Showing papers on "Nonlinear programming published in 2006"
TL;DR: A comprehensive description of the primal-dual interior-point algorithm with a filter line-search method for nonlinear programming is provided, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix.
Abstract: We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.
7,966 citations
01 Jan 2006
TL;DR: The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings, and it is effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming.
Abstract: This paper describes Knitro 5.0, a C-package for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving large-scale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an active-set method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings.
1,022 citations
TL;DR: An interior-point method for nonlinear programming that enjoys the flexibility of switching between a line search method that computes steps by factoring the primal-dual equations and a trust region method that uses a conjugate gradient iteration.
Abstract: An interior-point method for nonlinear programming is presented It enjoys the flexibility of switching between a line search method that computes steps by factoring the primal-dual equations and a trust region method that uses a conjugate gradient iteration Steps computed by direct factorization are always tried first, but if they are deemed ineffective, a trust region iteration that guarantees progress toward stationarity is invoked To demonstrate its effectiveness, the algorithm is implemented in the Knitro [6,28] software package and is extensively tested on a wide selection of test problems
997 citations
Book•
01 Jan 2006
TL;DR: This book will help readers to understand the mathematical foundations of the modern theory and methods of nonlinear optimization and to analyze new problems, develop optimality theory for them, and choose or construct numerical solution methods.
Abstract: Optimization is one of the most important areas of modern applied mathematics, with applications in fields from engineering and economics to finance, statistics, management science, and medicine. While many books have addressed its various aspects, Nonlinear Optimization is the first comprehensive treatment that will allow graduate students and researchers to understand its modern ideas, principles, and methods within a reasonable time, but without sacrificing mathematical precision. Andrzej Ruszczynski, a leading expert in the optimization of nonlinear stochastic systems, integrates the theory and the methods of nonlinear optimization in a unified, clear, and mathematically rigorous fashion, with detailed and easy-to-follow proofs illustrated by numerous examples and figures. The book covers convex analysis, the theory of optimality conditions, duality theory, and numerical methods for solving unconstrained and constrained optimization problems. It addresses not only classical material but also modern topics such as optimality conditions and numerical methods for problems involving nondifferentiable functions, semidefinite programming, metric regularity and stability theory of set-constrained systems, and sensitivity analysis of optimization problems. Based on a decade's worth of notes the author compiled in successfully teaching the subject, this book will help readers to understand the mathematical foundations of the modern theory and methods of nonlinear optimization and to analyze new problems, develop optimality theory for them, and choose or construct numerical solution methods. It is a must for anyone seriously interested in optimization.
913 citations
TL;DR: In this article, the authors show that the load flow problem of a radial distribution system can be modeled as a convex optimization problem, particularly a conic program, which facilitates the inclusion of the distribution power flow equations in radial system optimization problems.
Abstract: This paper shows that the load flow problem of a radial distribution system can be modeled as a convex optimization problem, particularly a conic program. The implications of the conic programming formulation are threefold. First, the solution of the distribution load flow problem can be obtained in polynomial time using interior-point methods. Second, numerical ill-conditioning can be automatically alleviated by the use of scaling in the interior-point algorithm. Third, the conic formulation facilitates the inclusion of the distribution power flow equations in radial system optimization problems. A state-of-the-art implementation of an interior-point method for conic programming is used to obtain the solution of nine different distribution systems. Comparisons are carried out with a previously published radial load flow program by R. Cespedes
592 citations
TL;DR: This tutorial gives an interpretation of Lagrangian duality in a multiuser multi-antenna communication problem; it illustrates the role of semidefinite relaxation inMultiuser detection problems; and methods to formulate robust optimization problems via second-order cone programming techniques are reviewed.
Abstract: Convex optimization methods are widely used in the design and analysis of communication systems and signal processing algorithms. This tutorial surveys some of recent progress in this area. The tutorial contains two parts. The first part gives a survey of basic concepts and main techniques in convex optimization. Special emphasis is placed on a class of conic optimization problems, including second-order cone programming and semidefinite programming. The second half of the survey gives several examples of the application of conic programming to communication problems. We give an interpretation of Lagrangian duality in a multiuser multi-antenna communication problem; we illustrate the role of semidefinite relaxation in multiuser detection problems; we review methods to formulate robust optimization problems via second-order cone programming techniques
513 citations
Book•
10 Jan 2006
TL;DR: Decomposition in Linear Programming: Complicating Constraints, Duality, and Local Sensitivity Analysis.
