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This paper undertakes to show that detrimental externalities tend to induce non-convexity of the social production possibility set. In particular we show that if externalities are sufficiently strong, convexity conditions must break down.
It is not our objective here to review in any detail the difficulties caused by non-convexity. Some of these consequences have long been recognized and are widely known.[2] However, until the recent appearance of papers by Starett [8], Portes [5], Kolm [3] and Baumol [2], it was apparently not recognized that externalities themselves are a source of non-convexity. These more recent writings suggest more than one connection between the two phenomena. However, one particularly straight-forward relationship seems to have received little or no attention. With sufficiently strong interactive effects non-convexity follows from the simple fact that if either of two mutually interfering activities is operated at zero level the other suffers no hindrance. The goal of this paper is to explore this phenomenon and show very dearly how it is that sufficiently severe detrimental externalities of the form described and non-convexity necessarily go together. [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 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).

This article evaluates some new developments in economic theory. According to economist Tjalling C. Koopmans, the theory of convex sets, relevant in activity analysis, can be applied to reveal the common logical structure of received economic theories of quite diverse origin. Objecting to the production function because it already presupposes output maximization, he recommends that one should start with the range of technological alternatives. This is a criticism, which rebounds, for his process also presumes a fast relation between input and output, almost regardless of organization, managerial acumen or labor discipline. On these hoary matters, not to mention capital aspects, the mathematical statement may be less flexible than verbal accounts. Activity analysts must live with the charge of being mechanistic, even if it is not always a harsh indictment. For mathematical convenience, Koopmans postulates constant returns to scale. Yet activity concepts have been used to urge that divisibility postulates do not insure constant returns to scale.