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Showing papers on "Linear programming published in 1994"


Journal ArticleDOI
TL;DR: An implementation of an interior point method to the optimal reactive dispatch problem is described in this article, which is based on the primal-dual algorithm and the numerical results in large scale networks (1832 and 3467 bus systems) have shown that this technique can be very effective to some optimal power flow applications.
Abstract: An implementation of an interior point method to the optimal reactive dispatch problem is described. The interior point method used is based on the primal-dual algorithm and the numerical results in large scale networks (1832 and 3467 bus systems) have shown that this technique can be very effective to some optimal power flow applications. >

842 citations


Book
01 Jan 1994
TL;DR: In this paper, the authors present a model-based approach to solving linear programming problems, which is based on the Gauss-Jordan method for solving systems of linear equations, and the Branch-and-Bound method for solving mixed integer programming problems.
Abstract: 1. INTRODUCTION TO MODEL BUILDING. An Introduction to Modeling. The Seven-Step Model-Building Process. Examples. 2. BASIC LINEAR ALGEBRA. Matrices and Vectors. Matrices and Systems of Linear Equations. The Gauss-Jordan Method for Solving Systems of Linear Equations. Linear Independence and Linear Dependence. The Inverse of a Matrix. Determinants. 3. INTRODUCTION TO LINEAR PROGRAMMING. What is a Linear Programming Problem? The Graphical Solution of Two-Variable Linear Programming Problems. Special Cases. A Diet Problem. A Work-Scheduling Problem. A Capital Budgeting Problem. Short-term Financial Planning. Blending Problems. Production Process Models. Using Linear Programming to Solve Multiperiod Decision Problems: An Inventory Model. Multiperiod Financial Models. Multiperiod Work Scheduling. 4. THE SIMPLEX ALGORITHM AND GOAL PROGRAMMING. How to Convert an LP to Standard Form. Preview of the Simplex Algorithm. The Simplex Algorithm. Using the Simplex Algorithm to Solve Minimization Problems. Alternative Optimal Solutions. Unbounded LPs. The LINDO Computer Package. Matrix Generators, LINGO, and Scaling of LPs. Degeneracy and the Convergence of the Simplex Algorithm. The Big M Method. The Two-Phase Simplex Method. Unrestricted-in-Sign Variables. Karmarkar"s Method for Solving LPs. Multiattribute Decision-Making in the Absence of Uncertainty: Goal Programming. Solving LPs with Spreadsheets. 5. SENSITIVITY ANALYSIS: AN APPLIED APPROACH. A Graphical Introduction to Sensitivity Analysis. The Computer and Sensitivity Analysis. Managerial Use of Shadow Prices. What Happens to the Optimal z-value if the Current Basis is No Longer Optimal? 6. SENSITIVITY ANALYSIS AND DUALITY. A Graphical Introduction to Sensitivity Analysis. Some Important Formulas. Sensitivity Analysis. Sensitivity Analysis When More Than One Parameter is Changed: The 100% Rule. Finding the Dual of an LP. Economic Interpretation of the Dual Problem. The Dual Theorem and Its Consequences. Shadow Prices. Duality and Sensitivity Analysis. 7. TRANSPORTATION, ASSIGNMENT, AND TRANSSHIPMENT PROBLEMS. Formulating Transportation Problems. Finding Basic Feasible Solutions for Transportation Problems. The Transportation Simplex Method. Sensitivity Analysis for Transportation Problems. Assignment Problems. Transshipment Problems. 8. NETWORK MODELS. Basic Definitions. Shortest Path Problems. Maximum Flow Problems. CPM and PERT. Minimum Cost Network Flow Problems. Minimum Spanning Tree Problems. The Network Simplex Method. 9. INTEGER PROGRAMMING. Introduction to Integer Programming. Formulation Integer Programming Problems. The Branch-and-Bound Method for Solving Pure Integer Programming Problems. The Branch-and-Bound Method for Solving Mixed Integer Programming Problems. Solving Knapsack Problems by the Branch-and-Bound Method. Solving Combinatorial Optimization Problems by the Branch-and-Bound Method. Implicit Enumeration. The Cutting Plane Algorithm. 10. ADVANCED TOPICS IN LINEAR PROGRAMMING. The Revised Simplex Algorithm. The Product Form of the Inverse. Using Column Generation to Solve Large-Scale LPs. The Dantzig-Wolfe Decomposition Algorithm. The Simplex Methods for Upper-Bounded Variables. Karmarkar"s Method for Solving LPs. 11. NONLINEAR PROGRAMMING. Review of Differential Calculus. Introductory Concepts. Convex and Concave Functions. Solving NLPs with One Variable. Golden Section Search. Unconstrained Maximization and Minimization with Several Variables. The Method of Steepest Ascent. Lagrange Multiples. The Kuhn-Tucker Conditions. Quadratic Programming. Separable Programming. The Method of Feasible Directions. Pareto Optimality and Tradeoff Curves. 