Showing papers on "Linear programming published in 1995"

Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications

Abstract: Introduction 1. Convex Analysis 2. Fundamentals of Nonlinear Operations 3. Duality Theory 4. Mixed-Integer Linear Optimization 5. Mixed-Integer Nonlinear Optimization 6. Process Synthesis 7. Heat Exchanger Network Synthesis 8. Distillation-Based Separation Systems Synthesis 9. Synthesis of Reactor Networks and Reactor-Separator-Recycle Systems

Positive Mathematical Programming

Abstract: A method for calibrating models of agricultural production and resource use using nonlinear yield or cost functions is developed. The nonlinear parameters are shown to be implicit in the observed land allocation decisions at a regional or farm level. The method is implemented in three stages and initiated by a constrained linear program. The procedure automatically calibrates the model in terms of output, input use, objective function values and dual values on model constraints. The resulting nonlinear models show smooth responses to parameterization and satisfy the Hicksian conditions for competitive firms.

Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

Abstract: This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

Global Sensitivity Analysis

Abstract: In applications of operations research models, decision makers must assess the sensitivity of outputs to imprecise values for some of the model's parameters. Existing analytic approaches for classic optimization models rely heavily on duality properties for assessing the impact of local parameter variations, parametric programming for examining systematic variations in model coefficients, or stochastic programming for ascertaining a robust solution. This paper accommodates extensive simultaneous variations in any of an operations research model's parameters. For constrained optimization models, the paper demonstrates practical approaches for determining relative parameter sensitivity with respect to a model's optimal objective function value, decision variables, and other analytic functions of a solution. Relative sensitivity is assessed by assigning a portion of variation in an output value to each parameter that is imprecisely specified. The computing steps encompass optimization, Monte Carlo sampling, ...

Recent Developments in Frontier Modelling and Efficiency Measurement

Abstract: In this paper recent developments in the estimation of frontier functions and the measurement of efficiency are surveyed, and the potential applicability of these methods in agricultural economics is discussed. Frontier production, cost and profit functions are discussed, along with the construction of technical, allocative, scale and overall efficiency measures relative to these estimated frontiers. The two primary methods of frontier estimation, econometric and linear programming, are compared. A survey of recent applications of frontier methods in agriculture is also provided. (This abstract was borrowed from another version of this item.)

New conditions for global stability of neural networks with application to linear and quadratic programming problems

Abstract: In this paper, we present new conditions ensuring existence, uniqueness, and Global Asymptotic Stability (GAS) of the equilibrium point for a large class of neural networks. The results are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous nondecreasing neuron activation functions. Such functions may be unbounded (but not necessarily surjective), may have infinite intervals with zero slope as in a piece-wise-linear model, or both. The conditions on GAS rely on the concept of Lyapunov Diagonally Stable (or Lyapunov Diagonally Semi-Stable) matrices and are proved by employing a class of Lyapunov functions of the generalized Lur'e-Postnikov type. Several classes of interconnection matrices of applicative interest are shown to satisfy our conditions for GAS. In particular, the results are applied to analyze GAS for the class of neural circuits introduced for solving linear and quadratic programming problems. In this application, the principal result here obtained is that these networks are GAS also when the constraint amplifiers are dynamical, as it happens in any practical implementation. >

Control of Uncertain Systems: A Linear Programming Approach

Abstract: In tackling the problem of robust controller design this volume brings together three branches of mathematics: operator theory, optimisation theory and algebraic theory of rational marix functions. Together, these techniques enable readers to capture the fundamental limitations of design in a quantitative way, and provide computable methods for analysis and synthesis of control systems.

Economic load dispatch multiobjective optimization procedures using linear programming techniques

Abstract: This paper outlines the optimization problem of real and reactive power, and presents the new algorithm for studying the load shedding and generation reallocation problem in emergencies where a portion of the transmission system is disabled and an AC power solution cannot be found for the overloaded system. The paper describes a novel and efficient method and algorithm to obtain the optimal shift in power dispatch related to contingency states or overload situations in power system operation and planning phases under various objectives such as economy, reliability and environmental conditions. The optimization procedures basically utilize linear programming with bounded variables and it incorporates the techniques of the Section Reduction Method and the Third Simplex Method. The validity and effectiveness of the algorithm is verified by means of two examples: a 10-bus system and the IEEE 30-Bus, six generators system. >

The fleet assignment problem: Solving a large-scale integer program

Abstract: Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.

