Showing papers on "Linear programming published in 2000"

Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models

Abstract: 1 Introduction- I Optimization Techniques- 2 Introduction to Optimization- 3 Linear Optimization- 4 Nonlinear Local Optimization- 5 Nonlinear Global Optimization- 6 Unsupervised Learning Techniques- 7 Model Complexity Optimization- II Static Models- 9 Introduction to Static Models- 10 Linear, Polynomial, and Look-Up Table Models- 11 Neural Networks- 12 Fuzzy and Neuro-Fuzzy Models- 13 Local Linear Neuro-Fuzzy Models: Fundamentals- 14 Local Linear Neuro-Fuzzy Models: Advanced Aspects- III Dynamic Models- 16 Linear Dynamic System Identification- 17 Nonlinear Dynamic System Identification- 18 Classical Polynomial Approaches- 19 Dynamic Neural and Fuzzy Models- 20 Dynamic Local Linear Neuro-Fuzzy Models- 21 Neural Networks with Internal Dynamics- IV Applications- 22 Applications of Static Models- 23 Applications of Dynamic Models- 24 Applications of Advanced Methods- A Vectors and Matrices- A1 Vector and Matrix Derivatives- A2 Gradient, Hessian, and Jacobian- B Statistics- B1 Deterministic and Random Variables- B2 Probability Density Function (pdf)- B3 Stochastic Processes and Ergodicity- B4 Expectation- B5 Variance- B6 Correlation and Covariance- B7 Properties of Estimators- References

The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm

Abstract: The purpose of this work is to present the MOSEK optimizer intended for solution of large-scale sparse linear programs. The optimizer is based on the homogeneous interior-point algorithm which in contrast to the primal-dual algorithm detects a possible primal or dual infeasibility reliably. It employs advanced (parallelized) linear algebra, it handles dense columns in the constraint matrix efficiently, and it has a basis identification procedure.

Observability and controllability of piecewise affine and hybrid systems

Abstract: We prove, in a constructive way, the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics, automata, and propositional logic. By focusing our investigation on the former class, we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems. Instead, we propose practical numerical tests based on mixed-integer linear programming.

Optimal response of a thermal unit to an electricity spot market

Abstract: This paper addresses the optimal response of a thermal unit to an electricity spot market. The objective is to maximize the unit profit from selling both energy and spinning reserve in the spot market. The paper proposes a 0/1 mixed-integer linear programming approach that allows a rigorous modeling of (i) nonconvex and nondifferentiable operating costs, (ii) exponential start-up costs, (iii) available spinning reserve taking into account ramp rate restrictions, and (iv) minimum up and down time constraints. This approach overcomes the modeling limitations of dynamic programming approaches and is computationally efficient. Results from realistic case studies are reported.

A Linear Programming Model for the Single Destination System Optimum Dynamic Traffic Assignment Problem

Abstract: Recently, Daganzo introduced the cell transmission model--a simple approach for modeling highway traffic flow consistent with the hydrodynamic model. In this paper, we use the cell transmission model to formulate the single destination System Optimum Dynamic Traffic Assignment (SO DTA) problem as a Linear Program (LP). We demonstrate that the model can obtain insights into the DTA problem, and we address various related issues, such as the concept of marginal travel time in a dynamic network and system optimum necessary and sufficient conditions. The model is limited to one destination and, although it can account for traffic realities as they are captured by the cell transmission model, it is not presented as an operational model for actual applications. The main objective of the paper is to demonstrate that the DTA problem can be modeled as an LP, which allows the vast existing literature on LP to be used to better understand and compute DTA. A numerical example illustrates the simplicity and applicability of the proposed approach.

The Discrepancy Method: Randomness and Complexity

Abstract: This book tells the story of the discrepancy method in a few short independent vignettes. It is a varied tale which includes such topics as communication complexity, pseudo-randomness, rapidly mixing Markov chains, points on the sphere and modular forms, derandomization, convex hulls, Voronoi diagrams, linear programming and extensions, geometric sampling, VC-dimension theory minimum spanning trees, linear circuit complexity, and multidimensonal searching.