Abstract: Motivation and Introduction.- Motivating Examples: Models with Decomposable Structure.- Decomposition Techniques.- Decomposition in Linear Programming: Complicating Constraints.- Decomposition in Linear Programming: Complicating Variables.- Duality.- Decomposition in Nonlinear Programming.- Decomposition in Mixed-Integer Programming.- Other Decomposition Techniques.- Local Sensitivity Analysis.- Local Sensitivity Analysis.- Applications.- Applications.- Computer Codes.- Some GAMS Implementations.- Solution to Selected Exercises.- Exercise Solutions.
457 citations
TL;DR: This work exemplifies its method by studies of steady‐state Navier–Stokes flow problems, thus extending the work by Borrvall and Petersson on topology optimization of fluids in Stokes flow.
Abstract: We present a versatile high-level programming-language implementation of non-linear topology optimization. Our implementation is based on the commercial software package FEMLAB, and it allows a wide range of optimization objectives to be dealt with easily. We exemplify our method by studies of steady-state Navier–Stokes flow problems, thus extending the work by Borrvall and Petersson on topology optimization of fluids in Stokes flow (Int. J. Num. Meth. Fluids 2003; 41:77–107). We analyse the physical aspects of the solutions and how they are affected by different parameters of the optimization algorithm. A complete example of our implementation is included as FEMLAB code in an appendix. Copyright © 2005 John Wiley & Sons, Ltd.
377 citations
TL;DR: The novel method requires few control variables, is relatively easy to implement and use, effective, and efficient, which makes it an attractive and widely applicable approach for solving practical engineering problems.
324 citations
TL;DR: The proposed coordination problem is formulated as a mixed integer nonlinear problem to take into account the discrete values for the pickup current settings and a modified particle swarm optimization (PSO) algorithm is proposed to calculate the optimal relay settings.
09 Oct 2006
TL;DR: A calibration method for eye-in-hand systems in order to estimate the hand-eye and the robot-world transformations in terms of a parametrization of a stochastic model and a novel metric is proposed for nonlinear optimization.
Abstract: This paper presents a calibration method for eye-in-hand systems in order to estimate the hand-eye and the robot-world transformations. The estimation takes place in terms of a parametrization of a stochastic model. In order to perform optimally, a metric on the group of the rigid transformations SE(3) and the corresponding error model are proposed for nonlinear optimization. This novel metric works well with both common formulations AX=XB and AX=ZB, and makes use of them in accordance with the nature of the problem. The metric also adapts itself to the system precision characteristics. The method is compared in performance to earlier approaches.
TL;DR: A novel metaheuristic, inspired by recent developments in the field of operations research, was applied to a set of three challenging parameter estimation problems of nonlinear dynamic biological systems, outperforming very significantly all the methods previously used for these benchmark problems.
Abstract: We consider the problem of parameter estimation (model calibration) in nonlinear dynamic models of biological systems. Due to the frequent ill-conditioning and multi-modality of many of these problems, traditional local methods usually fail (unless initialized with very good guesses of the parameter vector). In order to surmount these difficulties, global optimization (GO) methods have been suggested as robust alternatives. Currently, deterministic GO methods can not solve problems of realistic size within this class in reasonable computation times. In contrast, certain types of stochastic GO methods have shown promising results, although the computational cost remains large. Rodriguez-Fernandez and coworkers have presented hybrid stochastic-deterministic GO methods which could reduce computation time by one order of magnitude while guaranteeing robustness. Our goal here was to further reduce the computational effort without loosing robustness. We have developed a new procedure based on the scatter search methodology for nonlinear optimization of dynamic models of arbitrary (or even unknown) structure (i.e. black-box models). In this contribution, we describe and apply this novel metaheuristic, inspired by recent developments in the field of operations research, to a set of complex identification problems and we make a critical comparison with respect to the previous (above mentioned) successful methods. Robust and efficient methods for parameter estimation are of key importance in systems biology and related areas. The new metaheuristic presented in this paper aims to ensure the proper solution of these problems by adopting a global optimization approach, while keeping the computational effort under reasonable values. This new metaheuristic was applied to a set of three challenging parameter estimation problems of nonlinear dynamic biological systems, outperforming very significantly all the methods previously used for these benchmark problems.