On computing three-finger force-closure grasps of polygonal objects

Abstract: This paper addresses the problem of computing stable grasps of 2-D polygonal objects. We consider the case of a hand equipped with three hard fingers and assume point contact with friction. We prove new sufficient conditions for equilibrium and force closure that are linear in the unknown grasp parameters. This reduces computing the stable grasp regions in configuration space to constructing the three-dimensional projection of a five-dimensional polytope. We present an efficient projection algorithm based on linear programming and variable elimination among linear constraints. Maximal object segments where fingers can be positioned independently while ensuring force closure are found by linear optimization within the grasp regions. The approach has been implemented and several examples are presented.

Stability of queueing networks and scheduling policies

Abstract: We develop a programmatic procedure for establishing the stability of queueing networks and scheduling policies. The method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a Lyapunov function. If the underlying system is Markovian, our method establishes not only positive recurrence and the existence of a steady-state probability distribution, but also the geometric convergence of an exponential moment. We illustrate this method on several example problems. >

Linear programming, complexity theory and elementary functional analysis

Abstract: We propose analyzing interior-point methods using notions of problem-instance size which are direct generalizations of the condition number of a matrix. The notions pertain to linear programming quite generally; the underlying vector spaces are not required to be finite-dimensional and, more importantly, the cones defining nonnegativity are not required to be polyhedral. Thus, for example, the notions are appropriate in the context of semi-definite programming. We prove various theorems to demonstrate how the notions can be used in analyzing interior-point methods. These theorems assume little more than that the interiors of the cones (defining nonnegativity) are the domains of self-concordant barrier functions.

Las Vegas algorithms for linear and integer programming when the dimension is small

Abstract: This paper gives an algorithm for solving linear programming problems. For a problem with n constraints and d variables, the algorithm requires an expected O(d2n) + (log n)O(d)d/2+O(1) + O(d4√nlog n) arithmetic operations, as n→∞. The constant factors do not depend on d. Also, an algorithm is given for integer linear programming. Let φ bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector. Let n and d be as before. Then, the algorithm requires expected O(2d dn + 8dd√n ln ln n) + dO(d)φ ln n operations on numbers with dO(1)φ bits, as n→∞, where the constant factors do not depend on d or φ to other convex programming problems. For example, an algorithm for finding the smallest sphere enclosing a set of n points in Ed has the same time bound.

Linear Optimization and Extensions

Abstract: The Linear Programming Problem.- Basic Concepts.- Five Preliminaries.- Simplex Algorithms.- Primal-Dual Pairs.- Analytical Geometry.- Projective Algorithms.- Ellipsoid Algorithms.- Combinatorial Optimization: An Introduction.

A primal-dual potential reduction method for problems involving matrix inequalities

Abstract: We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worst-case analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interior-point methods the overall computational effort is therefore dominated by the least-squares system that must be solved in each iteration. A type of conjugate-gradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugate-gradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration. We describe in detail how the algorithm works for optimization problems withL Lyapunov inequalities, each of sizem. We prove an overallworst-case operation count of O(m 5.5L1.5). Theaverage-case complexity appears to be closer to O(m 4L1.5). This estimate is justified by extensive numerical experimentation, and is consistent with other researchers' experience with the practical performance of interior-point algorithms for linear programming. This result means that the computational cost of extending current control theory based on the solution of Lyapunov or Riccatiequations to a theory that is based on the solution of (multiple, coupled) Lyapunov or Riccatiinequalities is modest.

Modeling and Solving the Two-Facility Capacitated Network Loading Problem

Abstract: This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-to-point communication demand in a network must be met by installing loading capacitated facilities on the arcs: Loading a facility incurs an arc specific and facility dependent cost. This paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost. We consider two approaches for solving the underlying mixed integer program: a Lagrangian relaxation strategy, and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem. We show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the Lagrangian relaxation bound. Our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation: from an average of 25% to an average of 8%. These results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry.

