains something of an art. In this manner bugs take treacherous disguises; such a crept into the class of HIPF. This article hereby proceed to bug extermination. The end empirical results obtained in that paper are not substantially changed; however, the test for convexity is altered.

This note focuses on two new and related cut strategies for integer programming: the ‘convexity-cut’ and ‘cut-search’ strategies. The fundamental notions underlying the convexity-cut approach are due to RICHARD D. YOUNG and EGON BALAS, whose ‘hypercylindrical’ and ‘intersection’ cuts provide the conceptual starting points for the slightly more general framework developed here. We indicate the utility of our framework (and hence the importance of the original Young and Balas ideas) by specifying a variety of new cuts that can be obtained from it. The second new strategy, cut search, shares with the convexity-cut strategy the notion of generating a cut by passing a hyperplane through the terminal endpoints of edges extended from the vertex of a cone. However, whereas the convexity-cut approach determines the extensions of these edges by reference to a convex set that contains the vertex of the cone in its interior, the cut-search approach determines these extensions by reference to associations between certain ‘proxy’ sets of points (e.g., collections of hyperplanes) and ‘candidate solutions’ to the integer program. The cut-search approach typically involves more work than the convexity-cut approach, but offers the chance to identify feasible solutions in the process, and can sometimes also yield somewhat stronger cuts than the convexity cuts. [ABSTRACT FROM AUTHOR]

The article presents a mathematical analysis pertaining to the average number of backorders incurred per year under an economic order policy. Specifically the author expands on theorems relative to inventory systems proposed by G. Hadley and T. M. Whitin. The author contends that the functions represented are not generally convex and therefore can lead to a failure of the optimization techniques proposed. The author presents an alternate optimization technique to correct the problem. Several equations are presented to support his theory.