Showing papers on "Nonlinear programming published in 1999"
TL;DR: The design and implementation of a new algorithm for solving large nonlinear programming problems follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration.
Abstract: The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primal-dual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests.
1,605 citations
TL;DR: A new approach for solving constrained numerical optimization problems which incorporates a homomorphous mapping between n-dimensional cube and a feasible search space and constitutes an example of the fifth decoder-based category of constraint handling techniques.
Abstract: During the last five years, several methods have been proposed for handling nonlinear constraints using evolutionary algorithms (EAs) for numerical optimization problems. Recent survey papers classify these methods into four categories: preservation of feasibility, penalty functions, searching for feasibility, and other hybrids.
In this paper we investigate a new approach for solving constrained numerical optimization problems which incorporates a homomorphous mapping between n-dimensional cube and a feasible search space. This approach constitutes an example of the fifth decoder-based category of constraint handling techniques. We demonstrate the power of this new approach on several test cases and discuss its further potential.
778 citations
TL;DR: Two suboptimal MPC schemes are presented and analyzed that are guaranteed to be stabilizing, provided an initial feasible solution is available and for which the computational requirements are more reasonable.
Abstract: Practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known. Stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex, nonlinear optimization problems are possible in limited computational time. In the paper, we first establish conditions under which suboptimal model predictive control (MPC) controllers are stabilizing; the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost. Second, we present and analyze two suboptimal MPC schemes that are guaranteed to be stabilizing, provided an initial feasible solution is available and for which the computational requirements are more reasonable.
641 citations
TL;DR: Numerical comparisons with MINOS and LANCELOT show that the interior-point algorithm for nonconvex nonlinear programming is efficient, and has the promise of greatly reducing solution times on at least some classes of models.
Abstract: The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.
567 citations
TL;DR: A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser (1994) and Donoho (1994), which include the p-norm-like (l/sub (p/spl les/1)/) diversity measures and the Gaussian and Shannon entropies.
Abstract: A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser (1994) and Donoho (1994). These measures include the p-norm-like (l/sub (p/spl les/1)/) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the affine scaling transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the (l/sub (p/spl les/1)/) diversity measures are equivalent to a previously developed class of algorithms called focal underdetermined system solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a well-behaved p=0 norm-like optimization algorithm. Computer experiments demonstrate that the p-norm-like and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution.
554 citations
TL;DR: This paper describes a software package, called LOQO, which implements a primal-dual interior-point method for general nonlinear programming, and shows that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness.
Abstract: This paper describes a software package, called LOQO, which implements a primal-dual interior-point method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions was published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems.
514 citations
TL;DR: This work proves global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.
Abstract: We present a convergence theory for pattern search methods for solving bound constrained nonlinear programs. The analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints. This analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems. We prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.
512 citations
TL;DR: The key is to observe that the trust-region problem within the currently generated Krylov subspace has a very special structure which enables it to be solved very efficiently.
Abstract: The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust-region boundary. In this paper, we investigate ways of continuing the process once the boundary has been encountered. The key is to observe that the trust-region problem within the currently generated Krylov subspace has a very special structure which enables it to be solved very efficiently. We compare the new strategy with existing methods. The resulting software package is available as HSL_VF05 within the Harwell Subroutine Library.
300 citations
Journal Article•
TL;DR: It is proved that, under reasonable conditions and for every possible choice of the starting point, the sequence of iterates has at least one first-order critical accumulation point.
Abstract: A trust-region SQP-filter algorithm of the type introduced by Fletcher and Leyffer [Math. Program., 91 (2002), pp. 239--269] that decomposes the step into its normal and tangential components allows for an approximate solution of the quadratic subproblem and incorporates the safeguarding tests described in Fletcher, Leyffer, and Toint [On the Global Convergence of an SLP-Filter Algorithm, Technical Report 98/13, Department of Mathematics, University of Namur, Namur, Belgium, 1998; On the Global Convergence of a Filter-SQP Algorithm, Technical Report 00/15, Department of Mathematics, University of Namur, Namur, Belgium, 2000] is considered. It is proved that, under reasonable conditions and for every possible choice of the starting point, the sequence of iterates has at least one first-order critical accumulation point.
273 citations
TL;DR: The algorithm employs a symbolic reformulation step that brings the original MINLP problem to an equivalent standard form for which a convex relaxation can be constructed and is solved using a spatial branch-and-bound algorithm which branches on both integer and continuous variables.
261 citations
TL;DR: In this paper, a parameterization of the Pareto set based on the recently developed normal-boundary intersection technique is used to formulate a subproblem, the solution of which yields the point of "maximum bulge", often referred to as the "knee of the pareto curve".
