Concave Objective Research Articles

Common Terminal Multicommodity Flow with a Concave Objective Function

We consider graphs in which there are n source nodes, each a source of a different commodity, and a common terminal node. Associated with each commodity is a nonincreasing function $\alpha _i (.),i = 1, \cdots ,n$. An algorithm is presented which yields a flow pattern that maximizes $Z = \Sigma _{i = 1}^n \int_0^{fi} {\alpha _i (x)dx} $, where $f_i $ is the flow value of the ith commodity.

Entrepreneurial Choice over Time under Conditions of Uncertainty

IN THIS PAPER we analyze the way that a firm which operates under uncertainty will behave during the current period and over time. To do so, we develop a new multi-stage model of the firm which has-as we show belowinteresting implications both for economic theory and for econometric research. The paper's main results are summarized in two theorems in Section 2. The first of these theorems shows that an entrepreneur who operates under uncertainty will behave in each period as if he were attempting to maximizesubject to production and financial constraints-a strictly concave one-stage objective function whose arguments represent: (1) the dividends which the firm pays during the period, and (2) the firm's end-of-period asset-and-financing structure. The second theorem shows that, if it is possible to represent the behavior over time of prices and certain other parameters by random processes, then the firm's behavior over time (as reflected during each period in the inputs it uses, the output it produces, the dividends it pays, and the end-ofperiod asset-and-financing structure it selects) can also be represented by well-defined random processes. The proofs of both theorems are given in the Appendix.

Solving Bicriterion Mathematical Programs

It often happens in applications of mathematical programming that there are two incommensurate objective functions to be extremized, rather than just one. One thus encounters bicriterion programs of the form of equation (1), [Formula: see text] where h is an increasing utility function, preferably quasiconcave, defined on outcomes of the concave objective functions f1 and f2, and x is a decision n-vector constrained to the convex set X. It is shown how such programs can be numerically solved if a parametric programming algorithm is available for the parametric subproblem [Formula: see text] A natural byproduct of the calculations is a relevant portion of the “tradeoff curve” between f1 and f2. Outlines of several algorithms for solving equation (1) under various special assumptions and a numerical example are presented to illustrate the application of the theory developed herein. A useful extension is presented that permits nonlinear scale changes to be made on the fı.