TL;DR: This paper examines the local convergence properties of SQP methods applied to MPECs and SQP is shown to converge superlinearly under reasonable assumptions near a strongly stationary point.
Abstract: Recently, nonlinear programming solvers have been used to solve a range of mathematical programs with equilibrium constraints (MPECs). In particular, sequential quadratic programming (SQP) methods have been very successful. This paper examines the local convergence properties of SQP methods applied to MPECs. SQP is shown to converge superlinearly under reasonable assumptions near a strongly stationary point. A number of examples are presented that show that some of the assumptions are difficult to relax.
TL;DR: Using perspective cuts substantially improves the performance of Branch & Cut approaches for at least two models that have the required structure: the Unit Commitment problem in electrical power production and the Mean-Variance problem in portfolio optimization.
Abstract: We show that the convex envelope of the objective function of Mixed-Integer Programming problems with a specific structure is the perspective function of the continuous part of the objective function. Using a characterization of the subdifferential of the perspective function, we derive “perspective cuts”, a family of valid inequalities for the problem. Perspective cuts can be shown to belong to the general family of disjunctive cuts, but they do not require the solution of a potentially costly nonlinear programming problem to be separated. Using perspective cuts substantially improves the performance of Branch & Cut approaches for at least two models that, either “naturally” or after a proper reformulation, have the required structure: the Unit Commitment problem in electrical power production and the Mean-Variance problem in portfolio optimization.
TL;DR: This work describes techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program and shows that the number of vertices is computationally tractable yielding exact separation algorithms.
Abstract: A gas network basically consists of a set of compressors and valves that are connected by pipes. The problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost-efficiently such that all demands of the gas network are satisfied. This problem leads to a complex mixed integer nonlinear optimization problem. We describe techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program. We study sub-polyhedra linking these piece-wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms. Suitable branching strategies complementing the separation algorithms are also presented. Our computational results demonstrate the success of this approach.
TL;DR: This overview paper first survey numerical approaches to solve nonlinear optimal control problems, and second, it presents the most recent algorithmic developments for real-time optimization in nonlinear model predictive control.
Abstract: In this overview paper, we first survey numerical approaches to solve nonlinear optimal control problems, and second, we present our most recent algorithmic developments for real-time optimization in nonlinear model predictive control. In the survey part, we discuss three direct optimal control approaches in detail: (i) single shooting, (ii) collocation, and (iii) multiple shooting, and we specify why we believe the direct multiple shooting method to be the method of choice for nonlinear optimal control problems in robotics. We couple it with an efficient robot model generator and show the performance of the algorithm at the example of a five link robot arm. In the real-time optimization part, we outline the idea of nonlinear model predictive control and the real-time challenge it poses to numerical optimization. As one solution approach, we discuss the real-time iteration scheme.
TL;DR: Comparing the properties of solutions generated by simultaneous approaches to those of classical variational methods, for a variety of problem classes, shows why simultaneous strategies are especially beneficial for dynamic systems with unstable modes, with path constraints and for large-scale, structured problems.
TL;DR: A synchronous variant of the NIMBUS method, which can handle versatile types of multiobjective optimization problems and includes new desirable features increasing its user-friendliness is described.
30 Nov 2006
TL;DR: In this article, a modified differential evolution (MDE) algorithm is presented for solving short-term hydrothermal scheduling problem, which is an improved version of a genetic algorithm, is a very simple, fast and robust global optimisation technique.
