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The article presents a solution to a large-scale traveling-salesman problem which shows that a certain tour of 49 cities, one in each of the 48 states and Washington, D.C., has the shortest road distance. A brief history of the problem is provided. A city in each state including the District of Columbia was selected to try the optimal path method on a large problem. The devices in obtaining solutions include the use of undirected tours, the imposition of linear conditions to characterize the tours, the speeding up of the iterative process and the use of a combinatorial approach.

A supplement to the article "Smooth Patterns of Production," written by A.J. Hoffman and Walter Jacobs, which appeared in the first volume of the periodical "Management Science" published in 1954, is provided. The author suggests a method of interpreting the mathematical theorem used by Hoffman and Jacob to measure the cost of speeding up industrial product production. It is suggested that the conclusions reached in the initial article depend on the cumulative delivery requirement variable. The author also critiques an algorithm included in theorem 4, which deals with product management and scheduling and is also discussed in the aforementioned article by Hoffman and Jacobs.

The article discusses the use of linear programming in U.S. industry. Three aspects of the application of linear programming are presented, including model building, mathematics, and computation. Many real-world industry cases have been related to finding the most economical way of transporting materials between various sources and destinations, and have been solved by special algorithms. In the mathematics of linear programming there are four main areas of research, including computational techniques that take advantage of special features of problems, methods of performing sensitivity analyses for the coefficients of the problem, methods that relate to dynamic problems, and stochastic techniques for handling uncertainty.

The article discusses the "transportation problem," which is a class of linear programming problems involving the selection of the most economical shipping routes for a commodity from given sources to given destinations. The transportation problem has been typically approached either as a specialized type of linear programming problem or as a method of representation of some combinatorial problems. The author asserts that a third method is possible, which is applicable for the determination of optimum linked paths over a series of points, or "transhipment." The transhipment problem is more complex than shipping an item from one point to another, because points can now be both origins and destinations. The question resembles a combinatorial problem but may be solved by linear programming.

In this paper, it is shown that the "caterer" problem, a problem in mathematical economics and logistics which has been discussed by Jacobs, Gaddum, Hoffman and Sokolowsky, and Prager, can be reduced to the problem of determining the maximum of the linear form Ln = ∑I=1nv1, subject to a series of constraints of the form V1 ≦ b1, v1 +v2 ≦ b2, v1 + v2 +v3 ≦ b2, ߪ, v1 + v2 + … + vK ≦ bK, v2 + v3 + … + vK+1 ≦ bK+1, …, vb-h+1 + … + Vn ≦ bn, 0 ≦ rI, I = 1, 2, … , n, under an assumption concerning the nonaccumulation of dirty laundry. This maximization problem is solved explicitly, using the functional equation technique of dynamic programming. [ABSTRACT FROM AUTHOR]

The article reports on the computational solution of linear programming equations involving almost-block-diagonal matrices. The author focuses on dynamic programming methods that can be used to solve weakly coupled and multi-stage economic systems. He suggests that most economic and industrial problems can be reduced to the problem of maximizing the linear form. It is suggested that finding the mathematical solutions to multi-dimensional problems is important because of the wide range of values of the solution's parameters. Dantzig's simplex algorithm enables programming problems of this type to be solved using digital computers.

A concave programming problem is presented, which is a model for an operations-research problem arising in both military and commercial applications It is amenable to linear programming methods and can he solved by a sequence of applications of the simplex algorithm [ABSTRACT FROM AUTHOR]

While the linear programming concept and the formulation of a specific model have value in themselves, the technique would fall short of its promise without efficient computing procedures. The simplex method has proved to be the most successful approach but it still leaves a wide range for detailed algorithms. At first, the method was applied by hand or with the aid of punched-card equipment. With the advent of internally-stored-program computers, automatic routines were designed. However, the early machines were adequate only for small problems and the original simplex method was not efficient for many LP models. A program of development was started at RAND to provide better computing techniques. Using first the IBM Model II CEC, then the 701 and now the 704, general LP routines of considerable power and flexibility have been developed. Future improvements will probably be possible only from higher level abstractions which take advantage of structure in the model matrix or which produce more efficient forms of the inverse of a sparse matrix. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to illustrate some applications of the functional equation technique of the theory of dynamic programming to a general class of problems arising in the study of networks, particularly those arising in transportation theory. [ABSTRACT FROM AUTHOR]

