scispace - formally typeset
Search or ask a question

Showing papers on "Nonlinear programming published in 1998"


Journal ArticleDOI
TL;DR: If U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods.
Abstract: We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods.

2,501 citations



Journal Article
TL;DR: This work identifies a template for scatter search and path relinking methods that provides a convenient and user friendly basis for their implementation and describes Illustrative forms of these subroutines that make it possible to create methods for a wide range of optimization problems.
Abstract: Scatter search and its generalized form called path relinking are evolutionary methods that have recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in the 1960s for combining decision rules and problem constraints, these methods use strategies for combining solution vectors that have proved effective for scheduling, routing, financial product design, neural network training, optimizing simulation and a variety of other problem areas. These approaches can be implemented in multiple ways, and offer numerous alternatives for exploiting their basic ideas. We identify a template for scatter search and path relinking methods that provides a convenient and user friendly basis for their implementation. The overall design can be summarized by a small number of key steps, leading to versions of scatter search and path relinking that are fully specified upon providing a handful of subroutines. Illustrative forms of these subroutines are described that make it possible to create methods for a wide range of optimization problems.

711 citations


Journal ArticleDOI
TL;DR: The deterministic global optimization algorithm, αBB (α-based Branch and Bound) is presented, which offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twice-differentiable NLPs.

503 citations


Journal ArticleDOI
TL;DR: In this paper, a Legendre pseudospectral method for directly estimating the costates of the Bolza problem encountered in optimal control theory is presented. But the method is based on calculating the state and control variables at the Legendre-Gauss-Lobatto (LGL) points.
Abstract: We present a Legendre pseudospectral method for directly estimating the costates of the Bolza problem encountered in optimal control theory. The method is based on calculating the state and control variables at the Legendre‐Gauss‐Lobatto (LGL) points. An Nth degree Lagrange polynomial approximation of these variables allows a conversion of the optimal control problem into a standard nonlinear programming (NLP) problem with the state and control values at the LGL points as optimization parameters. By applying theKarush ‐Kuhn‐Tucker (KKT) theorem to the NLP problem, we show that the KKT multipliers satisfy a discrete analog of the costate dynamics including the transversality conditions. Indeed, we prove that the costates at the LGL points are equal to the KKT multipliers divided by the LGL weights. Hence, the direct solution by this method also automatically yields the costates by way of the Lagrange multipliers that can be extracted from an NLP solver. One important advantage of this technique is that it allows a very simple way to check the optimality of the direct solution. Numerical examples are included to demonstrate the method.

463 citations


Journal ArticleDOI
TL;DR: In this article, the authors describe the solution of an optimal power flow (OPF) problem in rectangular form by an interior-point method (IPM) for nonlinear programming.
Abstract: The paper describes the solution of an optimal power flow (OPF) problem in rectangular form by an interior-point method (IPM) for nonlinear programming. Some OPF variants when formulated in rectangular form have quadratic objective and quadratic constraints. Such quadratic features allow for ease of matrix setup, and inexpensive incorporation of higher-order information in a predictor-corrector procedure that generally improves IPM performance. The mathematical development of the IPM in the paper is based on a general nonlinear programming problem. Issues in implementation to solve the rectangular OPF are discussed. Computational tests apply the IPM to both the rectangular and polar OPF versions. Test results show that both algorithms perform extremely well.

376 citations


Journal ArticleDOI
TL;DR: The emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty.
Abstract: This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems. These methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty. We give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.

340 citations


Journal ArticleDOI
TL;DR: The performance of the proposed algorithm and its alternative underestimators is studied through their application to a variety of problems and a number of rules for branching variable selection and variable bound updates are shown to enhance convergence rates.

324 citations


Journal ArticleDOI
TL;DR: In this article, a simulated annealing algorithm (SAA) was used to solve the unit commitment problem (UCP) and new rules for randomly generating feasible solutions were introduced.
Abstract: This paper presents a simulated annealing algorithm (SAA) to solve the unit commitment problem (UCP). New rules for randomly generating feasible solutions are introduced. The problem has two subproblems: a combinatorial optimization problem; and a nonlinear programming problem. The former is solved using the SAA while the latter problem is solved via a quadratic programming routine. Numerical results showed an improvement in the solutions costs compared to previously obtained results.

