Showing papers on "Linear programming published in 1990"

Approximation algorithms for scheduling unrelated parallel machines

Abstract: We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep ij . The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

A two‐phase decomposition method for optimal design of looped water distribution networks

Abstract: A two-phase decomposition method is proposed for the optimal design of new looped water distribution networks as well as for the parallel expansion of existing ones. The main feature of the method is that it generates a sequence of improving local optimal solutions. The first phase of the method takes a gradient approach with the flow distribution and pumping heads as decision variables and is an extension of the linear programming gradient method proposed by Alperovits and Shamir (1977) for nonlinear modeling. The technique is iterative and produces a local optimal solution. In the second phase the link head losses of this local optimal solution are fixed, and the resulting concave program is solved for the link flows and pumping heads; these then serve to restart the first phase to obtain an improved local optimal solution. The whole procedure continues until no further improvement can be achieved. Some applications and extensions of the method are also discussed.

Further developments in LP-based optimal power flow

Abstract: The authors describe developments that have transformed the LP (linear programming) approach into a truly general-purpose OPF (optimal power flow) solver, with computational and other advantages over even recent nonlinear programming (NLP) methods. it is pointed out that the nonseparable loss-minimization problem can now be solved, giving the same results as NLP on power systems of any size and type. Coupled formulations, where for instance voltages and VAr become constraints on MW scheduling, are handled. Former limitations on the modeling of generator cost curves have been eliminated. In addition, the approach accommodates a large variety of power system operating limits, including the very important category of contingency constraints. All of the reported enhancements are fully implemented in the production OPF software described here, and most have already been utilized within the industry. >

A Branch and Bound Algorithm for the Bilevel Programming Problem

Abstract: The bilevel programming problem is a static Stackelberg game in which two players try to maximize their individual objective functions. Play is sequential and uncooperative in nature. This paper presents an algorithm for solving the linear/quadratic case. In order to make the problem more manageable, it is reformulated as a standard mathematical program by exploiting the follower's Kuhn–Tucker conditions. A branch and bound scheme suggested by Fortuny-Amat and McCarl is used to enforce the underlying complementary slackness conditions. An example is presented to illustrate the computations, and results are reported for a wide range of problems containing up to 60 leader variables, 40 follower variables, and 40 constraints. The main contributions of the paper are in the step-by-step details of the implementation, and in the scope of the testing.

Computational Difficulties of Bilevel Linear Programming

Abstract: We show, using small examples, that two algorithms previously published for the Bilevel Linear Programming problem BLP may fail to find the optimal solution and thus must be considered to be heuristics. A proof is given that solving BLP problems is NP-hard, which makes it unlikely that there is a good, exact algorithm.

The Mixed Integer Linear Bilevel Programming Problem

Abstract: A two-person, noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem. In this paper, we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron. The decision variables are variously partitioned into continuous and discrete sets. The leader goes first, and through his choice may influence but not control the responses available to the follower. For two reasons the resultant problem is extremely difficult to solve, even by complete enumeration. First, it is not possible to obtain tight upper bounds from the natural relaxation; and second, two of the three standard fathoming rules common to branch and bound cannot be applied fully. In light of these limitations, we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations. A series of heuristics are then proposed in an effort to strike a balance between accuracy and speed. The computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables.

Linear reactive power optimization in a large power network using the decomposition approach

Abstract: A mathematical framework is presented for the solution of the economic dispatch problem The application of the Dantzig-Wolfe decomposition method for the solution of this problem is emphasized The system's optimization problem is decomposed into several subproblems corresponding to specific areas in the power system The upper bound technique along with the decomposition method are applied to a 16-bus system and a modified IEEE 30-bus system, and numerical results are presented for larger systems The results indicate that the presented formulation of the reactive power optimization and the application of the decomposition procedure will facilitate the solution of the problem The algorithm can be applied to a large-scale power network, where its solution represents a significant reduction in the number of iterations and the required computation time >