Abstract: A modified differential evolution (MDE) algorithm, for solving short-term hydrothermal scheduling problem is presented. Hydrothermal scheduling involves the optimisation of a nonlinear objective function with a set of operational and physical constraints. Differential evolution, an improved version of a genetic algorithm, is a very simple, fast and robust global optimisation technique. The differential evolution algorithm is modified in order to handle the reservoir end volume constraints in the hydrothermal scheduling. The transmission losses are also accounted for through the use of loss coefficients. The study is extended for the combined economic emission dispatch. The performance of the proposed approach is validated by illustration with two test systems. The results of the proposed approach are compared with those of dynamic programming, nonlinear programming, genetic algorithm and evolutionary programming techniques. From the numerical results, it is found that the modified DE based approach is able to provide a better solution at a lesser computational effort.
28 Jun 2006
TL;DR: A survey of control allocation methods for overactuated vessels is presented in this paper, where the objective is to minimize the use of control effort (or power) subject to actuator rate and position constraints, power constraints as well as other operational constraints.
Abstract: Control allocation problems for marine vessels can be formulated as optimization problems, where the objective typically is to minimize the use of control effort (or power) subject to actuator rate and position constraints, power constraints as well as other operational constraints. In addition, singularity avoidance for vessels with azimuthing thrusters represent a challenging problem since a non-convex nonlinear program must be solved. This is useful to avoid temporarily loss of controllability in some cases. In this paper, a survey of control allocation methods for overactuated vessels are presented.
TL;DR: The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions, and the results are then extended to an interior-relaxation approach.
Abstract: This paper studies theoretical and practical properties of interior-penalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then extended to an interior-relaxation approach. Superlinear convergence to strongly stationary points is also established. Two strategies for updating the penalty parameter are proposed, and their efficiency and robustness are studied on an extensive collection of test problems.
TL;DR: A Genetic Algorithm (GA), as a soft computing approach, is a powerful tool for solving various reliability optimization problems, such as reliability optimization of redundant system, reliability optimized with alternative design, reliability optimization with time-dependent reliability, reliability Optimization with interval coefficients, bicriteria reliability optimization, and reliability optimize with fuzzy goals.
TL;DR: A specialized solution method is developed, based on a nonlinear column generation technique, and it is proved that it converges to the globally optimal solution.
Abstract: We consider the problem of finding the jointly optimal end-to-end communication rates, routing, power allocation and transmission scheduling for wireless networks. In particular, we focus on finding the resource allocation that achieves fair end-to-end communication rates. Using realistic models of several rate and power adaption schemes, we show how this cross-layer optimization problem can be formulated as a nonlinear mathematical program. We develop a specialized solution method, based on a nonlinear column generation technique, and prove that it converges to the globally optimal solution. We present computational results from a large set of networks and discuss the insight that can be gained about the influence of power control, spatial reuse, routing strategies and variable transmission rates on network performance.
TL;DR: This paper aims at investigating the performance of multiobjective evolutionary algorithms (MOEAs) on solving large instances of the mapping problem and shows that MOEAs provide the designer with a highly accurate set of solutions in a reasonable amount of time.
Abstract: Sesame is a software framework that aims at developing a modeling and simulation environment for the efficient design space exploration of heterogeneous embedded systems. Since Sesame recognizes separate application and architecture models within a single system simulation, it needs an explicit mapping step to relate these models for cosimulation. The design tradeoffs during the mapping stage, namely, the processing time, power consumption, and architecture cost, are captured by a multiobjective nonlinear mixed integer program. This paper aims at investigating the performance of multiobjective evolutionary algorithms (MOEAs) on solving large instances of the mapping problem. With two comparative case studies, it is shown that MOEAs provide the designer with a highly accurate set of solutions in a reasonable amount of time. Additionally, analyses for different crossover types, mutation usage, and repair strategies for the purpose of constraints handling are carried out. Finally, a number of multiobjective optimization results are simulated for verification.
TL;DR: By integrating a GA with a nonlinear interior point method (IPM), a novel hybrid method for the optimal reactive power flow (ORPF) problem is proposed in this article, which can be mainly divided into two parts.