Scheduling Aircraft Landings--The Static Case

Abstract: In this paper, we consider the problem of scheduling aircraft (plane) landings at an airport. This problem is one of deciding a landing time for each plane such that each plane lands within a predetermined time window and that separation criteria between the landing of a plane and the landing of all successive planes are respected. We present a mixed-integer zero--one formulation of the problem for the single runway case and extend it to the multiple runway case. We strengthen the linear programming relaxations of these formulations by introducing additional constraints. Throughout, we discuss how our formulations can be used to model a number of issues (choice of objective function, precedence restrictions, restricting the number of landings in a given time period, runway workload balancing) commonly encountered in practice. The problem is solved optimally using linear programming-based tree search. We also present an effective heuristic algorithm for the problem. Computational results for both the heuristic and the optimal algorithm are presented for a number of test problems involving up to 50 planes and four runways.

Using Branch-and-Price-and-Cut to Solve Origin-Destination Integer Multicommodity Flow Problems

Abstract: We present a column-generation model and branch-and-price-and-cut algorithm for origin-destination integer multicommodity flow problems. The origin-destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity (defined in this case by an origin-destination pair) may use only one path from origin to destination. Branch-and-price-and-cut is a variant of branch-and-bound, with bounds provided by solving linear programs using column-and-cut generation at nodes of the branch-and-bound tree. Because our model contains one variable for each origin destination path, for every commodity, the linear programming relaxations at nodes of the branch-and-bound tree are solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. We devise a new branching rule that allows columns to be generated efficiently at each node of the branch-and-bound tree. Then, we describe cuts (cover inequalities) that can be generated at each node of the branch-and-bound tree. These cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry. We detail the implementation of our combined column and- cut generation method and present computational results for a set of test problems arising from telecommunications applications. We illustrate the value of our branching rule when used to find a heuristic solution and compare branch-and-price and branch-and-price-and-cut methods to find optimal solutions for highly capacitated problems.

Electric Power System Applications of Optimization

Abstract: Electric power system models power flow computation constrained optimization and applications linear programming and applications interior point methods non-linear programming dynamic programming Lagrange relaxation decomposition method optimal power flow unit commitment genetic algorithm epilogue.

Linear programming with interval coefficients

Abstract: In order to solve a linear programme, the model coefficients must be fixed at specific values, which implies that the coefficients are perfectly accurate. In practice, however, the coefficients are generally estimates. The only way to deal with uncertain coefficients is to test the sensitivity of the model to changes in their values, either singly or in very small groups. We propose a new approach in which some or all of the coefficients of the LP are specified as intervals. We then find the best optimum and the worst optimum for the model, and the point settings of the interval coefficients that yield these two extremes. This provides the range of the optimised objective function, and the coefficient settings give some insight into the likelihood of these extremes.

The volume algorithm: producing primal solutions with a subgradient method

Abstract: We present an extension to the subgradient algorithm to produce primal as well as dual solutions. It can be seen as a fast way to carry out an approximation of Dantzig-Wolfe decomposition. This gives a fast method for producing approximations for large scale linear programs. It is based on a new theorem in linear programming duality. We present successful experience with linear programs coming from set partitioning, set covering, max-cut and plant location.

A Discrete Strategy Improvement Algorithm for Solving Parity Games

Abstract: A discrete strategy improvement algorithm is given for constructing winning strategies in parity games, thereby providing also a new solution of the model-checking problem for the modal μ-calculus Known strategy improvement algorithms, as proposed for stochastic games by Hoffman and Karp in 1966, and for discounted payoff games and parity games by Puri in 1995, work with real numbers and require solving linear programming instances involving high precision arithmetic In the present algorithm for parity games these difficulties are avoided by the use of discrete vertex valuations in which information about the relevance of vertices and certain distances is coded An efficient implementation is given for a strategy improvement step Another advantage of the present approach is that it provides a better conceptual understanding and easier analysis of strategy improvement algorithms for parity games However, so far it is not known whether the present algorithm works in polynomial time The long standing problem whether parity games can be solved in polynomial time remains open

