Showing papers on "Linear programming published in 1993"

Nonlinear Programming: Theory and Algorithms

Abstract: COMPREHENSIVE COVERAGE OF NONLINEAR PROGRAMMING THEORY AND ALGORITHMS, THOROUGHLY REVISED AND EXPANDED"Nonlinear Programming: Theory and Algorithms"--now in an extensively updated Third Edition--addresses the problem of optimizing an objective function in the presence of equality and inequality constraints. Many realistic problems cannot be adequately represented as a linear program owing to the nature of the nonlinearity of the objective function and/or the nonlinearity of any constraints. The "Third Edition" begins with a general introduction to nonlinear programming with illustrative examples and guidelines for model construction.Concentration on the three major parts of nonlinear programming is provided: Convex analysis with discussion of topological properties of convex sets, separation and support of convex sets, polyhedral sets, extreme points and extreme directions of polyhedral sets, and linear programmingOptimality conditions and duality with coverage of the nature, interpretation, and value of the classical Fritz John (FJ) and the Karush-Kuhn-Tucker (KKT) optimality conditions; the interrelationships between various proposed constraint qualifications; and Lagrangian duality and saddle point optimality conditionsAlgorithms and their convergence, with a presentation of algorithms for solving both unconstrained and constrained nonlinear programming problemsImportant features of the "Third Edition" include: New topics such as second interior point methods, nonconvex optimization, nondifferentiable optimization, and moreUpdated discussion and new applications in each chapterDetailed numerical examples and graphical illustrationsEssential coverage of modeling and formulating nonlinear programsSimple numerical problemsAdvanced theoretical exercisesThe book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques. The logical and self-contained format uniquely covers nonlinear programming techniques with a great depth of information and an abundance of valuable examples and illustrations that showcase the most current advances in nonlinear problems.

AMPL: A Modeling Language for Mathematical Programming

Abstract: Practical large-scale mathematical programming involves more than just the application of an algorithm to minimize or maximize an objective function. Before any optimizing routine can be invoked, considerable effort must be expended to formulate the underlying model and to generate the requisite computational data structures. AMPL is a new language designed to make these steps easier and less error-prone. AMPL closely resembles the symbolic algebraic notation that many modelers use to describe mathematical programs, yet it is regular and formal enough to be processed by a computer system; it is particularly notable for the generality of its syntax and for the variety of its indexing operations. We have implemented an efficient translator that takes as input a linear AMPL model and associated data, and produces output suitable for standard linear programming optimizers. Both the language and the translator admit straightforward extensions to more general mathematical programs that incorporate nonlinear expressions or discrete variables.

Semi-infinite programming: theory, methods, and applications

Abstract: Starting from a number of motivating and abundant applications in §2, including control of robots, eigenvalue computations, mechanical stress of materials, and statistical design, the authors describe a class of optimization problems which are referred to as semi-infinite, because their constraints bound functions of a finite number of variables on a whole region. In §§3–5, first- and second-order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints. Another main effort for achieving simplification is through duality in §6. There, algebraic properties of finite linear programming are brought to bear on duality theory in semi-infinite programming. Section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent (SQP-type) methods. Taking this differentiable point of view, this paper can be considered to be complementar...

A lift-and-project cutting plane algorithm for mixed 0-1 programs

Abstract: We propose a cutting plane algorithm for mixed 0---1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point. This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LP's needed to generate the cuts. An additional strengthening step suggested by Balas and Jeroslow is then applied. We report our computational experience with a preliminary version of the algorithm. This approach is related to the work of Balas on disjunctive programming, the matrix cone relaxations of Lovasz and Schrijver and the hierarchy of relaxations of Sherali and Adams.

Solving airline crew scheduling problems by branch-and-cut

Abstract: The crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline's fleets. The problem is especially important today since costs for flying personnel of major U.S. carriers have grown and now often exceed $1.3 billion a year and are the second largest item next to fuel cost of the total operating cost of major U.S. carriers. Thus even small percentage savings amount to substantial dollar amounts. We present a branch-and-cut approach to solving to proven optimality large set partitioning problems arising within the airline industry. We first provide some background related to this important application and then describe the approach for solving representative problems in this problem class. The branch-and-cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree-search algorithm that uses automatic reformulation procedures, heuristics and linear programming technology to assist in the solution. Numerical experiments are reported for a sample of 68 large-scale real-world crew scheduling problems. These problems include both pure set partitioning problems and set partitioning problems with side constraints. These "base constraints" represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality. An interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground.