Presolving in linear programming

Abstract: Most modern linear programming solvers analyze the LP problem before submitting it to optimization. Some examples are the solvers WHIZARD (Tomlin and Welch, 1983), OB1 (Lustig et al., 1994), OSL (Forrest and Tomlin, 1992), Sciconic (1990) and CPLEX (Bixby, 1994). The purpose of the presolve phase is to reduce the problem size and to discover whether the problem is unbounded or infeasible. In this paper we present a comprehensive survey of presolve methods. Moreover, we discuss the restoration procedure in detail, i.e., the procedure that undoes the presolve. Computational results on the NETLIB problems (Gay, 1985) are reported to illustrate the efficiency of the presolve methods.

Incorporating Condition Measures into the Complexity Theory of Linear Programming

Abstract: This work is an attempt, among other things, to begin developing a complexity theory in which problem instance data is allowed to consist of real, even irrational, numbers and yet computations are of finite precision.Complexity theory generally assumes that the exact data specifying a problem instance is used by algorithms. The efficiency of an algorithm is judged relative to the size of the input. For the Turing model of computation, size refers to the bit-length of the input, which is required to consist of integers (or rational numbers separated into numerators and denominators).We replace customary measures of size with condition measures. These measures reflect the amount of data accuracy necessary to achieve the desired computational goal. The measures are similar in spirit, and closely related, to condition numbers.

Optimization method for reactive power planning by using a modified simple genetic algorithm

Abstract: This paper presents an improved simple genetic algorithm developed for reactive power system planning. Successive linear programming is used to solve operational optimization sub-problems. A new population selection and generation method which makes the use of Benders' cut is presented in this paper. It is desirable to find the optimal solution in few iterations, especially in some test cases where the optimal results are expected to be obtained easily. However, the simple genetic algorithm has failed in finding the solution except through an extensive number of iterations. Different population generation and crossover methods are also tested and discussed. The method has been tested for 6 bus and 30 bus power systems to show its effectiveness. Further improvement for the method is also discussed.

Randomized rounding without solving the linear program

Abstract: We introduce a new technique called oblivious rounding a variant of randomized rounding that avoids the bottleneck of first solving the linear program. Avoiding this bottleneck yields more efficient algorithms and brings probabilistic methods to bear on a new class of problems. We give oblivious rounding algorithms that approximately solve general packing and covering problems, including a parallel algorithm to find sparse strategies for matrix games.

A calibration method for agricultural economic production models

Abstract: A method for calibrating agricultural production models is presented. The data requirements are those for a linear programming model with the addition of elasticities of substitution. Using these data, production models with a CES production function can be simply and automatically calibrated using small computers. The resulting models are shown to satisfy the standard microeconomic conditions. When used for analysis of policy changes, the CES models are able to respond smoothly to changes in prices or constraints. Prior estimates of elasticities of substitution, supply or demand can be incorporated in the models.

Controlled Queueing Systems

Abstract: Semi-Regenerative Decision Models Description of Basic Decision Model Rigorous Definitions and Assumptions Examples of Controlled Queues Optimization Problems Renewal Kernels of the Decision Model Special Classes of Strategies Sufficiency of Markov Strategies Dynamic Programming Discounting in Continuous Time Dynamic Programming Equation Bellman Functions Finite-Horizon Problem Infinite-Horizon Discounted-Cost Problem Random-Horizon Problem Average Cost Criterion Preliminaries: Weak Topology, Limit Passages Preliminaries: Taboo Probabilities, Limit Theorems for Markov Renewal Processes Notation, Recurrence-Communication Assumptions, Examples Existence of Optimal Policies Existence of Optimal Strategies: General Criterion Existence of Optimal Strategies: Sufficient Conditions Optimality Equation Constrained Average-Cost Problem Average-Cost Optimality as Limiting Case of Discounted-Cost Optimality Continuously Controlled Markov Jump Processes Facts About Measurability of Stochastic Processes Marked Point Processes and Random Measures The Predictable s-Algebra Dual Predictable Projections of Random Measures Definition of Controlled Markov Jump Process An M/M/1 Queue With Controllable Input and Service Rate Dynamic Programming Optimization Problems Structured Optimization Problems for Decision Processes Convex Regularization Submodular and Supermodular Functions Existence of Monotone Solutions for Optimization Problems Processes with Bounded Drift Birth and Death Processes Control of Arrivals The Model Description Finite-Horizon Discounted-Cost Problem Cost Functionals Infinite-Horizon Case with and without Discounting Optimal Dynamic Pricing Policy: Model Results Control of Service Mechanism Description of the System Static Optimization Problem Optimal Policies for the Queueing Process Service System with Two Interacting Servers Analysis of Optimality Equation Optimal Control in Models with Several Classes of Customers Description of Models and Processes Associated Controlled Processes Existence of Optimal Simple Strategies for the Systems with Alternating Priority Existence of Optimal Simple Strategy for the System with Feedback Equations for Stationary Distributions Stationary Characteristics of the Systems with Alternating Priority Stationary Characteristics of the System with Feedback Models with Alternating Priority: Linear Programming Problem Linear Programming Problem in the Model with Feedback Model with Periods of Idleness and Discounted-Cost Criterion Basic Formulas Construction of Optimal Modified Priority Discipline Bibliography Index Each chapter also includes an Introduction, and a Remarks and Exercises section