Restless Bandits, Linear Programming Relaxations, and a Primal-Dual Index Heuristic

Abstract: We develop a mathematical programming approach for the classicalPSPACE-hard restless bandit problem in stochastic optimization. We introduce a hierarchy ofN (whereN is the number of bandits) increasingly stronger linear programming relaxations, the last of which is exact and corresponds to the (exponential size) formulation of the problem as a Markov decision chain, while the other relaxations provide bounds and are efficiently computed. We also propose a priority-index heuristic scheduling policy from the solution to the firstorder relaxation, where the indices are defined in terms of optimal dual variables. In this way we propose a policy and a suboptimality guarantee. We report results of computational experiments that suggest that the proposed heuristic policy is nearly optimal. Moreover, the second-order relaxation is found to provide strong bounds on the optimal value.

Linear complementarity systems

Abstract: We introduce a new class of dynamical systems called "linear complementarity systems." The time evolution of these systems consists of a series of continuous phases separated by "events" which cause a change in dynamics and possibly a jump in the state vector. The occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming. The framework we describe is suitable for certain situations in which both differential equations and inequalities play a role; for instance, in mechanics, electrical networks, piecewise linear systems, and dynamic optimization. We present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions.

A homogeneous linear programming algorithm for the security constrained economic dispatch problem

Abstract: This paper presents a study of the simplified homogeneous and self-dual (SHSD) linear programming (LP) interior point algorithm applied to the security constrained economic dispatch (SCED) problem. Unlike other interior point SCED applications that consider only the N security problem, this paper considers both (N-1) and (N-2) network security conditions. An important feature of the optimizing interior point LP algorithm is that it can detect infeasibility of the SCED problem reliably. This feature is particularly important in SCED applications since line overloading following a contingency often results in an infeasible schedule. The proposed method is demonstrated on the IEEE 24 bus test system and a practical 175 bus network. A comparison is carried out with the predictor-corrector interior point algorithm for the SCED problem presented previously (see ibid., vol. 12, no.2, p.803-10, 1997).

An Algorithm for the Solution of Multiparametric Mixed Integer Linear Programming Problems

Abstract: In this paper, we present an algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems involving (i) 0-1 integer variables, and, (ii) more than one parameter, bounded between lower and upper bounds, present on the right hand side (RHS) of constraints. The solution is approached by decomposing the mp-MILP into two subproblems and then iterating between them. The first subproblem is obtained by fixing integer variables, resulting in a multiparametric linear programming (mp-LP) problem, whereas the second subproblem is formulated as a mixed integer linear programming (MILP) problem by relaxing the parameters as variables.

A fuzzy programming method for deriving priorities in the analytic hierarchy process

Abstract: The estimation of the priorities from pairwise comparison matrices is the major constituent of the Analytic Hierarchy Process (AHP). The priority vector can be derived from these matrices using different techniques, as the most commonly used are the Eigenvector Method (EVM) and the Logarithmic Least Squares Method (LLSM). In this paper a new Fuzzy Programming Method (FPM) is proposed, based on geometrical representation of the prioritisation process. This method transforms the prioritisation problem into a fuzzy programming problem that can easily be solved as a standard linear programme. The FPM is compared with the main existing prioritisation methods in order to evaluate its performance. It is shown that it possesses some attractive properties and could be used as an alternative to the known prioritisation methods, especially when the preferences of the decision-maker are strongly inconsistent.