12. REVIEW OF CALCULUS AND PROBABILITY. Review of Integral Calculus. Differentiation of Integrals. Basic Rules of Probability. Bayes" Rule. Random Variables. Mean Variance and Covariance. The Normal Distribution. Z-Transforms. Review Problems. 13. DECISION MAKING UNDER UNCERTAINTY. Decision Criteria. Utility Theory. Flaws in Expected Utility Maximization: Prospect Theory and Framing Effects. Decision Trees. Bayes" Rule and Decision Trees. Decision Making with Multiple Objectives. The Analytic Hierarchy Process. Review Problems. 14. GAME THEORY. Two-Person Zero-Sum and Constant-Sum Games: Saddle Points. Two-Person Zero-Sum Games: Randomized Strategies, Domination, and Graphical Solution. Linear Programming and Zero-Sum Games. Two-Person Nonconstant-Sum Games. Introduction to n-Person Game Theory. The Core of an n-Person Game. The Shapley Value. 15. DETERMINISTIC EOQ INVENTORY MODELS. Introduction to Basic Inventory Models. The Basic Economic Order Quantity Model. Computing the Optimal Order Quantity When Quantity Discounts Are Allowed. The Continuous Rate EOQ Model. The EOQ Model with Back Orders Allowed. Multiple Product Economic Order Quantity Models. Review Problems. 16. PROBABILISTIC INVENTORY MODELS. Single Period Decision Models. The Concept of Marginal Analysis. The News Vendor Problem: Discrete Demand. The News Vendor Problem: Continuous Demand. Other One-Period Models. The EOQ with Uncertain Demand: the (r, q) and (s,S models). The EOQ with Uncertain Demand: the Service Level Approach to Determining Safety Stock Level. Periodic Review Policy. The ABC Inventory Classification System. Exchange Curves. Review Problems. 17. MARKOV CHAINS. What is a Stochastic Process. What is a Markov Chain? N-Step Transition Probabilities. Classification of States in a Markov Chain. Steady-State Probabilities and Mean First Passage Times. Absorbing Chains. Work-Force Planning Models. 18.DETERMINISTIC DYNAMIC PROGRAMMING. Two Puzzles. A Network Problem. An Inventory Problem. Resource Allocation Problems. Equipment Replacement Problems. Formulating Dynamic Programming Recursions. The Wagner-Whitin Algorithm and the Silver-Meal Heuristic. Forward Recursions. Using Spreadsheets to Solve Dynamic Programming Problems. Review Problems. 19. PROBABILISTIC DYNAMIC PROGRAMMING. When Current Stage Costs are Uncertain but the Next Period"s State is Certain. A Probabilistic Inventory Model. How to Maximize the Probability of a Favorable Event Occurring. Further Examples of Probabilistic Dynamic Programming Formulations. Markov Decision Processes. Review Problems. 20. QUEUING THEORY. Some Queuing Terminology. Modeling Arrival and Service Processes. Birth-Death Processes. M/M/1/GD/o/o Queuing System and the Queuing Formula L=o W, The M/M/1/GD/o Queuing System. The M/M/S/ GD/o/o Queuing System. The M/G/ o/GD/oo and GI/G/o/GD/o/oModels. The M/ G/1/GD/o/o Queuing System. Finite Source Models: The Machine Repair Model. Exponential Queues in Series and Opening Queuing Networks. How to Tell whether Inter-arrival Times and Service Times Are Exponential. The M/G/S/GD/S/o System (Blocked Customers Cleared). Closed Queuing Networks. An Approximation for the G/G/M Queuing System. Priority Queuing Models. Transient Behavior of Queuing Systems. Review Problems. 21.SIMULATION. Basic Terminology. An Example of a Discrete Event Simulation. Random Numbers and Monte Carlo Simulation. An Example of Monte Carlo Simulation. Simulations with Continuous Random Variables. An Example of a Stochastic Simulation. Statistical Analysis in Simulations. Simulation Languages. The Simulation Process. 22.SIMULATION WITH PROCESS MODEL. Simulating an M/M/1 Queuing System. Simulating an M/M/2 System. A Series System. Simulating Open Queuing Networks. Simulating Erlang Service Times. What Else Can Process Models Do? 23. SPREADSHEET SIMULATION WITH @RISK. Introduction to @RISK: The Newsperson Problem. Modeling Cash Flows From A New Product. Bidding Models. Reliability and Warranty Modeling. Risk General Function. Risk Cumulative Function. Risktrigen Function. Creating a Distribution Based on a Point Forecast. Forecasting Income of a Major Corporation. Using Data to Obtain Inputs For New Product Simulations. Playing Craps with @RISK. Project Management. Simulating the NBA Finals. 24. FORECASTING. Moving Average Forecasting Methods. Simple Exponential Smoothing. Holt"s Method: Exponential Smoothing with Trend. Winter"s Method: Exponential Smoothing with Seasonality. Ad Hoc Forecasting, Simple Linear Regression. Fitting Non-Linear Relationships. Multiple Regression. Answers to Selected Problems. Index.