Stochastic vs. Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty

Abstract: I. The General Framework.- 1. Multiobjective programming under uncertainty : scope and goals of the book.- 2. Multiobjective programming : basic concepts and approaches.- 3. Stochastic programming : numerical solution techniques by semi-stochastic approximation methods.- 4. Fuzzy programming : a survey of recent developments.- II. The Stochastic Approach.- 1. Overview of different approaches for solving stochastic programming problems with multiple objective functions.- 2. "STRANGE" : an interactive method for multiobjective stochastic linear programming, and "STRANGE-MOMIX" : its extension to integer variables.- 3. Application of STRANGE to energy studies.- 4. Multiobjective stochastic linear programming with incomplete information : a general methodology.- 5. Computation of efficient solutions of stochastic optimization problems with applications to regression and scenario analysis.- III. The Fuzzy Approach.- 1. Interactive decision-making for multiobjective programming problems with fuzzy parameters.- 2. A possibilistic approach for multiobjective programming problems. Efficiency of solutions.- 3. "FLIP" : an interactive method for multiobjective linear programming with fuzzy coefficients.- 4. Application of "FLIP" method to farm structure optimization under uncertainty.- 5. "FULPAL" : an interactive method for solving (multiobjective) fuzzy linear programming problems.- 6. Multiple objective linear programming problems in the presence of fuzzy coefficients.- 7. Inequality constraints between fuzzy numbers and their use in mathematical programming.- 8. Using fuzzy logic with linguistic quantifiers in multiobjective decision making and optimization: A step towards more human-consistent models.- IV. Stochastic Versus Fuzzy Approaches and Related Issues.- 1. Stochastic versus possibilistic multiobjective programming.- 2. A comparison study of "STRANGE" and "FLIP".- 3. Multiobjective mathematical programming with inexact data.

Convex separable optimization is not much harder than linear optimization

Abstract: The polynomiality of nonlinear separable convex (concave) optimization problems, on linear constraints with a matrix with “small” subdeterminants, and the polynomiality of such integer problems, provided the inteter linear version of such problems ins polynomial, is proven. This paper presents a general-purpose algorithm for converting procedures that solves linear programming problems. The conversion is polynomial for constraint matrices with polynomially bounded subdeterminants. Among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix, to integer-separable convex programming. An algorithm for finding a e-accurate optimal continuous solution to the nonlinear problem that is polynomial in log(1/e) and the input size and the largest subdeterminant of the constraint matrix is also presented. These developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function. The practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function, only a polynomial number of function evaluations on a prespecified grid.

MSLiP: a computer code for the multistage stochastic linear programming problem

Abstract: This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.

Improved linear programming models for discriminant analysis

Abstract: Discriminant analysis is an important tool for practical problem solving. Classical statistical applications have been joined recently by applications in the fields of management science and artificial intelligence. In a departure from the methodology of statistics, a series of proposals have appeared for capturing the goals of discriminant analysis in a collection of linear programming formulations. The evolution of these formulations has brought advances that have removed a number of initial shortcomings and deepened our understanding of how these models differ in essential ways from other familiar classes of LP formulations. We will demonstrate, however, that the full power of the LP discriminant analysis models has not been achieved, due to a previously undetected distortion that inhibits the quality of solutions generated. The purpose of this paper is to show how to eliminate this distortion and thereby increase the scope and flexibility of these models. We additionally show how these outcomes open the door to special model manipulations and simplifications, including the use of a successive goal method for establishing a series of conditional objectives to achieve improved discrimination.

Efficient algorithm for optimal force distribution-the compact-dual LP method

Abstract: An efficient algorithm, the compact-dual linear programming (LP) method, is presented to solve the force distribution problem. In this method, the general solution of the linear equality constraints is obtained by transforming the underspecified matrix into row-reduced echelon form; then, the linear equality constraints of the force distribution problem are eliminated. In addition, the duality theory of linear programming is applied. The resulting method is applicable to a wide range of systems, constraints, and objective functions and yet is computationally efficient. The significance of this method is demonstrated by solving the force distribution problem of a grasping system under development at Ohio State called DIGITS. With two fingers grasping an object and hard point contact with friction considered, the CPU time on a VAX-11/785 computer is only 1.47 ms. If four fingers are considered and a linear programming package in the IMSL library is utilized, the CPU time is then less than 45 ms. >

An algorithm for linear programming which requires O(( m + n ) n 2 + ( m + n ) 1.5 n ) L ) arithmetic operations

Abstract: We present an algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a precision of O(L) bits.L is bounded by the number of bits in the input. The worst-case running time of the algorithm is better than that of Karmarkar's algorithm by a factor of $$\sqrt {m + n} $$ .

Linear programming and convex hulls made easy

Abstract: We present two randomized algorithms. One solves linear programs involving m constraints in d variables in expected time O(m). The other constructs convex hulls of n points in Rd, d > 3, in expected time O(n⌈d/2⌉). In both bounds d is considered to be a constant. In the linear programming algorithm the dependence of the time bound on d is of the form d!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.

Parallel processors for planning under uncertainty

Abstract: Our goal is to demonstrate for an important class of multistage stochastic models that three techniques — namely nested decomposition, Monte Carlo importance sampling, and parallel computing — can be effectively combined to solve this fundamental problem of large-scale linear programming.