Reactive power optimization by genetic algorithm

Abstract: This paper presents a new approach to optimal reactive power planning based on a genetic algorithm. Many outstanding methods to this problem have been proposed in the past. However, most these approaches have the common defect of being caught to a local minimum solution. The integer problem which yields integer value solutions for discrete controllers/banks still remain as a difficult one. The genetic algorithm is a kind of search algorithm based on the mechanics of natural selection and genetics. This algorithm can search for a global solution using multiple paths and treat integer problems naturally. The proposed method was applied to practical 51-bus and 224-bus systems to show its feasibility and capabilities. Although this method is not as fast as sophisticated traditional methods, the concept is quite promising and useful in the coming computer age. >

A direct nonlinear predictor-corrector primal-dual interior point algorithm for optimal power flows

Abstract: A new algorithm using the primal-dual interior point method with the predictor-corrector for solving nonlinear optimal power flow (OPF) problems is presented. The formulation and the solution technique are new. Both equalities and inequalities in the OPF are considered and simultaneously solved in a nonlinear manner based on the Karush-Kuhn-Tucker (KKT) conditions. The major computational effort of the algorithm is solving a symmetrical system of equations, whose sparsity structure is fixed. Therefore only one optimal ordering and one symbolic factorization are involved. Numerical results of several test systems ranging in size from 9 to 2423 buses are presented and comparisons are made with the pure primal-dual interior point algorithm. The results show that the predictor-corrector primal-dual interior point algorithm for OPF is computationally more attractive than the pure primal-dual interior point algorithm in terms of speed and iteration count. >

On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming

Abstract: We describe several adaptive-step primal-dual interior point algorithms for linear programming. All have polynomial time complexity while some allow very long steps in favorable circumstances. We provide heuristic reasoning for expecting that the algorithms will perform much better in practice than guaranteed by the worst-case estimates, based on an analysis using a nonrigorous probabilistic assumption.

Weak sharp minima in mathematical programming

Abstract: The notion of a sharp, or strongly unique, minimum is extended to include the possibility of a nonunique solution set. These minima will be called weak sharp minima. Conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases. These conditions are also shown to be sufficient under the appropriate convexity hypotheses. The existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem. In particular, a result of Mangasarian and Meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty. Consequences for the convergence theory of algorithms are also examined, especially conditions yielding finite termination.

A signal-dependent time-frequency representation: optimal kernel design

Abstract: A new time-frequency distribution (TFD) that adapts to each signal and so offers a good performance for a large class of signals is introduced. The design of the signal-dependent TFD is formulated in Cohen's class as an optimization problem and results in a special linear program. Given a signal to be analyzed, the solution to the linear program yields the optimal kernel and, hence, the optimal time-frequency mapping for that signal. A fast algorithm has been developed for solving the linear program, allowing the computation of the signal-dependent TFD with a time complexity on the same order as a fixed-kernel distribution. Besides this computational efficiency, an attractive feature of the optimization-based approach is the ease with which the formulation can be customized to incorporate application-specific knowledge into the design process. >

Continuation and path following

Abstract: The main ideas of path following by predictor–corrector and piecewise-linear methods, and their application in the direction of homotopy methods and nonlinear eigenvalue problems are reviewed. Further new applications to areas such as polynomial systems of equations, linear eigenvalue problems, interior methods for linear programming, parametric programming and complex bifurcation are surveyed. Complexity issues and available software are also discussed.

A penalty function approach for solving bi-level linear programs

Abstract: The paper presents an approach to bi-level programming using a duality gap—penalty function format. A new exact penalty function exists for obtaining a global optimal solution for the linear case, and an algorithm is given for doing this, making use of some new theoretical properties. For each penalty parameter value, the central optimisation problem is one of maximising a convex function over a polytope, for which a modification of an algorithm of Tuy (1964) is used. Some numerical results are given. The approach has other features which assist the actual decisionmaking process, which make use of the natural roles of duality gaps and penalty parameters. The approach also allows a natural generalization to nonlinear problems.

