Jamie J. Goode

On Symmetric Duality in Nonlinear Programming

In this study we generalize the formulation of symmetric duality introduced by Dantzig, Eisenberg, and Cottle to include the case where the constraints of the inequality type are defined via closed convex cones and their polars. The new formulation retains the symmetric properties of the original programs. Under suitable convexity/concavity assumptions we generalize the known results about symmetric duality. The case where the function involved is strongly convex/strongly concave is also treated and Karamardians result in this case is generalized. As a result, we show that every strongly convex function achieves a minimum value over any closed convex cone at a unique point. Some special cases of symmetric programs are then considered, leading to generalizations of Wolfes duality as well as generalizations of quadratic and linear programming formulations.

A survey of various tactics for generating Lagrangian multipliers in the context of Lagrangian duality

Abstract This study concerns itself with Lagrangian duality for continuous and discrete mathematical programming problems. Properties of the dual function, including subdifferentiability, differentiability, ascent, and steepest ascent directions are discussed. We show the relationship between directions of steepest ascent and shortest subgradients under different normalization constraints. We then discuss various strategies for generating and updating the Lagrangian multiplier vectors in the course of dual optimization.

An algorithm for solving linearly constrained minimax problems

Abstract In this study, we propose an algorithm for solving a minimax problem over a polyhedral set defined in terms of a system of linear inequalities. At each iteration a direction is found by solving a quadratic programming problem and then a suitable step size along that direction is taken through an extension of Armijos approximate line search technique. We show that each accumulation point is a Kuhn-Tucker solution and give a condition that guarantees convergence of the whole sequence of iterations. Through the use of an exact penalty function, the algorithm can be used for solving constrained nonlinear programming. In this case, our algorithm resembles that of Han, but differs from it both in the direction-finding and the line search steps.

A Cutting-Plane Algorithm for the Quadratic Set-Covering Problem

This paper develops an algorithm to solve certain quadratic set-covering problems where the constraint set is of the inequality type. It extends one of Bellmore and Ratliff for linear set-covering problems with involutory bases where cutting planes that exclude both integer and noninteger solutions are generated at each iteration. The new algorithm can be used to solve problems of the equality and mixed types by introducing a penalty term in the objective function. Computational experience with the new algorithm is presented.

Necessary optimality criteria in mathematical programming in the presence of differentiability

Abstract We consider the problem of minimizing a function over a region defined by an arbitrary set, equality constraints, and constraints of the inequality type defined via a convex cone. Under some moderate convexity assumptions on the arbitrary set we develop Optimality criteria of the minimum principle type which generalize the Fritz John Optimality conditions. As a consequence of this result necessary Optimality criteria of the saddle point type drop out. Here convexity requirements on the functions are relaxed to convexity at the point under investigation. We then present the weakest possible constraint qualification which insures positivity of the lagrangian multiplier corresponding to the objective function.

The traveling salesman problem: A duality approach

In this study we formulate the dual of the traveling salesman problem, which extends in a natural way the dual problem of Held and Karp to the nonsymmetric case. The dual problem is solved by a subgradient optimization technique. A branch and bound scheme with stepped fathoming is then used to find optimal and suboptimal tours. Computational experience for the algorithm is presented.

An algorithm for finding the shortest element of a polyhedral set with application to lagrangian duality

Abstract An algorithm is developed which finds the point in a compact polyhedral set with smallest Euclidean norm. At each iteration the algorithm requires knowledge of only those vertices of the set which are adjacent to a current reference vertex. This feature is shown to permit the adoption of the technique to find iteratively the shortest subgradient (i.e. the direction of steepest ascent) of the lagrangian dual function for large scale linear programs. Procedures are presented for finding the direction of steepest ascent in both the equality constraint and the inequality constraint cases of lagrangian duality.

Nonlinear programming without differentiability in banach spaces: necessary and sufficient constraint qualification

The problem under consideration is a maximization problem over a constraint set defined by a finite number of inequality and equality constraints over an arbitrary set in a reflexive Banach space. A generalization of the Kuhn-Tucker necessary conditions is developed where neither the objective function nor the constraint functions are required to be differentiable. A new constraint qualification is imposed in order to validate the optimality criteria. It is shown that this qualification is the weakest possible in the sense that it is necessary for the optimality criteria to hold at the point under investigation for all families of objective functions having a constrained local maximum at this point

OPTIMALITY CRITERIA IN NONLINEAR PROGRAMMING WITHOUT DIFFERENTIABILITY

This paper discusses stationary-point optimality conditions for inequality-constrained nonlinear programming problems where the functions involved are continuous but not necessarily differentiable. We obtain generalizations of the well known Fritz John and Kuhn-Tucker necessary conditions. We also discuss the sufficient conditions for optimality where the usual convexity assumption is replaced by a weaker assumption of “supportability.”

Analysis of inhalation rCBF data.

Relative to other approaches that have been recommended, fitting all head data, solving for a time shift, and including an air passage artifact term in the model significantly improved the estimate of gray matter blood flow by the inhalation technique. A robust algorithm, which incorporates these features, has been developed. Formulas which facilitate implementation of this algorithm are reported. An artifact from large scalp arteries was not significant and does not need to be included in the model.