Mokhtar S. Bazaraa

On the choice of step size in subgradient optimization

Abstract This paper recommends some procedures for the selection of step sizes in the context of subgradient optimization. The first of these procedures is developed in detail in this study and is a theoretically convergent scheme. This method has two phases, the first phase is designed to accelerate the solution procedure towards an optimal solution, while the second phase helps to close in on an optimal solution. A second technique recommended is a simple-minded scheme which, although not theoretically convergent, seems to be computationally very efficient. These two methods are shown to compare favorably with Held, Wolfe and Crowders scheme for prescribing step sizes. We also suggest some modifications of the latter scheme to make it computationally more efficient.

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.

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.”

On the convergence of a class of algorithms using linearly independent search directions

Powell has shown that the cyclic coordinate method with exact searches may not converge to a stationary point. In this note we consider a more general class of algorithms for unconstrained minimization, and establish their convergence under the assumption that the objective function has a unique minimum along any line.

A property regarding degenerate pivots for linear assignment networks

Given an extreme point of an assignment polytope of size m, there exist 2m−1mm−2 alternative basis representations according to the choice of the m - 1 degenerate basic variables. This note establishes the basis structure which admits the maximum degenerate pivots and that which admits the minimum degenerate pivots. It is shown that the upper bound is achieved if and only if the basis is a chain graph and that the lower bound is achieved if the degenerate basic arcs all originate at some facility or all terminate at some location.

An algorithm for linearly constrained nonlinear programming problems

Abstract In this paper an algorithm for solving a linearly constrained nonlinear programming problem is developed. Given a feasible point, a correction vector is computed by solving a least distance programming problem over a polyhedral cone defined in terms of the gradients of the “almost” binding constraints. Mukais approximate scheme for computing the step size is generalized to handle the constraints. This scheme provides an estimate for the step size based on a quadratic approximation of the function. This estimate is used in conjunction with Armijo line search to calculate a new point. It is shown that each accumulation point is a Kuhn-Tucker point to a slight perturbation of the original problem. Furthermore, under suitable second order optimality conditions, it is shown that eventually only one trial is needed to compute the step size.