A summary of the important variations on the simplex method is presented, with an emphasis placed on the connection between these techniques, the revised simplex method, and the dual theorem. Only elementary matrix algebra is employed to enable the nonmathematician to utilize the various algorithms. Specifically, the dual simplex algorithm, parametric programming, and techniques for additional and secondary constraints, and for upper bounded variables are discussed. [ABSTRACT FROM AUTHOR]

The well known chemical equilibrium problem is expressed in the form of minimizing the free energy of a mixture in order to compute the chemical composition at equilibrium. By piece-wise linear approximations to the free energy function, the problem becomes a linear program which can be solved by a standard code on a computing machine. Successive approximations give any degree of accuracy. [ABSTRACT FROM AUTHOR]

A simplex computation for an arc-chain formulation of the maximal multi-commodity network flow problem is proposed. Since the number of variables in this formulation is too large to be dealt with explicitly, the computation treats non-basic variables implicitly by replacing the usual method of determining a vector to enter the basis with several applications of a combinatorial algorithm for finding a shortest chain joining a pair of points in a network. [ABSTRACT FROM AUTHOR]

A linear programming model is set forth for maximizing a vector of final demand deliveries under both regional production constraints and transportation network constraints. The model is essentially designed to study short-run effects, and hence no provision is made for the expansion of either industrial or transportation capacities. Within available capacities, however, the location of production is chosen so as to maximize final demand deliveries. A rough dynamic version of the model is presented as well as an algorithm for reaching an approximation of the optimum solution. [ABSTRACT FROM AUTHOR]

This note is concerned with two target assignment models An optimum assignment is one which maximizes the expected value of targets destroyed The first model, which admits an explicit solution, associates values only with the number of targets destroyed An algorithm which enjoys a computational nicety is established when the values of the individual targets are assumed known This latter model is a special case of Flood's target- assignment model [ABSTRACT FROM AUTHOR]

This paper studies the close relation between the Gass-Saaty parametric programming algorithm and the `primal-dual' procedures recently exploited by DANTZIG, FORD, AND FULKERSON lit is shown that the two procedures are equivalent The possibility of eliminating the two-phase character of the simplex method using these techniques is discussed Finally, the application of the techniques to problems with special structure is considered [ABSTRACT FROM AUTHOR]

The problem considered in this paper is that of allocating a budget of resources among the links of a network for the purpose of increasing its flow capacity relative to given sources and sinks. On the assumption that the cost of increasing each link capacity is linear, a labeling algorithm is described that permits rapid calculation of optimal allocations for all budgets. [ABSTRACT FROM AUTHOR]

A tentative solution to the general schedule-sequencing problem is presented in a linear-programming form. Feasible solutions hinge on work recently presented on integer solutions to linear-programming problems. As yet, the computation involved for a practical problem would be quite large. [ABSTRACT FROM AUTHOR]

The problem of specifying an allowance for defects in a production lot is that of balancing the cost of producing too many items against the risk of not having enough to meet requirements. A model of these costs is here proposed. Sufficient conditions are developed on the probability distribution of defectives for total cost to have a single minimum with respect to the allowance. A sequential algorithm is investigated and shown to produce an optimum allowance if certain further conditions on the probability are met. Next, it is shown for a special class of probability distributions that the above conditions are satisfied. This class is that for which the probability of an item being defective is independent of previous defects in the lot, and includes the binomial distribution. Finally some computational aspects of this algorithm are discussed, and an easily computable starting value is given. [ABSTRACT FROM AUTHOR]

In network problems such as the maximum-flow problem and the minimum-route problem, it is often desirable to attempt to simplify the given network before applying the various algorithms available for its solution. This is especially true when the maximum flow (or minimum route) between a number of different pairs of points is desired. Various transformations are discussed that can lead to considerable simplification. In particular, it is shown that wye-delta transformations analogous to those used with electrical networks are available. The application of these transformations to the maximum-flow problem (both with and without node capacities) is discussed and it is shown that for the minimum-route problem dual transformations apply. The effect of the topological properties of a network on the usefulness of these transformations is examined briefly. The application of the transformations to two networks in the literature is shown [ABSTRACT FROM AUTHOR]

The article describes two linear programming solutions to the assembly-line balancing problem. The balancing problem arises from a number of operations that must be performed sequentially with partial ordering constraints. The linear constraints are in variables that can be evaluated in linear objective functions. The linear programs developed to solve the balancing problem involve large numbers of constraint equations, variables, special integer variables. This makes complex the computations to develop an algorithm needed to solve the problem.