Abstract: By integrating a genetic algorithm (GA) with a nonlinear interior point method (IPM), a novel hybrid method for the optimal reactive power flow (ORPF) problem is proposed in this paper. The proposed method can be mainly divided into two parts. The first part is to solve the ORPF with the IPM by relaxing the discrete variables. The second part is to decompose the original ORPF into two sub-problems: continuous optimization and discrete optimization. The GA is used to solve the discrete optimization with the continuous variables being fixed, whereas the IPM solves the continuous optimization with the discrete variables being constant. The optimal solution can be obtained by solving the two sub-problems alternately. A dynamic adjustment strategy is also proposed to make the GA and the IPM to complement each other and to enhance the efficiency of the hybrid proposed method. Numerical simulations on the IEEE 30-bus, IEEE 118-bus and Chongqing 161-bus test systems illustrate that the proposed hybrid method is efficient for the ORPF problem
01 Jan 2006
TL;DR: Fast Linear Algebra for Multiarc Trajectory Optimization and Parametric Sensitivity Analysis for Optimal Boundary Control of a 3D Reaction-Diffusion System.
Abstract: Fast Linear Algebra for Multiarc Trajectory Optimization (Nicolas Berend, J. Frederic Bonnans, Julien Laurent-Varin, MounirHaddou, Christophe Talbot).- Lagrange Multipliers with Optimal Sensitivity Properties in Constrained Optimization (Dimitri P. Bertsekas).- n O(n^2) Algorithm for Isotonic Regression (Oleg Burdakov, Oleg Sysoev, Anders Grimvall, Mohamed Hussian).- Knitro: An Integrated Package for Nonlinear Optimization (Richard H. Byrd, Jorge Nocedal, Richard A. Waltz).- On implicit-factorization constraint preconditioners (H. Sue Dollar, Nicholas I. M. Gould, Andrew J. Wathen).- Optimal algorithms for large sparse quadratic programming problems with uniformly bounded spectrum (Zdenek Dostal).- Numerical methods for separating two polyhedra (Yury G. Evtushenko, Alexander I. Golikov, Saed Ketabchi).- Exact penalty functions for generalized Nash problems (Francisco Facchinei, Jong-Shi Pang).- Parametric Sensitivity Analysis for Optimal Boundary Control of a 3D Reaction-Diffusion System (Roland Griesse, Stefan Volkwein).- Projected Hessians for Preconditioning in One-Step One-Shot Design Optimization (Andreas Griewank).- Conditions and parametric representations of approximate minimal elements of a set through scalarization (Cesar Gutierrez, Bienvenido Jimenez, Vicente Novo).- Efficient methods for large-scale unconstrained optimization (Ladislav Luksan, Jan Vlcek).- A variational approach for minimum cost flow problems (Giandomenico Mastroeni).- Multi-Objective Optimisation of Expensive Objective Functions with Variable Fidelity Models (Daniele Peri, Antonio Pinto, Emilio F. Campana).-Towards the Numerical Solution of a Large Scale PDAE Constrained Optimization Problem Arising in Molten Carbonate Fuel Cell Modeling (Hans Josef Pesch, Kati Sternberg, Kurt Chudej).- The NEWUOA software for unconstrained optimization without derivatives (M.J.D. Powell).
TL;DR: In this article, the fundamental elements of a simulation-based design environment for shape optimization are presented and analyzed, and a new Verification and Validation (V&V) methodology for assessing errors and uncertainties in simulation based optimization is also introduced based on the trends.
TL;DR: A novel modified differential evolution (MDE) is used for solving process synthesis and design problems in chemical engineering, and the performance of MDE is compared with that of Genetic Algorithm, Evolution Strategy, and MINLP-Simplex Simulated Annealing (M-SIMPSA).
21 Aug 2006
TL;DR: In this paper, a tool developed for the preliminary design of low-thrust trajectories is described, where the trajectory is discretized into segments and a nonlinear programming method is used for optimization.
Abstract: A tool developed for the preliminary design of low-thrust trajectories is described. The trajectory is discretized into segments and a nonlinear programming method is used for optimization. The tool is easy to use, has robust convergence, and can handle many intermediate encounters. In addition, the tool has a wide variety of features, including several options for objective function and different low-thrust propulsion models (e.g., solar electric propulsion, nuclear electric propulsion, and solar sail). High-thrust, impulsive trajectories can also be optimized.