427 citations


Journal ArticleDOI
TL;DR: This work uses time-domain input-output data to validate uncertainty models and develops algorithms that are computationally tractable and reduce to (generally nondifferentiable) convex feasibility problems or to linear programming problems.
Abstract: In this paper we offer a novel approach to control-oriented model validation problems. The problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input-output data. Our approach directly uses time-domain input-output data to validate uncertainty models. The algorithms we develop are computationally tractable and reduce to (generally nondifferentiable) convex feasibility problems or to linear programming problems. In special cases, we give analytical solutions to these problems. >

372 citations


Journal ArticleDOI
TL;DR: This paper focused on two kinds of linear programmings with fuzzy numbers, called interval number and fuzzy number linearprogrammings, respectively, and gave the numerical solutions of the illustrative examples.

367 citations


Journal ArticleDOI
TL;DR: This work presents an O√nL-iteration homogeneous and self-dual linear programming LP algorithm, which solves the linear programming problem without any regularity assumption concerning the existence of optimal, feasible, or interior feasible solutions.
Abstract: We present an O√nL-iteration homogeneous and self-dual linear programming LP algorithm. The algorithm possesses the following features: • It solves the linear programming problem without any regularity assumption concerning the existence of optimal, feasible, or interior feasible solutions. • It can start at any positive primal-dual pair, feasible or infeasible, near the central ray of the positive orthant cone, and it does not use any big M penalty parameter or lower bound. • Each iteration solves a system of linear equations whose dimension is almost the same as that solved in the standard primal-dual interior-point algorithms. • If the LP problem has a solution, the algorithm generates a sequence that approaches feasibility and optimality simultaneously; if the problem is infeasible or unbounded, the algorithm will correctly detect infeasibility for at least one of the primal and dual problems.

