A new algorithm for solving the general quadratic programming problem

For the general quadratic programming problem (including an equivalent form of the linear complementarity problem) a new solution method of branch and bound type is proposed. The branching procedure uses a well-known simplicial subdivision and the bound estimation is performed by solving certain linear programs.     

A dual-active-set algorithm for positive semi-definite quadratic programming

Because of the many important applications of quadratic programming, fast and efficient methods for solving quadratic programming problems are valued. Goldfarb and Idnani (1983) describe one such method. Well known to be efficient and numerically stable, the Goldfarb and Idnani method suffers only from the restriction that in its original form it cannot be applied to problems which are positive semi-definite rather than positive definite. In this paper, we present a generalization of the Goldfarb and Idnani method to the positive semi-definite case and prove finite termination of the generalized algorithm. In our generalization, we preserve the spirit of the Goldfarb and Idnani method, and extend their numerically stable implementation in a natural way. Supported in part by ATERB, NSERC and the ARC. Much of this work was done in the Department of Mathematics at the University of Western Australia and in the Department of Combinatorics and Optimization at the University of Waterloo.     

Application of average dynamic programming to inventory systems

We show the existence ofaverage cost (AC-) optimal policy for an inventory system withuncountable state space; in fact, the AC-optimal cost and an AC-optimal stationary policy areexplicitly computed. In order to do this, we use a variant of thevanishing discount factor approach, which have been intensively studied in recent years but the available results not cover the inventory problem we are interested in.The work of the first author (OVA) was partially supported by Fondo del Sistema de Investigación del Mar de Cortéz under grant SIMAC/94/CT-005. The work of the second author (RMdO) was partially supported by Consejo Nacional de Ciencia y Tecnologia (CONACyT) under grant 0635P-E9506.     

Discrete linear bilevel programming problem

In this paper, we analyze some properties of the discrete linear bilevel program for different discretizations of the set of variables. We study the geometry of the feasible set and discuss the existence of an optimal solution. We also establish equivalences between different classes of discrete linear bilevel programs and particular linear multilevel programming problems. These equivalences are based on concave penalty functions and can be used to design penalty function methods for the solution of discrete linear bilevel programs.Support of this work has been provided by the INIC (Portugal) under Contract 89/EXA/5, by INVOTAN, FLAD, and CCLA (Portugal), and by FCAR (Québec), NSERC, and DND-ARP (Canada).     

Composite mix design in the production of asbestos/cement pipes

A mathematical model is presented for the determination of optimal mix of asbestos fibres to satisfy a set of desired pipe properties which vary with pipe size, class, and material availability. The approach utilizes a linear/separable programming model which is aimed at minimizing the total cost of fibres used. The fibre properties, such as reinforcing strength, processability, and length as well as cost and stocks available are taken into account in the model. Also represented are the profile of the ideal mix relative to the required composite strength, processability, and ‘dust content’. Asbestos mixes designed using the model have been implemented in the production of asbestos/cement pipes in one of the asbestos/cement pipe manufacturing plants in the Gulf area. Optimum model mixes have demonstrated significant technical and economic merits over the conventionally designed mixes.     

A comparative study of two learning rules for associative memory

This paper addresses itself to a practical problem encountered in using iterative learning rules for associative memory models. The performance of a learning rule based on linear programming which overcomes this problem is compared with that of a representative iterative rule by numerical simulation. Results indicate superior performance by the linear programming rule. An algorithm for computing radii of maximal hyperspheres around patterns in the state space of a model is presented. Fractional volumes of basins of attractions are computed for the representative iterative rule as well as the linear programming rule. With the radii of maximal hyperspheres as weight factors for corresponding patterns to be stored, the linear programming rule gives rise to the maximal utilisation of the state space.     

Multi-product production planning: A goal programming approach

Cost minimization multi-product production problems with static production resource usage and internal product flow requirements have been solved by linear programming (LP) with input/output analysis. If the problem is complicated by interval resource estimates, interval linear programming (ILP) can be used. The solution of realistic problems by the above method is cumbersome. This paper suggests that linear goal programming (LGP) can be used to model a multi-product production system. LGP's unique modeling capabilities are used to solve a production planning problem with variable resource parameters. Input/output analysis is used to determine the technological coefficients for the goal constraints and is also used to derive an information sub-model that is used to reduce the number of variable resource goal constraints. Preliminary findings suggest that the LGP approach is more cost-efficient (in terms of CPU time) and in addition provides valuable information for aggregate planning.     

Yet another geometric programming dual algorithm

We describe an algorithm for the geometric programming dual problem which uses an adaptation of the generalized LP algorithm, proposed by Dantzig et al. twenty-five years ago for the chemical equilibrium problem, and show the slack primal constraints pose no numerical difficulties for this algorithm as they do for previous dual-based algorithms.     

Current state of the art of algorithms and computer software for geometric programming

This paper attempts to consolidate over 15 years of attempts at designing algorithms for geometric programming (GP) and its extensions. The pitfalls encountered when solving GP problems and some proposed remedies are discussed in detail. A comprehensive summary of published software for the solution of GP problems is included. Also included is a numerical comparison of some of the more promising recently developed computer codes for geometric programming on a specially chosen set of GP test problems. The relative performance of these codes is measured in terms of their robustness as well as speed of computation. The performance of some general nonlinear programming (NLP) codes on the same set of test problems is also given and compared with the results for the GP codes. The paper concludes with some suggestions for future research.An earlier version of this paper was presented at the ORSA/TIMS Conference, Chicago, 1975.This work was supported in part by the National Research Council of Canada, Grant No. A-3552, Canada Council Grant No. S74-0418, and a research grant from the School of Organization and Management, Yale University. The author wishes to thank D. Himmelblau, T. Jefferson, M. Rijckaert, X. M. Martens, A. Templeman, J. J. Dinkel, G. Kochenberger, M. Ratner, L. Lasdon, and A. Jain for their cooperation in making the comparative study possible.     

Choice of optimal search strategy

The problem of searching the extremum of a scalar function of a vector argument is considered. It is assumed that a finite set of algorithms, each of which is capable of finding the extremum, is specified. Every algorithm is characterized by a given number of operators each of which is identified with the state of the system. Each algorithm is defined by a transition probability matrix over the possible states (operators) and a corresponding matrix of the respective changes in the value of the function toward the extremal point.A procedure is given for selecting an optimal sequence of algorithms which maximizes the total expected change in the value of the function toward the optimum.