299 citations


Book ChapterDOI
01 Jan 1998
TL;DR: The resource allocation problem seeks to find an optimal allocation of a fixed amount of resources to activities so as to minimize the cost incurred by the allocation.
Abstract: The resource allocation problem seeks to find an optimal allocation of a fixed amount of resources to activities so as to minimize the cost incurred by the allocation. A simplest form of the problem is to minimize a separable convex function under a single constraint concerning the total amount of resources to be allocated. The amount of resources to be allocated to each activity is treated as a continuous or integer variable, depending on the cases. This can be viewed as a special case of the nonlinear programming problem or the nonlinear integer programming problem.

294 citations


Book
31 Mar 1998
TL;DR: This book discusses the role of Ellipsoid Method for Complexity Analysis of Combinatorial Problems, and the importance of Semidefinite Programming Bounds for Extremal Graph Problems.
Abstract: Preface. 1. Elements of Convex Analysis, Linear Algebra, and Graph Theory. 2. Subgradient and epsilon-Subgradient Methods. 3. Subgradient-Type Methods with Space Dilation. 4. Elements of Information and Numerical Complexity of Polynomial Extremal Problems. 5. Decomposition Methods Based on Nonsmooth Optimization. 6. Algorithms for Constructing Optimal on Volume Ellipsoids and Semidefinite Programming. 7. The Role of Ellipsoid Method for Complexity Analysis of Combinatorial Problems. 8. Semidefinite Programming Bounds for Extremal Graph Problems. 9. Global Minimization of Polynomial Functions and 17-th Hilbert Problem. References.

Journal ArticleDOI
TL;DR: In this paper, a statistical inference is developed and applied to estimation of the error, validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm for two-stage stochastic programming with recourse where the random data have a continuous distribution.
Abstract: In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for two-stage stochastic programming with recourse where the random data have a continuous (multivariate normal) distribution. We think that the novelty of the numerical approach developed in this paper is twofold. First, various variance reduction techniques are applied in order to enhance the rate of convergence. Successful application of those techniques is what makes the whole approach numerically feasible. Second, a statistical inference is developed and applied to estimation of the error, validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Journal ArticleDOI
TL;DR: A new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints is described, which is much simpler to implement and allows any kind of changes of the working set in every step.
Abstract: In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [P. Spellucci, Han's method without solving QP, in: A. Auslender, W. Oettli, J. Stoer (Eds), Optimization and Optimal Control, Lecture Notes in Control and Information Sciences, vol. 30, Springer, Berlin, 1981, pp. 123–141.] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numcrical tests. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Journal ArticleDOI
01 Feb 1998
TL;DR: It is demonstrated that the genetic algorithm cannot only serve as a global search algorithm but by appropriately defining the objective function it can simultaneously achieve a parsimonious architecture.
Abstract: The recent surge in activity of neural network research in business is not surprising since the underlying functions controlling business data are generally unknown and the neural network offers a tool that can approximate the unknown function to any degree of desired accuracy. The vast majority of these studies rely on a gradient algorithm, typically a variation of backpropagation, to obtain the parameters (weights) of the model. The well-known limitations of gradient search techniques applied to complex nonlinear optimization problems such as artificial neural networks have often resulted in inconsistent and unpredictable performance. Many researchers have attempted to address the problems associated with the training algorithm by imposing constraints on the search space or by restructuring the architecture of the neural network. In this paper we demonstrate that such constraints and restructuring are unnecessary if a sufficiently complex initial architecture and an appropriate global search algorithm is used. We further show that the genetic algorithm cannot only serve as a global search algorithm but by appropriately defining the objective function it can simultaneously achieve a parsimonious architecture. The value of using the genetic algorithm over backpropagation for neural network optimization is illustrated through a Monte Carlo study which compares each algorithm on in-sample, interpolation, and extrapolation data for seven test functions.