363 citations


Journal ArticleDOI
TL;DR: The present paper is intended to review the existing literature on multi-objective combinatorial optimization (MOCO) problems and examines various classical combinatorials problems in a multi-criteria framework.
Abstract: In the last 20 years many multi-objective linear programming (MOLP) methods with continuous variables have been developed. However, in many real-world applications discrete variables must be introduced. It is well known that MOLP problems with discrete variables can have special difficulties and so cannot be solved by simply combining discrete programming methods and multi-objective programming methods. The present paper is intended to review the existing literature on multi-objective combinatorial optimization (MOCO) problems. Various classical combinatorial problems are examined in a multi-criteria framework. Some conclusions are drawn and directions for future research are suggested.

320 citations


Journal ArticleDOI
TL;DR: The primal-dual predictor-corrector (OB1) algorithm as mentioned in this paper is one of the state-of-the-art algorithms for linear programming with respect to interior point methods.
Abstract: A survey of the significant developments in the field of interior point methods for linear programming is presented, beginning with Karmarkar's projective algorithm and concentrating on the many variants that can be derived from logarithmic barrier methods. Full implementation details of the primal-dual predictor-corrector code OB1 are given, including preprocessing, matrix orderings, and matrix factorization techniques. A computational comparison of OB1 with a state-of-the-art simplex code using eight large models is given. In addition, computational results are presented where OB1 is used to solve two very large models that have never been solved by any simplex code INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

272 citations


Journal ArticleDOI
TL;DR: In this article, the adjoint variable is used for sensitivity analysis and the linear programming method is used to obtain the optimal topology, which can handle various problems, for example, multiple objective functions and multiple design criteria.

271 citations


Journal ArticleDOI
TL;DR: Computational results indicate that by using an appropriate combination of constraints, the gap between the lower and upper bounds at the root of the search tree can be reduced considerably.

243 citations



Book
31 Mar 1994
TL;DR: In this article, the authors introduce the logarithmic barrier method (LBP) and the center method (CPM) for reducing the complexity of LP and CPM, respectively.
Abstract: Glossary of Symbols and Notations. 1. Introduction of IPMs. 2. The logarithmic barrier method. 3. The center method. 4. Reducing the complexity for LP. 5. Discussion of other IPMs. 6. Summary, conclusions and recommendations. Appendices: A. Self-concordance proofs. B. General technical lemmas. Bibliography. Index.

Journal ArticleDOI
TL;DR: The proposed planning model is a two-stage stochastic linear program (SLP) with recourse, a sampling based algorithm called Stochastic decomposition (SD) for very large-scale SLPs, such as the ones solved in this application.
Abstract: We study a planning problem associated with networks for private line services. In these networks, demands are known to exhibit considerable variability, and as such, they should be treated as random variables. The proposed planning model is a two-stage stochastic linear program (SLP) with recourse. Due to the enormous size of the deterministic equivalent, we choose a sampling based algorithm calledstochastic decomposition (SD). For very large-scale SLPs, such as the ones solved in this application, SD provides an effective methodology. The model presented in this paper is validated by using a detailed simulation of the network. We report results with a network that has 86 demand pairs, 89 links and 706 potential routes.

Journal ArticleDOI
TL;DR: A new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies is introduced, and analytic bounds which improve upon Kingman's bound (1970) for E/sub 2//M/1 queues are obtained.
Abstract: Except for the class of queueing networks and scheduling policies admitting a product form solution for the steady-state distribution, little is known about the performance of such systems. For example, if the priority of a part depends on its class (e.g., the buffer that the part is located in), then there are no existing results on performance, or even stability. In most applications such as manufacturing systems, however, one has to choose a control or scheduling policy, i.e., a priority discipline, that optimizes a performance objective. In this paper the authors introduce a new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies. Assuming stability, and examining the consequence of a steady state for general quadratic forms, the authors obtain a set of linear equality constraints on the mean values of certain random variables that determine the performance of the system. Further, the conservation of time and material gives an augmenting set of linear equality and inequality constraints. Together, these allow the authors to bound the performance, either above or below, by solving a linear program. The authors illustrate this technique on several typical problems of interest in manufacturing systems. For an open re-entrant line modeling a semiconductor plant, the authors plot a bound on the mean delay (called cycle-time) as a function of line loading. It is shown that the last buffer first serve policy is almost optimal in light traffic. For another such line, it is shown that it dominates the first buffer first serve policy. For a set of open queueing networks, the authors compare their lower bounds with those obtained by another method of Ou and Wein (1992). For a closed queueing network, the authors bracket the performance of all buffer priority policies, including the suggested priority policy of Harrison and Wein (1990). The authors also study the asymptotic heavy traffic limits of the lower and upper bounds. For a manufacturing system with machine failures, it is shown how the performance changes with failure and repair rates. For systems with finite buffers, the authors show how to bound the throughput. Finally, the authors illustrate the application of their method to GI/GI/1 queues. The authors obtain analytic bounds which improve upon Kingman's bound (1970) for E/sub 2//M/1 queues. >