An algorithm for large scale 0-1 integer programming with application to airline crew scheduling

Abstract: We present an approximation algorithm for solving large 0–1 integer programming problems whereA is 0–1 and whereb is integer. The method can be viewed as a dual coordinate search for solving the LP-relaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working together with this method. The approximation scheme works by adjusting the costs as little as possible so that the new problem has an integer solution. The degree of approximation is determined by a parameter, and for different levels of approximation the resulting algorithm can be interpreted in terms of linear programming, dynamic programming, and as a greedy algorithm. The algorithm is used in the CARMEN system for airline crew scheduling used by several major airlines, and we show that the algorithm performs well for large set covering problems, in comparison to the CPLEX system, in terms of both time and quality. We also present results on some well known difficult set covering problems that have appeared in the literature.

On the existence of equilibria in noncooperative optimal flow control

Abstract: The existence of Nash equilibria in noncooperative flow control in a general product-form network shared by K users is investigated. The performance objective of each user is to maximize its average throughput subject to an upper bound on its average time-delay. Previous attempts to study existence of equilibria for this flow control model were not successful, partly because the time-delay constraints couple the strategy spaces of the individual users in a way that does not allow the application of standard equilibrium existence theorems from the game theory literature. To overcome this difficulty, a more general approach to study the existence of Nash equilibria for decentralized control schemes is introduced. This approach is based on directly proving the existence of a fixed point of the best reply correspondence of the underlying game. For the investigated flow control model, the best reply correspondence is shown to be a function, implicitly defined by means of K interdependent linear programs. Employing an appropriate definition for continuity of the set of optimal solutions of parameterized linear programs, it is shown that, under appropriate conditions, the best reply function is continuous. Brouwer's theorem implies, then, that the best reply function has a fixed point.

Stochastic user equilibrium assignment in networks with queues

Abstract: A stochastic user equilibrium assignment algorithm is presented for steady state store-and-forward networks. The links of the network have constant travel times and the links or nodes have finite capacities. When capacity is reached, delay sufficient to match demand to the available capacity is generated. It has been shown by others that the equilibrium assignment in networks of this kind is the solution to a particular linear programming problem. By adding an entropy term to the objective function, a convex nonlinear programming problem is formed which yields a stochastic user equilibrium assignment. For the case of link constraints, it is proven that the Lagrange multipliers of both the linear and the non-linear programming problems give the equilibrium delays in the network. The requirements for uniqueness are investigated. Iterative algorithms are formulated for solving the nonlinear programming problem with either link or node constraints and convergence is proven. For networks where path enumeration is likely to be a problem, a column generation technique is proposed. An illustrative example is presented.

On convergence of an augmented Lagrangian decomposition method for sparse convex optimization

Abstract: A decomposition method for large-scale convex optimization problems with block-angular structure and many linking constraints is analysed. The method is based on a separable approximation of the augmented Lagrangian function. Weak global convergence of the method is proved and speed of convergence analysed. It is shown that convergence properties of the method are heavily dependent on sparsity of the linking constraints. Application to large-scale linear programming and stochastic programming is discussed.