Journal ArticleDOI
01 Aug 1998
TL;DR: Two new solutions that attempt to solve the homogeneous matrix equation of the for, AX=ZB are presented: a closed-form method which uses quaternion algebra and a positive quadratic error function associated with this representation; and a method based on nonlinear constrained minimization and which simultaneously solves for rotations and translations.
Abstract: Zhuang et al. (1994) proposed a method that allows simultaneous computation of the rigid transformations from world frame to robot base frame and from hand frame to camera frame. Their method attempts to solve a homogeneous matrix equation of the for, AX=ZB. They use quaternions to derive explicit linear solution for X and Z. In this paper, we present two new solutions that attempt to solve the homogeneous matrix equation mentioned above: 1) a closed-form method which uses quaternion algebra and a positive quadratic error function associated with this representation; 2) a method based on nonlinear constrained minimization and which simultaneously solves for rotations and translations. These results may be useful to other problems that can be formulated in the same mathematical form. We perform a sensitivity analysis for both our two methods and the linear method developed by Zhuang et al. This analysis allows the comparison of the three methods. In the light of this comparison, the nonlinear optimization method, which solves for rotations and translations simultaneously, seems to be the most stable one with respect to noise and to measurement errors.

Journal ArticleDOI
TL;DR: A survey on results related to scheduling problems where machines are not continuously available for processing, NP-hardness results, polynomial optimization and approximation algorithms, and single and multi machine problems.
Abstract: We will give a survey on results related to scheduling problems where machines are not continuously available for processing. We will deal with single and multi machine problems and analyze their complexity. We survey NP-hardness results, polynomial optimization and approximation algorithms. We also distinguish between on-line and off-line formulations of the problems. Results are concerned with criteria on completion times and due dates.

Journal ArticleDOI
TL;DR: In this paper, the optimum design of a distributed wastewater network where multicomponent streams are considered that are to be processed by units for reducing the concentration of several contaminants is discussed.
Abstract: This paper deals with the optimum design of a distributed wastewater network where multicomponent streams are considered that are to be processed by units for reducing the concentration of several contaminants. The proposed model gives rise to a nonconvex nonlinear problem which often exhibits local minima and causes convergence difficulties. A search procedure is proposed in this paper that is based on the successive solution of a relaxed linear model and the original nonconvex nonlinear problem. Several examples are presented to illustrate that the proposed method often yields global or near global optimum solutions. The model is also extended for selecting different treatment technologies and for handling membrane separation modules.

Journal ArticleDOI
01 Jan 1998
TL;DR: In this article, an application of the tabu search (TS) method to solve the unit commitment problem (UCP) is presented, where the TS seeks to counter the danger of entrapment at a local optimum by incorporating a memory structure that forbids or penalises certain moves that would return to recently visited solutions.
Abstract: An application of the tabu search (TS) method to solve the unit commitment problem (UCP) is presented. The TS seeks to counter the danger of entrapment at a local optimum by incorporating a memory structure that forbids or penalises certain moves that would return to recently visited solutions. New rules for randomly generating feasible solutions for the UCP are introduced. The problem is divided into two subproblems: a combinatorial optimisation problem and a nonlinear programming problem. The former is solved using the tabu search algorithm (TSA) while the latter problem is solved via a quadratic programming routine. Numerical results show an improvement in the solution cost compared to previously obtained results.

Journal ArticleDOI
TL;DR: A Historical Sketch on Sensitivity Analysis and Parametric Programming T.J. Greenberg and the Optimal Set and Optimal Partition Approach.
Abstract: Foreword. Preface. 1. A Historical Sketch on Sensitivity Analysis and Parametric Programming T. Gal. 2. A Systems Perspective: Entity Set Graphs H. Muller-Merbach. 3. Linear Programming 1: Basic Principles H.J. Greenberg. 4. Linear Programming 2: Degeneracy Graphs T. Gal. 5. Linear Programming 3: The Tolerance Approach R.E. Wendell. 6. The Optimal Set and Optimal Partition Approach A.B. Berkelaar, et al. 7. Network Models G.L. Thompson. 8. Qualitative Sensitivity Analysis A. Gautier, et al. 9. Integer and Mixed-Integer Programming C. Blair. 10. Nonlinear Programming A.S. Drud, L. Lasdon. 11. Multi-Criteria and Goal Programming J. Dauer, Yi-Hsin Liu. 12. Stochastic Programming and Robust Optimization H. Vladimirou, S.A. Zenios. 13. Redundancy R.J. Caron, et al. 14. Feasibility and Viability J.W. Chinneck. 15. Fuzzy Mathematical Programming H.-J. Zimmermann. Subject Index.