Feedback control for linear time-invariant systems with state and control bounds in the presence of disturbances

Abstract: The problem of the stabilizing linear control synthesis in the presence of state and input bounds for systems with additive unknown disturbances is considered. The only information required about the disturbances is a finite convex polyhedral bound. Discrete- and continuous-time systems are considered. The property of positive D-invariance of a region is introduced, and it is proved that a solution of the problem is achieved by the selection of a polyhedral set S and the computation of a feedback matrix K such that S is positively D-invariant for the closed-loop system. It is shown that if polyhedral sets are considered, the solution involves simple linear programming algorithms. However, the procedure suggested requires a great amount of computational work offline if the state-space dimension is large, because the feedback matrix K is obtained as a solution of a large set of linear inequalities. All of the vertices of S are required. >

Piecewise linear risk function and portfolio optimization

Abstract: A new portfolio optimization model using a piecewise linear risk function is proposed. This model is similar to, but has several advantages over the classical Markowitz's quadratic risk model. First, it is much easier to generate an optimal portfolio since the problem to be solved is a linear program instead of a quadratic program. Second, integer constraints associated with real transaction can be incorporated without making the problem intractable. Third, it enables us to distinguish two distributions with the same first and second moment but with different third moment. Fourth, we can generate the capital-market line and derive CAPM type equilibrium relations. We compared the piecewise linear risk model with the quadratic risk model using historical data of Tokyo Stock Market, whose results partly support the claims stated above.

Gate sizing in MOS digital circuits with linear programming

Abstract: In this paper a solution is presented to tune the delay of a circuit composed of cells to a prescribed value, while minimizing power consumption. The tuning is performed by adapting the load drive capabilities of the cells. This optimization problem is mapped onto a linear program, which is then solved by the Simplex algorithm. This approach guarantees to find the global optimum, and has proven feasible for circuits of up to several thousand cells. The method can be used with any convex delay model. Results show that circuits can be speeded up by a factor of 2 at a cost of only 10 to 30% of extra power.

Pattern classification using linear programming

Abstract: A method and apparatus (system) is provided for the separation into and the identification of classes of events wherein each of the events is represented by a signal vector comprising the signals x1, x2, . . . , xj, . . . , xn. The system in principle is different from neural networks and statistical classifiers. The system comprises a plurality of assemblies. The training or adaptation module stores a set of training examples and has a set of procedures (linear programming, clustering and others) that operate on the training examples and determine a set of transfer function and threshold values. These transfer functions and threshold values are installed on a recognized module for use in the identification phase. The training module is extremely fast and can deal with very difficult pattern classification problems that arise in speech and image recognition, robotics, medical diagnosis, warfare systems and others. The systems also exploits parallelism in both the learning and recognition phases.

Feasibility issues in a primal-dual interior-point method for linear programming

Abstract: A new method for obtaining an initial feasible interior-point solution to a linear program is presented. This method avoids the use of a "big-M", and is shown to work well on a standard set of test problems. Conditions are developed for obtaining a near-optimal solution that is feasible for an associated problem, and details of the computational testing are presented. Other issues related to obtaining and maintaining accurate feasible solutions to linear programs with an interior-point method are discussed. These issues are important to consider when solving problems that have no primal or dual interior-point feasible solutions.

Incorporating the Decision-maker's Preferences in the Goal-programming Model

Abstract: Many algorithms have been developed for multiple-criteria decision-making problems. Goal programming (GP) is one of these algorithms. This model is a special extension of linear programming. Usually, it is not easy for the decision-maker to choose his aspiration levels a priori. Moreover, the incommensurability of the measurement units of the various objectives creates an aggregation problem. However, in the standard GP formulation, the decision-maker is not required to arbitrate among conflicting objectives. To deal with these difficulties, we explicitly introduce the structure of the decision-maker's preferences into the GP model in order to evaluate the impact of deviations from the decision-maker's aspirations levels. Easily and naturally, the idea of a generalized criterion, as introduced in the Promethee outranking method, will be used to build this structure of preferences.

A centered projective algorithm for linear programming

Abstract: We describe a projective algorithm for linear programming that shares features with Karmarkar's projective algorithm and its variants and with the path-following methods of Gonzaga, Kojima-Mizuno-Yoshise, Monteiro-Adler, Renegar, Vaidya and Ye. It operates in a primal-dual setting, stays close to the central trajectories, and converges in O√n L iterations like the latter methods. Here n is the number of variables and L the input size of the problem. However, it is motivated by seeking reductions in a suitable potential function as in projective algorithms, and the approximate centering is an automatic byproduct of our choice of potential function.

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

Abstract: A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.