Algorithms are developed for solving problems to minimize the length of production schedules. The algorithms generate anyone, or all, schedule(s) of a particular subset of all possible schedules, called the active schedules. This subset contains, in turn, a subset of the optimal schedules. It is further shown that every optimal schedule is equivalent to an active optimal schedule. Computational experience with the algorithms shows that it is practical, in problems of small size, to generate the complete bet of all active schedules and to pick the optimal schedules directly from this set and, when this is not practical, to random sample from the bet of all active schedules and, thus, to produce schedules that are optimal with a probability as close to unity as is desired. The basic algorithm can also generate the particular schedules produced by well known machine loading rules. [ABSTRACT FROM AUTHOR]

In the study of production scheduling to meet random fluctuations in supply and demand, a probabilistic measure of effectiveness can be used. This measure is a piecewise quadratic positive definite function having discontinuous derivatives at a finite number of points. The unique minimum of this function, which always exists, can be found by applying an algorithm resembling the simplex algorithm. The computations per iteration are longer and the logic more complicated than for the simplex algorithm. The method can be applied to any overdetermined system in which all of the critical variables are functions of a single adjustable variable. The system is assumed to be linear and the input disturbances are presumed Gaussian. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm for the solution of mathematical programming problems having a linear objective function and non-linear constraints. The algorithm is basically an adaptation of the simplex method to the case of non-linear constraints. Certain complications are, however, introduced through non-linearity in the constraints, and it is shown how these can be overcome. [ABSTRACT FROM AUTHOR]

Periodic budgeting, as practiced in industry and taught in the accounting curriculum, combines estimates by individual departments in a process of coordinative aggregation. The purpose is to supply management with a financial plan for future operations. Frequently, budgeting is charged with the more ambitious task of finding the most profitable course for an enterprise. If this means selecting a combination of managerial policies which optimizes the long-term profit of the enterprise, the above definition of the purpose of this discipline seems to overstate the potential of traditional budgeting activity. It makes the layman believe that this area of accounting is in a position to determine optimal solutions. Undoubtedly this is not the case since budgeting traditionally neither applies any algorism to optimize the long term profit function nor provides any means for determining and comparing all the alternatives resulting from the innumerable factor and policy combinations feasible for an enterprise.

An algorithm for the generation of feasible schedules and the computation of the completion times of the job operations of feasible schedule is presented. Using this algorithm, the distribution of schedule times over the set of feasible schedule--or a subset of feasible schedules--was determined for technological orderings that could occur in a general machine shop. These distributions are found to be approximately normal. Biasing techniques corresponding to "first come first serve," random choice of jobs ready at each machine and combinations of these two extremes were used to compute distributions of schedule times. In all the experiments "first come first serve" appears the best in the sense that convergence to the minimum is fastest and the smallest. [ABSTRACT FROM AUTHOR]

To what extent can a parent firm with many branches standardize the activities of its satellites and enforce such uniformity without causing undue strain on the organizational structure? This mathematical study, based on a linear model of the multiple branch organization constructed within a framework of activity analysis, outlines the parameters and postulates three hypotheses in regard to the behavior tendencies of such multibranch organizations. [ABSTRACT FROM AUTHOR]

Can demand and consequently the market for consumers' durable goods be reduced to a mathematical model? Yes, these business economists contend, provided the right framework is chosen and proper statistical correlations are made to compensate for those eternal marketing variables -- price, credit, income, and use-facilities. They illustrate their point by constructing an econometric model, applicable to any durable good, and demonstrate its use and accuracy by analyzing sales of washing machines. [ABSTRACT FROM AUTHOR]

Commentary is presented for an article published in the July 1960 issue of "Management Science," written by L. V. Kantorovich with an introductory note by T. C. Koopmans. According to the author, in his introduction Koopmans addresses in particular a linear programming problem and a matrix game presented by Kantorovich. Also presented is an interpretation of the article's treatment of decision variables and mathematical optimization. The author notes that the article presents algorithms that have been improved from their contemporary presentations.