Journal ArticleDOI
TL;DR: Computational tests of three approaches to feature selection algorithm via concave minimization on publicly available real-world databases have been carried out and compared with an adaptation of the optimal brain damage method for reducing neural network complexity.
Abstract: The problem of discriminating between two finite point sets in n-dimensional feature space by a separating plane that utilizes as few of the features as possible is formulated as a mathematical program with a parametric objective function and linear constraints. The step function that appears in the objective function can be approximated by a sigmoid or by a concave exponential on the nonnegative real line, or it can be treated exactly by considering the equivalent linear program with equilibrium constraints. Computational tests of these three approaches on publicly available real-world databases have been carried out and compared with an adaptation of the optimal brain damage method for reducing neural network complexity. One feature selection algorithm via concave minimization reduced cross-validation error on a cancer prognosis database by 35.4% while reducing problem features from 32 to 4.

Book
30 Nov 1998
TL;DR: This paper presents the results of a large-scale study of the convergence of the CA Algorithm for Nonlinear Programs with respect to Column Generation/Simplicial Decomposition Algorithm in the context of discrete-time decision-making.
Abstract: Preface. 1. Introduction. 2. Technical Preliminaries. 3. Instances of the Cost Approximation Algorithm. 4. Merit Functions for Variational Inequality Problems. 5. Convergence of the CA Algorithm for Nonlinear Programs. 6. Convergence of the CA Algorithm for Variational Inequality Problems. 7. Finite Identification of Active Constraints and of Solutions. 8. Parallel and Sequential Decomposition CA Algorithms. 9. A Column Generation/Simplicial Decomposition Algorithm. A. Definitions. References. Index.

Book ChapterDOI
01 Jan 1998
TL;DR: An algorithm for nonlinear optimization that employs both trust region techniques and line searches that performs a backtracking line search from the failed point, and preserves the strong convergence properties of trust region methods.
Abstract: We propose an algorithm for nonlinear optimization that employs both trust region techniques and line searches. Unlike traditional trust region methods, our algorithm does not resolve the subproblem if the trial step results in an increase in the objective function, but instead performs a backtracking line search from the failed point. Backtracking can be done along a straight line or along a curved path. We show that the new algorithm preserves the strong convergence properties of trust region methods. Numerical results are also presented.

Journal ArticleDOI
TL;DR: Large-scale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available is concerned, and a method suitable for large problems can be obtained.
Abstract: This paper concerns large-scale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penalty-barrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primal-dual system similar to that proposed for interior methods. The augmented penalty-barrier function may be interpreted as a merit function for values of the primal and dual variables. An inertia-controlling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penalty-barrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.

Journal ArticleDOI
TL;DR: A mathematical programming model is developed which minimizes the operating costs subject to service constraints and capacity requirements and optimizes on lines, line types, routes, frequencies and train lengths.

Journal ArticleDOI
TL;DR: In this article, a mixed-integer nonlinear programming (MINLP) model for performing structural and parameter optimization of utility plants that satisfy given electrical, mechanical and heating demands of industrial processes is presented.
Abstract: This paper presents a mixed-integer nonlinear programming (MINLP) model for performing structural and parameter optimization of utility plants that satisfy given electrical, mechanical and heating demands of industrial processes. In this model, nonlinear equations are extensively used for the cost of equipment and for the plant performance in terms of enthalpies, entropies and efficiencies. The proposed approach allows for the simultaneous optimization of the configuration, and selection of flowrates, enthalpies and steam turbine efficiencies. All major conventional utility plant equipment are included in the superstructure for the MINLP model. The proposed approach is not only useful for synthesis, but also for analysing different design alternatives. The model has been implemented in the computer package STEAM, and several applications are reported to illustrate the program capabilities, including a comparison with a simplified MILP model.