Journal ArticleDOI
TL;DR: Using a powerful technique involving quadratic or higher order potential functions, methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies are proposed and found that the first order approximation of the method is at least as good as simulation-based existing methods.
Abstract: We consider open and closed multiclass queueing networks, with Poisson arrivals (for open networks), exponentially distributed class dependent service times and class dependent deterministic or probabilistic routing. The performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies. By optimizing over these sets, we obtain lower bounds on achievable performance. In the special case of single station networks (multiclass queues and Klimov's model) and homogeneous multiclass networks, the polyhedron derived is exactly equal to the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. We apply the same approach to closed networks to obtain upper bounds on the optimal throughput. We check the tightness of our bounds by simulating heuristic policies and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with a number of variables and constraints that is polynomial (quadratic) in the number of classes in the network. The $i$th order approximation leads to a convex programming problem in dimension $O(R^{i+1})$, where $R$ is the number of classes in the network, and can be solved efficiently using techniques from semidefinite programming.

Journal ArticleDOI
TL;DR: MINTO is a software system that solves mixed-integer linear programs by a branch-and-bound algorithm with linear programming relaxations and provides automatic constraint classification, preprocessing, primal heuristics and constraint generation.

Journal ArticleDOI
01 Jan 1994
TL;DR: The inability of the classical technique to handle the line flow constraints so far is circumvented innovatively by expressing the line flows in terms of active power generations through distribution factors that are elegantly developed from existing load flow information using a perturbation technique.
Abstract: An attempt is made to explore the feasibility of developing an approach to solve the power system economic emission load dispatch (EELD) problem with line flow constraints using a classical technique based on co-ordination equations. The inability of the classical technique to handle the line flow constraints so far is circumvented innovatively by expressing the line flows in terms of active power generations through distribution factors. These distribution factors are elegantly developed from existing load flow information using a perturbation technique. The proposed model based on the classical technique for EELD is tested on IEEE 14- and 30-bus test systems and the results are compared with those obtained by quadratic programming, the Hessian method, the Ricochet gradient method and the linear programming method.

Journal ArticleDOI
TL;DR: In this article, an extended quadratic interior point (EQIP) method is proposed to solve power system optimization problem (PSOP), such as economic dispatch (ED) and VAr planning (VP) problems.
Abstract: Karmarkar's interior point method as a computation method for solving linear programming (LP) has attracted interest in the operation research community, due to its efficiency, reliability, and accuracy. This paper presents an extended quadratic interior point (EQIP) method, based on improvement of initial condition for solving both linear and quadratic programming problems, to solve power system optimization problem (PSOP), such as economic dispatch (ED) and VAr planning (VP) problems. The EQIP method is able to accommodate the nonlinearity in objectives and constraints. The scheme is demonstrated on several IEEE standard systems and is capable of achieving fast convergence and improvement in computational speed over an existing efficient Simplex, such as the MINOS code. The number of iterations during the computation is relatively insensitive to numbers of controls and constraints. Moreover, the proposed EQIP method guarantees a global optimum within the interior feasible region. >

