Elmor L. Peterson

Geometric programming with signomials

The difference of twoposynomials (namely, polynomials with arbitrary real exponents, but positive coefficients and positive independent variables) is termed asignomial.Each signomial program (in which a signomial is to be either minimized or maximized subject to signomial constraints) is transformed into an equivalent posynomial program in which a posynomial is to be minimized subject only to inequality posynomial constraints. The resulting class of posynomial programs is substantially larger than the class of (prototype)geometric programs (namely, posynomial programs in which a posynomial is to be minimized subject only to upper-bound inequality posynomial constraints). However, much of the (prototype) geometric programming theory is generalized by studying theequilibrium solutions to thereversed geometric programs in this larger class. Actually, some of this theory is new even when specialized to the class of prototype geometric programs. On the other hand, all of it can indirectly, but easily, be applied to the much larger class of well-posedalgebraic programs (namely, programs involving real-valued functions that are generated solely by addition, subtraction, multiplication, division, and the extraction of roots).

Computing Economic Equilibria on Affine Networks with Lemke's Algorithm

Consider a multicommodity transhipment problem where the prices at each location are an affine function of the supplies and demands at that location and the shipping costs are an affine function of the quantities shipped. A system of prices, supplies, demands, and shipments is defined to be an equilibrium, if there is a balance in the shipments, supplies, and demands of goods at each location, if local prices do not exceed the cost of importing, and if shipments are price efficient. Lemkes algorithm is used to compute an equilibrium.

Symmetric Duality for Generalized Unconstrained Geometric Programming

The conjugate transform is used to generalize, symmetrize, and further study Duffin’s original formulation of duality for unconstrained geometric programming. This study provides new economic interpretations for the geometric dual problem; and it yields new theorems concerning the existence,uniqueness and characterization of optimal solutions. The economic interpretations come from a new closed-form solution to a related economically interesting class of convex programming problems.

Geometric Programming: Duality in Quadratic Programming and

The duality theory of geometric programming as developed by Duflin, Peterson and Zener [7] is based on abstract properties shared by certain classical inequalities, such as Cauchy’s arithmetic-geometric mean inequality and Holder’s inequality. Inequalities with these abstract properties have been termed “geometric inequalities” [7, p. 195]. In this sequence of papers [15], [16], [17] a new geometric inequality is established and used to extend the “refined duality theory” for “posynomial” geometric programs [6] and [7, Chap. VI]. This extended duality theory treats both “quadratically-constrained quadratic programs” and “

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-Approximation II (Canonical Programs)

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The Duality between Suboptimization and Parameter Deletion

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Traffic Equilibria Analysed Via Geometric Programming

-approximation (regression) problems” through a rather novel and unified formulation of these two classes of programs. This work generalizes some of the work of others on (linearly-constrained) quadratic programs and provides a new explicit formulation of duality for constrained approximation problems. Duality theories have been developed for a large class of prog...

The proximity of (algebraic) geometric programming to linear programming

The main tool used in studying the influence of given perturbation parameters on a given optimization problem is, of course, the corresponding Rockafellar dual problem in which the dual variables are in a one-to-one correspondence with the parameters. However, even when the given optimization problem is defined in terms of simple formulas, there are many important cases where the corresponding dual problem cannot be computed explicitly in terms of simple formulas unless certain additional uninteresting perturbation parameters are included. Under a very weak hypothesis, the main theorem to be given here asserts that a suboptimization of the resulting dual problem over the additional uninteresting dual variables produces the desired dual problem i.e., the dual problem that corresponds to the original optimization problem without the additional perturbation parameters. In addition to its uses in parametric analysis this theorem can be used to show that various decomposition principles are dual to one another and hence are essentially equivalent.

Optimality Conditions in Generalized Geometric Programming

The “traffic-assignment problem” consists of predicting “Wardrop-equilibrium flows” on a roadway network when origin-to-destination “input flows” are specified. The “demand-equilibrium problem” consists of predicting those input flows that place the network in a state of “economic equilibrium” when the input flows are related via given travel-demand (feedback) curves to the resulting Wardrop-equilibrium origin-to-destination “travel costs”.