The special class of linear programs described as network flow problems includes many of the special structure optimization problems in such areas as assignment, transportation, catering, warehousing, production scheduling, etc Besides the physical motivation of the flow description, this class of problems possesses conceptually simple, and computationally elegant, solution techniques originated by DANTZIG, FORD, FULKERSON, ET AL This paper describes a generalization of this class of problems to 'process-flow networks,' or 'flow with gains,' in which flow in any branches of the network may be multiplied by an arbitrary constant, called the branch gain, before leaving the branch and flowing into the remainder of the network This generalization permits the description of networks in which different kinds of flow may be converted one to another, without 'constant returns to scale' Among other interesting problems, this model includes the metalprocessing problem, the machine-loading problem, financial budgeting, aircraft routing, warehousing with 'breeding' or 'evaporation,' the two-equation capacitated linear program, etc The solution method here described and proved is a natural extension of Ford and Fulkerson's technique (or specialization of the primal-dual method) and retains much of the physical motivation and easy computational aspect of their technique, a direct starting solution is usually available, and optimization of the imbedded linear program (the restricted primal problem) can be accomplished without the use of the simplex technique Conceptually, the solution procedure consists of the following steps (1) Find the inerementally cheapest loop in the network that will absorb flow (2) Establish as much additional flow into the network along this route as possible (3) Repeat Steps (1) and (2) until the desired flow is established, or until infeasibility is discovered The solution technique includes a maximal-flow procedure and produces a 'min-cut... [ABSTRACT FROM AUTHOR]

Efficient algorithms are developed optimizing an important class of capacity scheduling models The specific problem considered can be simply described in terms of contracting for warehousing capacity Contracts must be let for warehouse capacity over n time periods, with the minimum capacity to be provided in each time period being specified Savings may be achieved by long-term leasing arrangements or by contracting at favorable periods of time, even though this creates idle capacity at certain time periods A minimum cost solution to this problem is sought The mathematical model also applies to problems of equipment replacement and overhaul, checkout, repair, and replacement of stochastically failing equipment, determination of economic lot size, product assortment, and deterministic batch queuing policies, labor-force planning, and multi-commodity warehouse decisions For some of these problems, such as equipment replacement, the computing algorithms presented are even more efficient than schemes heretofore proposed for simpler versions of the same problem. [ABSTRACT FROM AUTHOR]

Some of the more recent theoretical and computational developments in non-linear programming are surveyed. The notions of Lagrange multipliers and duality are discussed together with applications of these ideas to scientific and business problems. Moreover, several algorithms for solving quadratic programming problems are reviewed. Explicit rules are given for two of these algorithms, and a simple example is solved by both methods. A large step gradient method for the solution of convex programs is given and one of Gomory's algorithms for integer programming is described. Simple examples are solved using both of these techniques. Linear fractional programming is also discussed briefly. [ABSTRACT FROM AUTHOR]

By constructing a new infinite dimensional space for which the extreme point--linear independence and opposite sign theorems of Charnes and Cooper continue to hold, and, building on a little-known work of Ear (herein presented), an extended dual theorem comparable in precision and exhaustiveness to the finite space theorem is developed. Building further on this a dual theorem is developed for arbitrary convex programs with convex constraints which subsumes in principle all characterizations of optimality or duality in convex programming. No differentiability or constraint qualifications are involved, and the theorem lends itself to new computational procedures. [ABSTRACT FROM AUTHOR]

This paper presents a new methodology for determining suboptimum relative location patterns for physical facilities. It presents a computer program governed by an algorithm which determines how relative location patterns should be altered to obtain sequentially the most improved pattern with each change, commands their alteration, evaluates the results of alterations, and identifies the sub-optimum relative location patterns. The computer output yields a block diagramatic layout of the facility areas, and the areas need not be equal. A manufacturing plant layout example is used because the methodology is well illustrated by reference to some kind of example, and this one is commonly known. The methodology itself, however, is general in nature and not restricted to such applications. [ABSTRACT FROM AUTHOR]

