Jay Rajasekera

Entropy Optimization and Mathematical Programming

Preface. 1. Introduction to Entropy and Entropy Optimization Principles. 2. Entropy Optimization Models. 3. Entropy Optimization Methods: Linear Case. 4. Entropy Optimization Methods: General Convex Case. 5. Entropic Perturbation Approach to Mathematical Programming. 6. Lp-Norm Perturbation Approach: A Generalization of Entropic Perturbation. 7. Extensions and Related Results. Bibliography. Index.

Controlled dual perturbations for posynomial programs

Abstract With geometric programs getting larger and larger, the dual based solution methods which optimize a less complicated problem than the original problem become more important. However there are two major difficulties related to the development of a dual method. One is the non-differentiability of the dual objective function over its feasible region and the other one is the conversion of a dual solution to a primal solution. In this paper we address these issues for the posynomial programs. A well-controlled dual perturbation method which guarantees ϵ-optimal primal dual solution pair is proposed to overcome the nondifferentiability difficulty and a simple linear programming method is introduced to generate a quick dual-to-primal conversion.

Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the “strong duality theorem” and derive a “dual-to-primal” conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find anε-optimal solution for the quadratically constrained minimum cross-entropy analysis.

Controlled perturbations for quadratically constrained quadratic programs

Consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function. The duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear. However, the objective function is not differentiable over the constraint region.In this paper, we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem. Based on this relationship, we establish a perturbation theory to display that a well-controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an ε-optimal dual solution for an arbitrarily small number ε. A simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding ε-optimal primal solution. Moreover, a numerical example is included to illustrate the potential of this controlled perturbation scheme.

On the convex programming approach to linear programming

For a general linear program in Karmarkars standard form, Fang recently proposed a new approach which would find an @e-optimal solution by solving an unconstrained convex dual program. The dual was constructed by applying generalized geometric programming theory to a linear programming problem. In this paper we show that Fangs results can be obtained directly using a simple geometric inequality. The new approach provides a better @e-optimal solution generation scheme in a simpler way.

Deriving an unconstrained convex program for linear programming

Consider a linear programming problem in Karmarkars standard form. By perturbing its linear objective function with an entropic barrier function and applying generalized geometric programming theory to it, Fang recently proposed an unconstrained convex programming approach to finding an epsilon-optimal solution. In this paper, we show that Fangs derivation of an unconstrained convex dual program can be greatly simplified by using only one simple geometric inequality. In addition, a system of nonlinear equations, which leads to a pair of primal and dual epsilon-optimal solutions, is proposed for further investigation.

A due-date assignment model for a flow shop with application in a lightguide cable shop

Due-date assignment is an important problem in a manufacturing facility known as a flow shop, which is capable of making a wide variety of products. In a CIM environment, a flow shop is integrated through a factory information system. In this paper, we describe a due-date assignment model using techniques from optimization and queuing theories. Optimization techniques are discussed for the models application in a lightguide cable manufacturing shop. This model, integrated with an information system, is used daily to assign due-dates for hundreds of cable orders. When the response time for a customer request must be short, this model, after surveying the shop conditions, provides fast due-date assignment.

A NEW APPROACH TO TOLERANCE ALLOCATION IN DESIGN COST ANALYSIS

When components are designed for manufacturing using computer aided design (CAD) tools, computing the minimum manufacturing cost while satisfying other manufacturing constraints becomes a very important issue. It is imperative that computer algorithms be integrated into the design tools in order to compute the costs at various stages of the design process. This paper focuses on the tolerance allocation stage. Assuming the manufacturing cost of a component is an exponential function of the design tolerance, an O(n log2n) algorithm is developed for optimally allocating the total design tolerance among the n constituent components of a product. This algorithm is of most value in CAD and design for manufacturability.

Minimum cross-entropy analysis with entropy-type constraints

Abstract In this paper, we study the cross-entropy optimization problem with entropy-type constraints. A simple geometric inequality is used to derive its dual problem and to show the “strong duality theorem”. We found this geometric dual is a computationally attractive canonical program that is always consistent. A “dual-to-primal” conversion formula and a “dual perturbation” algorithm are also derived for computations.

Outline of a quality plan for industrial research and development projects

Quality issues associated with projects in an industrial research laboratory are analyzed. By recognizing the common characteristics of projects in an industrial research laboratory, the problem of identifying the quality priorities is broken down into a series of levels. For each project at the top level are the personnel who play important roles for the success or failure of the project. At the second level are the stages through which the project progresses during its successful completion. At the third level are the factors contributing to the successful completion of the stages above. At the fourth level are the subfactors or the quality metrics that identify the level of success of the factors above. Through the application of this analytic hierarchy process (AHP), a planning and decision-making tool, the quality plan is constructed, and activities of industrial research projects are brought under a common umbrella to identify the most important quality aspects of customers. The scoring mechanism developed here is useful for quality monitoring purposes. It is noted that the way a quality plan is structured can be especially useful for computerizing activities in order to monitor the progress of a large number of projects. >