A primal-dual infeasible-interior-point algorithm for linear programming

Abstract: As in many primal—dual interior-point algorithms, a primal—dual infeasible-interior-point algorithm chooses a new point along the Newton direction towards a point on the central trajectory, but it does not confine the iterates within the feasible region. This paper proposes a step length rule with which the algorithm takes large distinct step lengths in the primal and dual spaces and enjoys the global convergence.

A linear programming approach for the weighted graph matching problem

Abstract: A linear programming (LP) approach is proposed for the weighted graph matching problem. A linear program is obtained by formulating the graph matching problem in L/sub 1/ norm and then transforming the resulting quadratic optimization problem to a linear one. The linear program is solved using a simplex-based algorithm. Then, approximate 0-1 integer solutions are obtained by applying the Hungarian method on the real solutions of the linear program. The complexity of the proposed algorithm is polynomial time, and it is O(n/sup 6/L) for matching graphs of size n. The developed algorithm is compared to two other algorithms. One is based on an eigendecomposition approach and the other on a symmetric polynomial transform. Experimental results showed that the LP approach is superior in matching graphs than both other methods. >

Multi-stage stochastic linear programs for portfolio optimization

Abstract: The paper demonstrates how multi-period portfolio optimization problems can be efficiently solved as multi-stage stochastic linear programs. A scheme based on a blending of classical Benders decomposition techniques and a special technique, called importance sampling, is used to solve this general class of multi-stochastic linear programs. We discuss the case where stochastic parameters are dependent within a period as well as between periods. Initial computational results are presented.

A parallel approximation algorithm for positive linear programming

Abstract: We introduce a fast parallel approximation algorithm for the positive linear programming optimization problem, i.e. the special case of the linear programming optimization problem where the input constraint matrix and constraint vector consist entirely of positive entries. The algorithm is elementary, and has a simple parallel implementation that runs in polylog time using a linear number of processors.

Loop Parallelization in the Polytope Model

Abstract: During the course of the last decade, a mathematical model for the parallelization of FOR-loops has become increasingly popular. In this model, a (perfect) nest of r FOR-loops is represented by a convex polytope in ℤr. The boundaries of each loop specify the extent of the polytope in a distinct dimension. Various ways of slicing and segmenting the polytope yield a multitude of guaranteed correct mappings of the loops' operations in space-time. These transformations have a very intuitive interpretation and can be easily quantified and automated due to their mathematical foundation in linear programming and linear algebra. With the recent availability of massively parallel computers, the idea of loop parallelization is gaining significance, since it promises execution speed-ups of orders of magnitude. The polytope model for loop parallelization has its origin in systolic design, but it applies in more general settings and methods based on it will become a part of future parallelizing compilers. This paper provides an overview and future perspective of the polytope model and methods based on it.

Monte Carlo (importance) sampling within a Benders decomposition algorithm for stochastic linear programs

Abstract: This paper focuses on Benders decomposition techniques and Monte Carlo sampling (importance sampling) for solving two-stage stochastic linear programs with recourse, a method first introduced by Dantzig and Glynn [7]. The algorithm is discussed and further developed. The paper gives a complete presentation of the method as it is currently implemented. Numerical results from test problems of different areas are presented. Using small test problems, we compare the solutions obtained by the algorithm with universe solutions. We present the solutions of large-scale problems with numerous stochastic parameters, which in the deterministic formulation would have billions of constraints. The problems concern expansion planning of electric utilities with uncertainty in the availabilities of generators and transmission lines and portfolio management with uncertainty in the future returns.

Solving Optimally the Static Ground-Holding Policy Problem in Air Traffic Control

Abstract: As air traffic congestion grows, ground-holding (or “gate-holding”) of aircraft is becoming increasingly common. The “ground-holding policy problem” (GHPP) consists of developing strategies for deciding which aircraft to hold on the ground and for how long. In this paper we present a stochastic linear programming solution to the static GHPP for a single airport. The computational complexity of existing solutions requires heuristic approaches in order to solve practical instances of the problem. The advantage of our solution is that, even for the largest airports, problem instances result in linear programs that can be solved optimally using just a personal computer. We present a set of algorithms and compare their performance to a deterministic solution and to the passive strategy of no ground-holds (i.e., to the strategy of taking all delays in the air) under different weather scenarios.