The onion technique: indexing for linear optimization queries

Abstract: This paper describes the Onion technique, a special indexing structure for linear optimization queries. Linear optimization queries ask for top-N records subject to the maximization or minimization of linearly weighted sum of record attribute values. Such query appears in many applications employing linear models and is an effective way to summarize representative cases, such as the top-50 ranked colleges. The Onion indexing is based on a geometric property of convex hull, which guarantees that the optimal value can always be found at one or more of its vertices. The Onion indexing makes use of this property to construct convex hulls in layers with outer layers enclosing inner layers geometrically. A data record is indexed by its layer number or equivalently its depth in the layered convex hull. Queries with linear weightings issued at run time are evaluated from the outmost layer inwards. We show experimentally that the Onion indexing achieves orders of magnitude speedup against sequential linear scan when N is small compared to the cardinality of the set. The Onion technique also enables progressive retrieval, which processes and returns ranked results in a progressive manner. Furthermore, the proposed indexing can be extended into a hierarchical organization of data to accommodate both global and local queries.

Optimum positioning of base stations for cellular radio networks

Abstract: Finding optimum base station locations for a cellular radio network is considered as a mathematical optimization problem. Dependent on the channel assignment policy, the minimization of interferences or the number of blocked channels, respectively, may be more favourable. In this paper, a variety of according analytical optimization problems are introduced. Each is formalized as an integer linear program, and in most cases optimum solutions can be given. Whenever by the complexity of the problem an exact solution is out of reach, simulated annealing is used as an approximate optimization technique. The performance of the different approaches is compared by extensive numerical tests.

Piecewise linear optimal controllers for hybrid systems

Abstract: We propose a procedure for synthesizing piecewise linear optimal controllers for discrete-time hybrid systems. A stabilizing controller is obtained by designing a model predictive controller, which is based on the minimization of a weighted l/sub 1///spl infin/-norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a multiparametric mixed-integer linear program, which avoids solving mixed-integer programs online. As the resulting control law is piecewise affine, online computation is drastically reduced to a simple linear function evaluation.

Concavity and Efficient Points of Discrete Distributions in Probabilistic Programming

Abstract: We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples.

Cell planning with capacity expansion in mobile communications: a tabu search approach

Abstract: The cell planning problem with capacity expansion is examined in wireless communications. The problem decides the location and capacity of each new base station to cover expanded and increased traffic demand. The objective is to minimize the cost of new base stations. The coverage by the new and existing base stations is constrained to satisfy a proper portion of traffic demands. The received signal power at the base station also has to meet the receiver sensitivity. The cell planning is formulated as an integer linear programming problem and solved by a tabu search algorithm. In the tabu search intensification by add and drop move is implemented by short-term memory embodied by two tabu lists. Diversification is designed to investigate proper capacities of new base stations and to restart the tabu search from new base station locations. Computational results show that the proposed tabu search is highly effective. A 10% cost reduction is obtained by the diversification strategies. The gap from the optimal solutions is approximately 1/spl sim/5% in problems that can be handled in appropriate time limits. The proposed tabu search also outperforms the parallel genetic algorithm. The cost reduction by the tabu search approaches 10/spl sim/20% in problems: with 2500 traffic demand areas (TDAs) in code division multiple access (CDMA).

A Linear Programming Approach to Novelty Detection

Abstract: Novelty detection involves modeling the normal behaviour of a system hence enabling detection of any divergence from normality. It has potential applications in many areas such as detection of machine damage or highlighting abnormal features in medical data. One approach is to build a hypothesis estimating the support of the normal data i.e. constructing a function which is positive in the region where the data is located and negative elsewhere. Recently kernel methods have been proposed for estimating the support of a distribution and they have performed well in practice - training involves solution of a quadratic programming problem. In this paper we propose a simpler kernel method for estimating the support based on linear programming. The method is easy to implement and can learn large datasets rapidly. We demonstrate the method on medical and fault detection datasets.

Brownian models of open processing networks: canonical representation of workload

Abstract: A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an “equivalent workload formulation” of a Brownian network model. Denoting by $Z(t)$ the state vector of the original Brownian network, one has a lower dimensional state descriptor $W(t) = MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of “heavy traffic ” for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix $M$. To be specific, rows of the canonical $M$ are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix $M$ is shown to be nonnegative, and another natural condition is identified which insures that $M$ admits a factorization related to the notion of resource pooling.