Journal ArticleDOI
TL;DR: In this article, a single linear program is proposed for discriminating between the elements of κ disjoint point sets in the n-dimensional real space Rn, where the conical hulls of the κ sets are not (κ;−1)-point disjoins in Rn + 1.
Abstract: A single linear program is proposed for discriminating between the elements of κ disjoint point sets in the n-dimensional real space Rn . When the conical hulls of the κ sets are (κ−1)-point disjoint in Rn +1, a κ-piece piecewise-linear surface generated by the linear program completely separates the κ sets. This improves on a previous linear programming approach which required that each set be linearly separable from the remaining κ−1 sets. When the conical hulls of the κ sets are not (κ;−1)-point disjoint, the proposed linear program generates an error-minimizing piecewise-linear separator for the κ Sets. For this case it is shown that the null solution is never a unique solver of the linear program and occurs only under the rather rare condition when the mean of each point set equals the mean of the means of the other κ−l sets. This makes the proposed linear computational programming formulation useful for approximately discriminating between κ sets that are not piecewise-linear separable. Computationa...

Journal ArticleDOI
TL;DR: In this paper, a rigorous mathematical method is proposed for dealing with the ramp-rate limits in unit commitment and the rotor fatigue effect in economic dispatch, where the Lagrangian relaxation method is used to generate the unit commitment schedule with relaxed power balance constraints and a network model is adopted to represent the dynamic process of operating a unit over the entire study time span.
Abstract: In this study, a rigorous mathematical method is proposed for dealing with the ramp-rate limits in unit commitment and the rotor fatigue effect in economic dispatch An iterative procedure is employed to coordinate the unit commitment and the power dispatch for obtaining an economical solution within a reasonable time The Lagrangian relaxation method is used to generate the unit commitment schedule with relaxed power balance constraints A network model is adopted to represent the dynamic process of operating a unit over the entire study time span, as the required unit commitment schedule can be achieved by searching for the shortest path in the network In order to find the global optimal solution for the economic dispatch problem within personal computer resources, a piecewise linear model is used for thermal units Furthermore, linear programming is used in optimizing the benefits of ramping the units, with low operating cost against the cost of shortening the service life of the turbine rotor In this regard, linear programming is used to dispatch the power generation among committed units by considering a ramping penalty for the fatigue effect in rotor shafts, while preserving the operational constraints of the system as well as the generating units >

Book
01 Jan 1994
TL;DR: A Numerical Study of Some Data Association Problems Arising in Multitarget Tracking and a Simple, Quadratically Convergent Interior Point Algorithm for Linear Programming and Convex Quadratic Programming.
Abstract: Preface. Restarting Strategies for the DQA Algorithm A.J. Berger, J.M. Mulvey, A. Ruszczynski. Mathematical Equivalence of the Auction Algorithm for Assignment and the epsilon-Relaxation (Preflow-Push) Method for Min Cost Flow D.P. Bertsekas. Preliminary Computational Experience with Modified Log-Barrier Functions for Large-Scale Nonlinear Programming M.G. Breitfeld, D.F. Shanno. A New Stochastic/Perturbation Method for Large-Scale Global Optimization and its Application to Water Cluster Problems R.H. Byrd, T. Derby, E. Eskow, K.P.B. Oldenkamp, R.B. Schnabel. Improving the Decomposition of Partially Separable Functions in the Context of Large-Scale Optimization: a First Approach A.R. Conn, N. Gould, P.L. Toint. Gradient-Related Constrained Minimization Algorithms in Function Spaces: Convergence Properties and Computational Implications J.C. Dunn. Some Reformulations and Applications of the Alternating Direction Method of Multipliers J. Eckstein, M. Fukushima. Experience with a Primal Presolve Algorithm R. Fourer, D.M. Gay. A Trust Region Method for Constrained Nonsmooth Equations S.A. Gabriel, Jong-Shi Pang. On the Complexity of a Column Generation Algorithm for Convex or Quasiconvex Feasibility Problems J.-L. Goffin, Zhi-Quan Luo, Yinyu Ye. Identificiation of the Support of Nonsmoothness C.T. Kelley. On Very Large Scale Assignment Problems Y. Lee, J.B. Orlin. Numerical Solution of Parabolic State Constrained Control Problems Using SQP and Interior-Point-Methods F. Leibfritz, E.W. Sachs. A Global Optimization Method for Weber's Problem with Attraction and Repulsion C.D. Maranas, C.A. Floudas. Large-Scale Diversity Minimization via Parallel Genetic Algorithms R.R. Meyer, J. Yackel. A Numerical Comparison of Barrier andModified Barrier Methods for Large-Scale Bound-Constrained Optimization S.G. Nash, R. Polyak, A. Sofer. A Numerical Study of Some Data Association Problems Arising in Multitarget Tracking A.B. Poore, N. Rijavec. Identifying the Optimal Face of a Network Linear Program with a Globally Convergent Interior Point Method M.G.C. Resende, T. Tsuchiya, G. Veiga. Solution of Large Scale Stochastic Programs with Stochastic Decomposition Algorithms S. Sen, J.Mai, J.L. Higle. A Simple, Quadratically Convergent Interior Point Algorithm for Linear Programming and Convex Quadratic Programming A.L. Tits, J.L. Zhou. On Two Algorithms for Nonconvex Nonsmooth Optimization Problems in Structural Mechanics M.Ap. Tzaferopoulos, E.S. Mistakidis, C.D. Bisbos, P.D. Panagiotopoulos.