A model and heuristic for solving very large item selection problems

Abstract: A model for solving very large item selection problems is presented. The model builds on previous work in binary programming applied to test con struction. Expert test construction practices are applied to situations in which all specifications for item selection cannot necessarily be met. A heuristic for selecting items that satisfy the constraints in the model also is presented. The heuristic is particu larly useful for situations in which the size of the test construction problem exceeds the limits of current implementations of linear programming algorithms. A variety of test construction problems involving real test specifications and item data from actual test assemblies were investigated using the model and the heuristic.

Survivable networks, linear programming relaxations and the parsimonious property

Abstract: We consider the survivable network design problem -- the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and thek-edge-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worst-case analyses of two heuristics for the survivable network design problem.

Production planning via scenario modelling

Abstract: Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.

Grey linear programming, its solving approach, and its application

Abstract: In systems analysis, uncertainties may exist in model parameters and input data. Those uncertainties can propagate through the analysis and generate uncertainty in the results. Grey systems theory offers a method for incorporating uncertainties into systems analysis. In this paper, a new method of solution for a grey linear programming (GLP) model is advanced. The GLP model allows grey messages concerning the model parameters and input data to be communicated into optimization processes and solutions. A new application field—grey systems analysis of water resource planning and decision making under uncertainty—is introduced, and a case study is reported of water quantity allocation and quality planning in a drainage basin area connected to a water delivery canal in Xiamen, China. The results indicate that the solutions derived are feasible for the study area. Sensitivity tests of the effects of grey inputs on grey outputs are reported. It is indicated that the grey degrees of the solutions increa...

Aquifer remediation design under uncertainty using a new chance constrained programming technique

Abstract: A new technique, called the mixed-integer-chance-constrained programming (MICCP) method is developed in this research. This technique considers uncertainty in all linear programming constraint coefficients and does not require a priori knowledge of the distribution. A groundwater remediation problem serves as an example. The method is developed to find the globally optimal trade-off curve for maximum reliability versus a minimum pumping objective. As the fields became more heterogeneous, the pumping rate of a reliable solution increases. Four simple “rule of thumb” methods are compared to the MICCP technique. In general, the performance of such methods decreases as the heterogeneity of the hydraulic conductivity field increases.

Stable Matchings, Optimal Assignments, and Linear Programming

Abstract: Vande Vate 1989 described the polytope whose extreme points are the stable core matchings in the Marriage Problem. Rothblum 1989 simplified and extended this result. This paper explores a corresponding linear program, its dual and consequences of the fact that the dual solutions have an unusually direct relation to the primal solutions. This close relationship allows us to provide simple proofs both of Vande Vate and Rothblum's results and of other important results about the core of marriage markets. These proofs help explain the structure shared by the marriage problem without sidepayments and the assignment game with sidepayments. The paper further explores "fractional" matchings, which may be interpreted as lotteries over possible matches or as time-sharing arrangements. We show that those fractional matchings in the Stable Marriage Polytope form a lattice with respect to a partial ordering that involves stochastic dominance. Thus, all expected utility functions corresponding to the same ordinal preferences will agree on the relevant comparisons. Finally, we provide linear programming proofs of slightly stronger versions of known incentive compatibility results.

The Quickest Flow Problem

Abstract: Consider a network $$\mathcal{N}$$ =(G, c, τ) whereG=(N, A) is a directed graph andc ij andτ ij , respectively, denote the capacity and the transmission time of arc (i, j) ∈A. The quickest flow problem is then to determine for a given valueυ the minimum numberT(υ) of time units that are necessary to transmit (send)υ units of flow in $$\mathcal{N}$$ from a given sources to a given sinks′. In this paper we show that the quickest flow problem is closely related to the maximum dynamic flow problem and to linear fractional programming problems. Based on these relationships we develop several polynomial algorithms and a strongly polynomial algorithm for the quickest flow problem. Finally we report computational results on the practical behaviour of our metholds. It turns out that some of them are practically very efficient and well-suited for solving large problem instances.