Showing papers in "Mathematical Programming in 1971"

The traveling-salesman problem and minimum spanning trees: Part II

Abstract: The relationship between the symmetric traveling-salesman problem and the minimum spanning tree problem yields a sharp lower bound on the cost of an optimum tour. An efficient iterative method for approximating this bound closely from below is presented. A branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it, ranging in size up to sixty-four cities. The bounds used are so sharp that the search trees are minuscule compared to those normally encountered in combinatorial problems of this type.

Linear programming with multiple objective functions: Step method (stem)

Abstract: This paper describes a solution technique for Linear Programming problems with multiple objective functions. In this type of problem it is often necessary to replace the concept of “optimum” with that of “best compromise”. In contrast with methods dealing with a priori weighted sums of the objective functions, the method described here involves a sequential exploration of solutions. This exploration is guided to some extent by the decision maker who intervenes by means of defined responses to precise questions posed by the algorithm. Thus, in this man-model symbiosis, phases of computation alternate with phases of decision. The process allows the decision-maker to “learn” to recognize good solutions and the relative importance of the objectives. The final decision (best compromise) furnished by the man-model system is obtained after a small number of successive phases.

Matroids and the greedy algorithm

Abstract: Linear-algebra rank is the solution to an especially tractable optimization problem This tractability is viewed abstractly, and extended to certain more general optimization problems which are linear programs relative to certain derived polyhedra

Blocking and anti-blocking pairs of polyhedra

Abstract: Some of the main notions and theorems about blocking pairs of polyhedra and antiblocking pairs of polyhedra are described. The two geometric duality theories conform in many respects, but there are certain important differences. Applications to various combinatorial extremum problems are discussed, and some classes of blocking and anti-blocking pairs that have been explicitly determined are mentioned.

Problems and methods with multiple objective functions

Abstract: LetA be a set of feasible alternatives or decisions, and supposen different indices, measures, or objectives are associated with each possible decision ofA. How can a “best” feasible decision be made? What methods can be used or experimented with to reach some decision?

On the basic theorem of complementarity

Abstract: Using a fixed point theorem of Browder, the basic existence theorem of Lemke in linear complementarity theory is generalized to the nonlinear case.

Experiments in mixed-integer linear programming

Abstract: This paper presents a “branch and bound” method for solving mixed integer linear programming problems. After briefly discussing the bases of the method, new concepts called pseudo-costs and estimations are introduced. Then, the heuristic rules for generating the tree, which are the main features of the method, are presented. Numerous parameters allow the user for adjusting the search strategy to a given problem.

Recent advances in unconstrained optimization

Abstract: We survey the development of algorithms and theory for the unconstrained optimization problem during the years 1967–1970. Therefore (except for one remark) the material is taken from papers that have already been published. This exception is an explanation of some numerical difficulties that can occur when using Davidon's (1959) variable metric algorithm.

Reinversion with the preassigned pivot procedure

Abstract: Mathematical programming computer systems using the product form in the inverse (PFI) must periodically resort to a reinversion with the current basis in order to reduce the amount of work to be done in the succeeding iterations. In this paper, we show the consequences of column, pivot selection and sequence upon the transformation vector (ETA) density and give an algorithm which will tend to minimize eta density and work done per iteration. The algorithm has been implemented and tested as a replacement for the previous inversion algorithm on the OPTIMA system for the CDC 6000 computers and on the MPS/III mathematical programming system for the IBM 360 computer. A comparative performance table is given.

Optimality conditions for quadratic programming

Abstract: This paper establishes a set of necessary and sufficient conditions in order that a vectorx be a local minimum point to the general (not necessarily convex) quadratic programming problem: minimizep T x + 1/2x T Qx, subject to the constraintsHx ≧ h.

A solvable case of the traveling salesman problem

Abstract: LetD = ‖dij‖ be the distance matrix defining a traveling salesman problem. IfD is upper triangular, i.e.dij = 0, fori ≥ j, then the traveling salesman problem can be solved with an amount of computation approximately equal to that required for an assignment problem of the same dimension.

Penalty function versus non-penalty function methods for constrained nonlinear programming problems

Abstract: The relative merits of using sequential unconstrained methods for solving: minimizef(x) subject togi(x) ź 0, i = 1, ź, m, hj(x) = 0, j = 1, ź, p versus methods which handle the constraints directly are explored. Nonlinearly constrained problems are emphasized. Both classes of methods are analyzed as to parameter selection requirements, convergence to first and second-order Kuhn-Tucker Points, rate of convergence, matrix conditioning problems and computations required.

On pseudo-convex functions of nonnegative variables

Abstract: : Two proofs are given for a conjecture of Bela Martos concerning conditions under which a quasi-convex quadratic function of nonnegative variables is actually a pseudo-convex function. (Author)

An “out-of-kilter” algorithm for solving minimum cost potential problems

Abstract: An algorithm is given for solving the optimum potential problem, which is the dual of the classical “out-of-kilter” algorithm for flow problems. Moreover, a new proof of finiteness is provided, which holds even for non-rational data; it applies to all the algorithms of network theory which include a labeling process.

Some recent results in n -person game theory

Abstract: The discussion will center mainly on some work on two solution concepts: the core for gaines without side payments and the nucleolus for games with side payments (characteristic funtion games). The core has become an important equilibrium concept in mathematical economics. The nucleolus is related to the theory of bargaining sets.

Linear inequalities, mathematical programming and matrix theory

Abstract: A survey is made of solvability theory for systems of complex linear inequalities. This theory is applied to complex mathematical programming and stability and inertia theorems in matrix theory.

On the solution of linear equation/inequality systems

Abstract: A pivotal algebra algorithm is given and finite convergence shown for finding a vector which satisfies an arbitrary system of linear equations and/or inequalities. A modified form of the algorithm, obtained by introducing a redundant equation, is then shown to be a way to describe phase I of the simplex method without reference to artificial variables or an artificial objective function. The hypothesis is introduced that in each pivot stage each row of the tableau has equal probability of being chosen as the pivot row. Under this assumption the expected value of the ratio of the number of pivot stages to the number of rows should grow with the natural Log of the number of rows. Use of the algorithm in proving theorems of the alternative is indicated.

A projective method for structured nonlinear programs

Abstract: This paper describes a partitioning method for solving a class of structured nonlinear programming problems with block diagonal constraints and a few coupling variables.

Applications of symmetric derivatives in mathematical programming

Abstract: In recent times the Kuhn—Tucker optimality conditions and the duality theorems for convex programming have been extended by generalizations of the convexity concept. In this paper the notion of a symmetric derivative for a function of several variables is introduced and used to provide extensions of some fundamental optimality and duality theorems of convex programming. Symmetric derivatives are also used to extend some optimality and duality theorems involving pseudoconvexity and differentiable quasiconvexity.

Discrete optimization on a multivariable boolean lattice

Abstract: Consider the problem of finding the minimum value of a scalar objective function whose arguments are theN components of 2N vector elements partially ordered as a Boolean lattice. If the function is strictly decreasing along any shortest path from the minimum point to its logical complement, then the minimum can be located precisely after sequential measurement of the objective function atN + 1 points. This result suggests a new line of research on discrete optimization problems.

Hybrid programs: Linear and least-distance

Abstract: We consider a quadratic program equivalent to the general problem of minimizing a convex quadratic function of many variables subject to linear inequality constraints.

Some problems in discrete optimization

Abstract: The present paper concentrates on several problems of network flows and discrete optimization. Progress has been made on some of the problems while little is known about others. Some of the problems discussed are shortest paths, multi-commodity flows, traveling salesman problems, m-center problem, telepak problems and binary trees.