For a project that consists of numerous jobs subject to technological ordering restrictions the polygon representing project cost versus completion time is to be determined when the normal and crash completion times are known for each job and the cost of doing the job varies linearly between these times. A linear programming formulation of this problem was given by Keliey [1] and a network flow formulation by Fulkerson [2]. Since the traditional mathematical background of civil engineers includes neither linear programming nor network flow theory, these methods are not as widely used in the building industry as they deserve. This paper shows that Fulkerson's algorithm can be given a structural interpretation using concepts that are familiar to civil engineers. [ABSTRACT FROM AUTHOR]

A network is a set of nodes Nt connected by arcs with nonnegative arc capacities btj, which indicates the maximum amount of flow that can pass through the arc from Nt to Nj. Given all bt,j, there is a maximum flow from Nt to Nj using all arcs. Under the assumption that btj = bjt, the present paper generalizes the max-flow min cut theorem of Ford and Fulkerson to the problem of finding the maximum simultaneous flows of two commodities and gives an algorithm similar to the labelling method for constructing the two flows. [ABSTRACT FROM AUTHOR]

The multi-index transportation problem is defined and a summary of the algorithm described in an earlier paper is given. This paper discusses the theorems justifying the use of the algorithm and gives a necessary condition for the existence of a solution. It also describes the formulation of three special-structure linear-programming problems as examples of multi-index problems. The three problems discussed are the capacitated Hitchcock problem, transportation where there is more than one method of transport available, and the transformation of one type of multi-index problem into another. A final section describes problems that have restrictions in the form of inequalities. [ABSTRACT FROM AUTHOR]

This paper approaches the assembly-line balancing problem as a sequencing problem involving precedence constraints that prohibit the occurrence of certain orderings. This approach permits the formulation of a dynamic programming algorithm for the exact solution of small assembly-line balancing problem. A generalization of this algorithm combined with a successive approximations technique is used for the solution of large problems. In addition, certain combinatorial problems associated with partially ordered sets are formulated and discussed in detail. These problems arise whenever sequencing problems with precedence constraints are treated by dynamic programming techniques. [ABSTRACT FROM AUTHOR]

The linear programing algorithms available for optimizing the routing of shipments in multi-plant, multi-destination systems cannot, in the current state of knowledge, be applied directly to the more general problem of determining the number and location of regional warehouses in large-scale distribution networks. This paper outlines a heuristic computer program for locating warehouses and compares it with recently published efforts at solving the problem either by means of simulation or as a variant of linear programing. The heuristic approach outlined in this paper appears to offer significant advantages in the solution of this class of problems in that it (1) provides considerable flexibility in the specification (modeling) of the problem to be solved, (2) can be used to study large-scale problems, that is, complexes with several hundred potential warehouse sites and several thousand shipment destinations, and (3) is economical of computer time. The results obtained in applying the program to small scale problems have been equal to or better than those provided by the alternative methods considered. [ABSTRACT FROM AUTHOR]

The 'explosion' problem is the problem of determining the required components from knowledge of demand for finished products. The 'netting' problem IS the problem of determining the net requirements of these components, for purposes of production or procurement from outside the firm, based on available inventory. This paper presents a simple algorithm which yields the 'netted explosion. 'The algorithm is suitable for digital computation in case of large inventories. [ABSTRACT FROM AUTHOR]

The optimal design of a module under a special set of circumstances is considered. These circumstances prevail in the design of a module for logistical applications and sometimes in modular equipment design in electronic technology. Mathematically, the problem considered is given Rαβ ≧0 find positive Bα, mβ such that Bαmβ≧Rαβ, α= l, ,2, ,n, β=1, 2, , k and such that W=Σmβ Bα, is a minimum. It is attacked by first examining the primal problem, then finding a dual problem, and finally finding an algorithm for solving the dual. [ABSTRACT FROM AUTHOR]

Describes an algorithm for computing interzonal transfers using the gravity model. Application of the gravity model to the problem of estimating the number of automobile trips between pairs of points of a highway network given the number of trips that arrive and depart each point.

Describes an algorithm for the traveling salesman problem. Breaking of the set of all tours into increasingly small subsets by a procedure called branching; Calculation of the lower bounds based on ideas frequently used in solving assignment problems.

