Ascent Procedure Research Articles

Sensitivity analysis for the matching problem and its use in solving matching problems with a single side constraint

This paper develops certain sensitivity analysis capabilities for use with a primaldual matching code. The specific problem addressed is reoptimizing after the costs of a subset of the edges have been increased by a constant amount. This capability is applied to a dual ascent procedure for a Lagrangian relaxation of a matching problem with a single generalized upper bound side constraint. Some of the sensitivity analysis capabilities should be useful in other contexts as well. In particular, we give a method for solving for a set of dual variables that satisfy the strong complementary conditions given a blossom structure.

On lower bounds for a class of quadratic 0, 1 programs

We present a new method of obtaining lower bounds for a class of quadratic 0, 1 programs that includes the quadratic assignment problem. The method generates a monotonic sequence of lower bounds and may be interpreted as a Lagrangean dual ascent procedure. We report on a computational comparison of our bounds with earlier work in [2] based on subgradient techniques.

Augmented Lagrangean Based Algorithms for Centralized Network Design

Capacitated spanning tree problems appear frequently as fundamental problems in many communication network design problems. An integer programming formulation and a new set of valid inequalities are presented for the linear characterization of the problem. A combination of a subgradient optimization procedure and an augmented Lagrangean-based procedure is used to generate tight lower bounds. The procedure begins with an explicit representation of a subset of the constraints, and the corresponding Lagrangean problem is solved. The solution is examined in order to identify implicit constraints that are violated. Those are added to the Lagrangean problem, forming an expanded problem, and an efficient dual ascent procedure is then applied. When no further improvement is possible through this procedure, a subgradient optimization procedure is invoked in order to further tighten the lower bound value. An exchange heuristic is applied to the nonfeasible Lagrangean solution, in an attempt to generate good feasible solutions to the problem. The procedure has been tested and has generated bounds that are significantly better than ones obtained through previously published procedures.

A branch-and-bound algorithm for the multi-level uncapacitated facility location problem

This paper discusses the problem of locating facilities over all levels in a multi-level distribution system where commodities are shipped from origin-level facilities to destination points via a number of intermediate level facilities. The objective is to determine the optimal set of facilities to open for each distribution level, which minimizes the total distribution costs including the fixed costs associated with opening them. The problem is formulated as a mixed integer linear program with the so-called ‘tight’ or ‘disaggregated’ constraints, by which in our branch-and-bound solution method, the dual based scheme of ‘dual ascent procedure’ by Erlenkotter, and Bilde and Krarup, is successfully applied. In addition, a heuristic ‘primal descent procedure’ which improves integer solutions already obtained, and some efficient node simplification procedure are introduced. Though our approach can be applied to any multi-level case, computational experiments are conducted mostly for two-level problems such that our approach can be compared with the other approaches. Computational experience with some small scale three-level problems is also reported on.

A dual ascent approach for steiner tree problems on a directed graph

The Steiner tree problem on a directed graph (STDG) is to find a directed subtree that connects a root node to every node in a designated node setV. We give a dual ascent procedure for obtaining lower bounds to the optimal solution value. The ascent information is also used in a heuristic procedure for obtaining feasible solutions to the STDG. Computational results indicate that the two procedures are very effective in solving a class of STDG's containing up to 60 nodes and 240 directed/120 undirected arcs.

Minimizing Net Interest Cost in Municipal Bond Bidding

The problem of minimizing net interest cost (NIC) on new issues of tax-exempt debt securities is formulated as an integer linear programming problem. The formulation (for one variant) is a p-median problem with one additional constraint. Other variants are also closely related to the p-median problem. A dual ascent procedure for solving this problem class is presented and is incorporated in a branch and bound algorithm. Computational results are presented for a number of real world problems.

Constructive Group Relaxations for Integer Programs

Recent work by Fisher and Shapiro has used Gomory’s group theoretic methods together with Lagrange multipliers to obtain bounds for the optimal value of integer programs. Here it is shown how an extension of the associated Abelian group can improve the bound without the artificiality of adding a cut. It is proved that there exists a finite group for which the dual ascent procedure of Fisher and Shapiro converges to the optimal integer solution. Constructive methods are given for finding this group. A worked example is included.

Conjugate gradient optimization applied to a copper converter model

Exothermic copper conversion reactions release heat energy which can be used to melt fluxing materials required in the reverberatory furnace. Oxygen enrichment of the air supplied to the converter results in shorter processing times and increased ability to smelt flux but with increased air supply costs. A conjugate gradient method in function space is employed to determine the optimal rate of flux addition to a calibrated mathematical model of the converter for various levels of enriched oxygen. The smelting of flotation concentrates in the converter and the effect of converter parameter variations in the optimal solutions are also considered. The mathematical model is derived from material and heat balance relationships and is calibrated using data from an operating converter. The paper also shows how the method of conjugate gradients in function space can be applied to a bounded control problem, in particular, one in which intervals of singular optimal control arise. The conjugate gradient method is found to converge substantially faster to a higher value of the performance index than a conventional steepest ascent procedure.

Simulation: A Tool for Farm Planning under Conditions of Weather Uncertainty

This paper attempts to determine the optimal organization and managerial policies of a farm operating under conditions of low and unstable rainfall. This objective was accomplished by computer simulation of the decision process. The performance characteristics of various decision rules were evaluated by simulating the farm over a set of sampled sequences of years with weather events constituting the main stochastic input. Optimal decision rules were defined in terms of the present value and the coefficient of variations of the income flows. The optimal decision rules were approximated by experimental variation of their parameters. A partial factorial design was first developed in order to determine the main features of the response surface. The optimum solution was then approached by the steepest ascent procedure. It is suggested that simulation may be useful in solving more general managerial problems arising under uncertainty.