Primal-dual Relationship Research Articles

Thirty years with spatial and intertemporal economics

In this paper the practical origin and development of the spatial and intertemporal price equilibrium (SIPE) models are chronologically followed around the personal life of the author. The fact that both the Marshallian (quantity) formulation and the Walrasian (price) formulation have been with the Takayama-Judge models in the primal-dual (purified-duality) relationship from the outset is pointed out. Historical use and abuse of large scale models in the past two decades, closely related to the SIPE models, are discussed. The author acknowledges recent developments of methodologies by younger generation researchers in this field and concludes that along with the recent development of high-powered computers and softwares, there will be a number of interesting and relevant SIPE model applications in the near future.

Maximum likelihood, entropy maximization, and the geometric programming approaches to the calibration of trip distribution models

Geometric programming is used to establish a formal primal-dual relationship between the maximum likelihood and the entropy maximization formulations of the trip distribution model. This relationship produces solution characteristics which agree with well known results. It provides an efficient procedure for interpreting and solving the model. A systematic method for conducting post-optimal sensitivity analysis is also available. The development is illustrated through a simple example.

Quadratic Posylognomials: An Extension of Posynomial Geometric Programming

Abstract The theory of geometric programming is extended to include a new function with logarithmic exponents. The function, defined as a Quadratic Posylognomial (QPL), is a series of nonlinear product terms with positive coefficients and positive variables. A QPL may be created by adding a linear function of the logarithms of the variables to the constant exponents of a posynomial. The logarithm of each nonlinear term is a quadratic form in the logarithms of the primal variables, whereas the logarithm of a general term of a posynomial is linear in the logarithms of the variables. The primal-dual relationships and necessary conditions are developed, and special cases where sufficient conditions exist are derived. The theory is the basis for solution of a machining economics problem using a more accurate QPL tool life equation and has the potential for solution of other engineering problems where a posynomial formulation is inadequate. The extended theory may also prove useful for condensing posynomial geo...

Complementary variational principles via generalised geometric programming

1. INTRODUCTION An effective way of solving physical problems is often provided by a variational representation. Such representations arise either naturally as physical principles e.g. Principle of Least Action, or as an alternative means to directly solving the dynamical equations of the system. In many cases one can associate with a variational principle an equivalent one, termed a complementary principle [l]. One then has the option of solving the problem via either principle or perhaps utilizing both jointly as an algorithmic stopping criteria i.e. the principles provide upper and lower bounds to the solution. For some time now, mathematical programmers have been interested in developing dual problems to various primal optimization problems. These are now a prominent part of optimization theory and are the basis for many efficient computational algorithms. In fact, the primal-dual relationships in function space studied by mathematical programmers are the same as the variational principle-complementary principle relationships studied by physicists. Whittle [2] has noted this relationship for a physical problem using Lagrangian type duality. Recently powerful and general approaches to duality theory for convex functionals have been developed by Rockafellar [3] and Scott and Jefferson [4]. Here we propose to illustrate the application of our theory to the generation of complementary variational principles for two different physical problems. A significant feature of our approach is a convenient method for handling inequality constraints by way of a cone.

Optimal Flexible Pavement Design

This paper proposes a model and solution procedure for the optimal design of flexible pavements that finds the best cost achievable for each possible combination of materials without actually computing the layer thicknesses. The primal-dual relationships and complementary slackness conditions of linear programming are used to reduce the required number of computations necessary to obtain the optimal design. The methodology can also be used for pavement structures with more than three layers and can be extended to cases in which there are also design specifications on the overall total thickness of the structure.

The discrete complementary variational principle and optimal control systems

A discrete complementary variational principle is developed and applied to linear and nonlinear discrete-time optimal control systems. Using the variational approach, a primal-dual relationship is established. This relationship provides a precise measure of system suboptimality independent of any a priori knowledge of the optimal solution.