Because the payload that an airplane can carry is a function of the longest segment of its route and differs from route to routes the usual algorithms of network flow theory will not yield the optimum flow of payload. In this paper two network algorithms are described. They determine the route of maximum payload flow per hour of flight time and the maximum steady- state payload flow through a network with capacity constraints on the nodes,. [ABSTRACT FROM AUTHOR]

This paper summarizes experimental results from applying several computational methods for solving the classic three-machine scheduling model. The basic mathematical problem is to select an optimal permutation of a items, where the objective function employed is the total amount of processing time elapsing for the completion of all n items on three machines. The methods tested are integer linear programming, ordinary linear programming with answers rounded to integers, a heuristic algorithm, and random sampling. Throughout the experimentation n=6. A dual integer programming code is tested on six trial problems. Consideration is given to studying the phenomenon of input-form sensitivity. The results are not encouraging and confirm previous experimental evidence as to the strong effect of varying the input form. However, the addition of a powerful bound on the objective function has a significant beneficial effect on the convergence of the dual algorithm. A sampling study of the statistical characteristics of this model is made with 100 sets of data randomly generated. Frequency distributions of minimum and mean total processing time are tabulated and analyzed. On this same set of problems, a linear programming approach is examined for effectiveness in producing nearly optimal schedules. Such an approach tends to give answers that are better than would be obtained by a single permutation drawn at random, but the LP rounded solutions average about an 11 per cent increase over the optimal processing time. A heuristic algorithm based on Johnson's method is tested on 20 of the 100 eases and gives excellent results. Finally attention is turned to the method of randomly sampling the permutations, possibly applying a simple heuristic gradient method to improve each sampled schedule. A feature of this method is that elementary probability theory can be applied to yield corresponding statements about the accuracy guaranteed by the approach. [ABSTRACT FROM AUTHOR]

A production and transportation system is considered in which the produced items successively are transported to n different selling places. Transportation to the N places may be ordered in NI different ways and the effects on the system of different orderings among the selling places when servicing them are studied. If the production capacity is K units per unit of time, and if the N places have associated with them times of transport t1,⋯tN and functions fl,⋯,fN expressing the demand as depending on the time of arrival, it is possible to obtain the total sale on the form KgN,{gN-1[A⋯g1(0)⋯]} where the functions gn are defined in terms of K, tn, and fn. Thus the problem to determine the order to serve that maximizes the total sale is reduced to the problem of finding the optimal permutation of gN {gN-1[ ⋯g1(0⋯). When deriving criterions of optimality for the total sale and related measures of effectiveness the following principle is used: Let E(1,⋯, N) be the effectiveness associated with the ordering 1,⋯, N and let kn. be some characteristic of place n. If k1≦⋯≦kn is sufficient (necessary) for E(l,⋯,N)≧E(l,⋯, n-1 , n+l, n, n+2,⋯N) for n=1⋯, N-1 then this condition is also sufficient (necessary) for E(1,⋯,N)≧E (i1, iN) for every permutation i1,⋯ of 1⋯,N. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm that will yield an optimum sequence for n jobs requiring processing through M machines when no passing is allowed. Theoretical development utilizing combinatorial analysis precedes the algorithm and an example problem. [ABSTRACT FROM AUTHOR]

The article presents information on literature related to the automatic stratification of descriptors. The exclusive stratification process and algorithms are defined in the paper entitled Automatic Stratification of Descriptors by David Lefkovitz. The computer program written to perform this process is coded in FORTRAN II and an object program has been produced. The Operation Manual for the IBM 7090 Exclusive Stratification Program gives a very brief description of the function that this program performs and indicates the manner in which it can be used for large scale processing of data, including such features as batch processing, manual interrupt and restart, English language or numerically coded output, and a few other printout and processing options.

In a previous article [7] the authors gave the theoretical validation of the sequential unconstrained minimization technique for solving the convex programming problem. The technique is based on an idea proposed by C. W. Carroll [2], [3]. The method has been implemented via an algorithm based on a second-order gradient technique that has proved extremely efficient on a considerable number of test problems of varying complexity. This paper explores the computational aspects of the method. Included are discussions of parameter selection, convergence criteria, and methods of minimizing an unconstrained function. It is shown that the problem variables, on the trajectory of minima of the sequence of unconstrained functions, can be developed as functions of a single parameter. This provides the theoretical basis for an extrapolation technique that significantly accelerates convergence in actual computations. The detailed computer solution of a small example is given to illustrate the typical convergence characteristics of the method. The speed and accuracy of the computational procedure are believed to be competitive with other known techniques for solving the convex programming problem. [ABSTRACT FROM AUTHOR]