Journal ArticleDOI
TL;DR: In this article, the authors gave an O(m^2 \log m) expected-time randomized algorithm for concurrent multicommodity flow with uniform capacities, where m is the number of wires to be routed in an n-node, m-edge network.
Abstract: This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniform-capacity concurrent flow problem has many interesting applications. Leighton and Rao used uniform-capacity concurrent flow to find an approximately "sparsest cut" in a graph and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. However, their method appeared to be impractical as it required solving a large linear program. This paper shows that their method might be practical by giving an $O(m^2 \log m)$ expected-time randomized algorithm for their concurrent flow problem on an $m$-edge graph. Raghavan and Thompson used uniform-capacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration. An $O(k^{3/2} (m + n \log n))$ expected-time randomized algorithm and an $O(k\min{n,k} (m+n\log n)\log k)$ deterministic algorithm is given for this problem when the channel width is $\Omega(\log n)$, where $k$ denotes the number of wires to be routed in an $n$-node, $m$-edge network.

Journal ArticleDOI
TL;DR: An optimal dynamic solution is presented that simplifies the structure of the control mechanism by exercising ground-holding on groups of aircraft instead of individual flights by using stochastic linear programming with recourse.
Abstract: Existing probabilistic solutions to the ground-holding problem in air traffic control are of a static nature, with ground-holds assigned to aircraft at the beginning of daily operations. In this paper we present an optimal dynamic solution that simplifies the structure of the control mechanism by exercising ground-holding on groups of aircraft instead of individual flights. Using stochastic linear programming with recourse, we have been able to solve problem instances for one of the largest airports in the U.S. with just a powerful PC. We illustrate the advantage of the probabilistic dynamic solution over: (a) the static solution; (b) a deterministic solution; and (c) the passive strategy of no ground-holding.

Journal ArticleDOI
TL;DR: This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems and shows that feasibility as well as min-mix resource-sharing problems for such constraints can be solved to a relative accuracy.
Abstract: This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems. The constraints of such problems consist of K disjoint convex compact sets $B^k $ called blocks, and M nonnegative-valued convex block-separable inequalities called coupling or resource constraints. The algorithms are based on an exponential potential function reduction technique. It is shown that feasibility as well as min-mix resource-sharing problems for such constraints can be solved to a relative accuracy $\varepsilon$ in $O( K\ln M ( \varepsilon^{ - 2} + \ln K ) )$ iterations, each of which solves K block problems to a comparable accuracy, either sequentially or in parallel. The same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration. An extension to objective and constraint functions of arbitrary sign is also presented. The above results yield fast approximatio...