Journal ArticleDOI
TL;DR: A global optimization algorithm is presented to rigorously solve the MINLP model by Yee and Grossmann (1990) for the synthesis of heat exchanger networks under the simplifying assumptions of linear area cost, arithmetic mean temperature difference driving forces and no stream splitting.

Journal ArticleDOI
TL;DR: A brief discussion of nonlinear least-squares optimization, its application within the context of some commercial spreadsheet software packages, and some of the capabilities of this approach are illustrated using examples from the authors' respective upper division Soil Physics lecture/laboratory courses.
Abstract: Many fundamental processes in the natural and physical sciences exhibit substantial nonlinear relations among the component variables. Instructing students on how to identify parameters in models for such relations based on measured data, and providing a means to illustrate the impacts of specific components on the behavior of nonlinear relationships, is therefore a critical element of university science training. Commercially available spreadsheet software programs offer convenient and effective means for performing nonlinear parameter estimation. Among the advantages of the spreadsheet approach to nonlinear optimization and parameter estimation are its relative ease of use, general applicability, demonstration or reinforcement of least-squares principles, and the ability to interactively alter selected parameter values and immediately view the results via embedded graphics. In this paper we provide a brief discussion of nonlinear least-squares optimization, demonstrate its application within the context of some commercial spreadsheet software packages, and illustrate some of the capabilities of this approach using examples from our respective upper division Soil Physics lecture/laboratory courses. Our experiences indicate that students learn spreadsheet techniques more readily than often specialized mathematical or statistical programs, and that they provide a valuable tool in their future endeavors.

Journal ArticleDOI
TL;DR: The differential-algebraic equation (DAE) optimization problem is transformed to a nonlinear programming problem by applying collocation on finite elements using a reduced space successive quadratic programming (rSQP) algorithm, which solves more than 150 DAEs in less than 7 CPU minutes.
Abstract: The differential-algebraic equation (DAE) optimization problem is transformed to a nonlinear programming problem by applying collocation on finite elements. The resulting problem is solved using a reduced space successive quadratic programming (rSQP) algorithm. Here, the variable space is partitioned into range and null spaces. Partitioning by choosing a pivot sequence for an LU factorization with partial pivoting allows us to detect unstable modes in the DAE system, which can now be stabilized without imposing new boundary conditions. As a result, the range space is decomposed in a single step by exploiting the overall sparsity of the collocation matrix; which performs better than the two-step condensation method used in standard collocation solvers. To deal with ill-conditioned constraints, we also extend the rSQP algorithm to include dogleg steps for the range space step that solves the collocation equations. The performance of this algorithm was tested on two well known unstable problems and on three chemical engineering examples including two reactive distillation columns and a plug-flow reactor with free radicals. One of these is u batch column where an equilibrium reaction takes place. The second reactive distillation problem is the startup of a continuous column with competitive reactions. These optimization problems, which include more than 150 DAEs, ure solved in less than 7 CPU minutes on workstation class computers.

Journal ArticleDOI
Moon-Won Park1, Yeong-Dae Kim1
TL;DR: A systematic procedure to find appropriate values for parameters quickly without much human intervention by using a nonlinear optimization method, the simplex method for nonlinear programming is suggested.

Journal ArticleDOI
TL;DR: In this article, a technique for changing the discretization in order to improve the accuracy of the approximation is described, and an integer programming technique is used to minimize the maximum error during the refinement iterations.
Abstract: SUMMARY The direct transcription method for solving optimal control problems involves the use of a discrete approximation to the original problem. This paper describes a technique for changing the discretization in order to improve the accuracy of the approximation. An integer programming technique is used to minimize the maximum error during the refinement iterations. The eƒciency of the method is illustrated for an application with path inequality constraints. ( 1998 John Wiley & Sons, Ltd.