Classifying convex extremum problems over linear topologies having separation properties

Mathematics Technical Repor

Fenchel-duality and separably-infinite programs

In two recent papers Chabnes, Gbibie, and Kortanek studied a special class of infinite linear programs where only a finite number of variables appear in an infinite number of constraints and where only a finite number of constraints have an infinite number of variables. Termed separably-infinite programs, their duality was used to characterize a class of saddle value problems as a uniextremai principle. We show how this characterization can be derived and extended within Fenchel and Rockafellar duality, and that the values of the dual separably-infinite programs embrace the values of the Fenchel dual pair within their interval. The development demonstrates that the general finite dimensional Fenchel dual pair is equivalent to a dual pair of separably-infinite programs when certain cones of coefficients are closed

Analytical properties of some multiple-source urban diffusion models

It is generally required that the concentration of a certain pollutant at any given point ( x, y ) shall be below a maximum amount defined by a known function S (x, y ). In this paper we analyze different ways of relating the emission rates of polluters with the resultant concentration Q (x, y ) by means of various transfer functions. We discuss the analytical properties of the transfer functions which can be derived from various well-known diffusion models. We also discuss a simple instance of optimization models of a type, introduced by Gorr and Kortanek (1970).

On a New Class of Combinatoric Optimizers for Multi-Product Single-Machine Scheduling

Many people have proposed objective functions, or optimizers, which guide one to schedule a multi-product single stage production system. In this paper we present a whole new class of optimizers, or solution concepts, which generalizes most of the well-known optimizers to date. Our combinatoric formulations are related to a new class of solution concepts for n-person games developed by Charnes-Kortanek [Charnes, A., K. O. Kortanek. 1967. On a class of convex and non-archimedean solution concepts for n-person games. Technical Report No. 22, Department of Operations Research, Cornell University, and Systems Research Memo No. 172, Northwestern University, Evanston, Illinois, March.]. By constructing a combinatoric linear programming problem, where some of the variables are determined by an arbitrary set of permutations, we encompass classical optimizers in one formulation including such concepts as (1) minimizing maximum lateness or tardiness, (2) maximizing minimum lateness or tardiness, (3) minimizing mean lateness or flow time, and (4) random sequencing. More generally, we characterize a new class of optimizers as optimal solutions to specially constructed combinatoric programming problems, including optimizers which are integer in character.

Maximum likelihood estimates with order restrictions on probabilities and odds ratios: A geometric programming approach

The problem of assigning cell probabilities to maximize a multinomial likelihood with order restrictions on the probabilies and/or restrictions on the local odds ratios is modeled as a posynomial geometric program (GP), a class of nonlinear optimization problems with a well-developed duality theory and collection of algorithms. (Local odds ratios provide a measure of association between categorical random variables.) A constrained multinomial MLE example from the literature is solved, and the quality of the solution is compared with that obtained by the iterative method of El Barmi and Dykstra, which is based upon Fenchel duality. Exploiting the proximity of the GP model of MLE problems to linear programming (LP) problems, we also describe as an alternative, in the absence of special-purpose GP software, an easily implemented successive LP approximation method for solving this class of MLE problems using one of the readily available LP solvers