Journal ArticleDOI
TL;DR: An approximation lemma is presented and it is shown that in all cases arbitrary probability vectors can be replaced by sparse ones (with only logarithmically many positive entries) without losing too much performance.

Journal ArticleDOI
TL;DR: The problem is formulated as a stochastic integer linear program, with first stage binary variables and second stage continuous variables, and solved to optimality by means of a branch and cut method.
Abstract: This paper considers a class of Capacitated Facility Location Problems in which customer demands are stochastic. The problem is formulated as a stochastic integer linear program, with first stage binary variables and second stage continuous variables. It is solved to optimality by means of a branch and cut method. Computational results are reported for problems involving up to 40 customers and 10 potential facility locations.

Journal ArticleDOI
TL;DR: This paper presents EVBDD-based algorithms for solving integer linear programs, computing spectral coefficients of Boolean functions, and performing function decomposition under the SIS environment.
Abstract: Edge-Valued Binary-Decision Diagrams (EVBDD's) are directed acyclic graphs that can represent and manipulate integer functions as effectively as Ordered Binary-Decision Diagrams OBDD's) do for Boolean functions. They have been used in logic verification for showing the equivalence between Boolean functions and arithmetic functions. In this paper, we present EVBDD-based algorithms for solving integer linear programs, computing spectral coefficients of Boolean functions, and performing function decomposition. These algorithms have been implemented in C under the SIS environment and experimental results are provided. >

Journal ArticleDOI
TL;DR: A nonproportional-assignment, user-equilibrium motivated, linear programming model for estimating origin-destination trip tables from available data on link traffic volumes and a column generation solution technique is presented to optimally solve the problem.
Abstract: We present a nonproportional-assignment, user-equilibrium motivated, linear programming model for estimating origin-destination (O-D) trip tables from available data on link traffic volumes. The model is designed to determine a traffic equilibrium network flow solution that reproduces the link volume data, if such a solution exists. However, it recognizes that due to incomplete information, the traffic may not conform to an equilibrium flow pattern, and moreover, there might be inconsistencies in the observed link flow data. Accordingly, the model permits violations in the equilibrium conditions as well as deviations from the observed link flows but at suitable incurred penalties in the objective function. A column generation solution technique is presented to optimally solve the problem. This methodology is extended to the situation in which a specified prior target trip table is available and one is required to find a solution that also has a tendency to match this table as closely as possible. Implementation strategies are discussed and the proposed method is illustrated using some sample test networks from the literature.

Journal ArticleDOI
TL;DR: The main focus of this part of the survey is on recent results that are related to the basic problems of computing, approximating, or measuring the convex sets which are the smallest that contain a given convex body K or are the largest contained in K.

Journal ArticleDOI
TL;DR: This paper proves that any feasible solution is associated with a valuation called a conservative height on the graph of constraints, and proposes a branch and bound enumeration procedure and two heuristics solving the problem.

Journal ArticleDOI
TL;DR: This paper shows that a modification of the Kojima—Megiddo—Mizuno algorithm “solves” the pair of problems in polynomial time without assuming the existence of the LP solution, and develops anO(nL)-iteration complexity result for a variant of the algorithm.
Abstract: Kojima, Megiddo, and Mizuno investigate an infeasible-interior-point algorithm for solving a primal--dual pair of linear programming problems and they demonstrate its global convergence. Their algorithm finds approximate optimal solutions of the pair if both problems have interior points, and they detect infeasibility when the sequence of iterates diverges. Zhang proves polynomial-time convergence of an infeasible-interior-point algorithm under the assumption that both primal and dual problems have feasible points. In this paper, we show that a modification of the Kojima--Megiddo--Mizuno algorithm "solves" the pair of problems in polynomial time without assuming the existence of the LP solution. Furthermore, we develop anO(nL)-iteration complexity result for a variant of the algorithm.