Abstract: This paper deals with the issue of generating one Pareto optimal point that is guaranteed to be in a “desirable” part of the Pareto set in a given multicriteria optimization problem. A parameterization of the Pareto set based on the recently developed normal-boundary intersection technique is used to formulate a subproblem, the solution of which yields the point of “maximum bulge”, often referred to as the “knee of the Pareto curve”. This enables the identification of the “good region” of the Pareto set by solving one nonlinear programming problem, thereby bypassing the need to generate many Pareto points. Further, this representation extends the concept of the “knee” for problems with more than two objectives. It is further proved that this knee is invariant with respect to the scales of the multiple objective functions. The generation of this knee however requires the value of each objective function at the minimizer of every objective function (the pay-off matrix). The paper characterizes situations when approximations to the function values comprising the pay-off matrix would suffice in generating a good approximation to the knee. Numerical results are provided to illustrate this point. Further, a weighted sum minimization problem is developed based on the information in the pay-off matrix, by solving which the knee can be obtained.
TL;DR: This paper introduces an unconstrained formulation of the nonlinear programming model and solves the estimation problem using a method based on repeated calls to a recently introduced unconStrained minimization algorithm.
TL;DR: In this paper, a mathematical programming model is proposed for determining the optimal water usage and treatment network (WUTN) in any chemical plant, which features the least amount of fresh water consumption and/or minimum wastewater treatment capacity.
Abstract: A mathematical programming model is proposed in this paper for determining the optimal water usage and treatment network (WUTN) in any chemical plant, which features the least amount of fresh water consumption and/or minimum wastewater treatment capacity. In particular, because design equations of all wastewater treatment facilities and all units which utilize either process or utility water are included in the model, more comprehensive integration on a plant-wide scale can be achieved. In comparison with the available technologies, the proposed method is more reliable, more accurate, and much faster in synthesizing the WUTNs. Furthermore, more cost-efficient alternatives may be identified in certain design cases.
TL;DR: In this article, a procedure for solving the power capacitor placement problem is presented, where the objective is to determine the minimum investment required to satisfy suitable reactive constraints, and optimal capacitor placement leads to a nonlinear programming problem with mixed (discrete and continuous) variables.
Abstract: A procedure for solving the power capacitor placement problem is presented. The objective is to determine the minimum investment required to satisfy suitable reactive constraints. Due to the discrete nature of reactive compensation devices, optimal capacitor placement leads to a nonlinear programming problem with mixed (discrete and continuous) variables. It is solved with an iterative algorithm based on successive linearizations of the original nonlinear model. The mixed integer linear programming problem to be solved at each iteration of the procedure is tackled by applying both a deterministic method (branch and bound) and genetic algorithm techniques. A hybrid procedure, aiming to exploit the best features of both algorithms is also considered. The proposed procedures are tested and compared with reference to a small CIGRE system and two actual networks derived from the Italian transmission and distribution system.
TL;DR: In this article, the identification of a linear parameter-varying syster whose parameter dependence can be written as a linear fractional transformation (LFT) is formulated as an output-error identification problem.
Abstract: This paper deals with the identification of a linear parameter-varying (LPV) syster whose parameter dependence can be written as a linear-fractional transformation (LFT). We formulate an output-error identification problem and present a parameter estimation scheme in which a prediction error-based cost function is minimized using nonlinear programming; its gradients and (approximate) Hessians can he computed using LPV filters and inne products, and identifiable model sets (i.e., local canonical forms) are obtained efficiently using a natural geometrical approach. Some computational issues and experiences are discussed, and a simple numerical example is provided fo r illustration.
TL;DR: A method of optimizing high-performance concrete mix proportioning for a given workability and compressive strength using artificial neural networks and nonlinear programming is described.
Abstract: A method of optimizing high-performance concrete mix proportioning for a given workability and compressive strength using artificial neural networks and nonlinear programming is described. The basic procedure of the methodology consists of three steps: (1) Build accurate models for workability and strength using artificial neural networks and experimental data; (2) incorporate these models in software allowing an evaluation of the specified properties for a given mix; and (3) incorporate the software in a nonlinear programming package allowing a search of the optimum proportion mix design. For performing optimum concrete mix design based on the proposed methodology, a software package has been developed. One can conduct mix simulations covering all the important properties of the concrete at the same time. To demonstrate the utility of the proposed methodology, experimental results from several different mix proportions based on various design requirements are presented.
TL;DR: It is proved that, for the multiple-quality case, the Lagrangian approach provides tighter lower bounds than the standard linear-programming relaxations used in global optimization algorithms.
Abstract: Pooling and blending problems occur frequently in the petrochemical industry where crude oils, procured from various sources, are mixed together to manufacture several end-products. Finding optimal solutions to pooling problems requires the solution of nonlinear optimization problems with multiple local minima. We introduce a new Lagrangian relaxation approach for developing lower bounds for the pooling problem. We prove that, for the multiple-quality case, the Lagrangian approach provides tighter lower bounds than the standard linear-programming relaxations used in global optimization algorithms. We present computational results on a set of 13 problems which includes four particularly difficult problems we constructed.
TL;DR: Three applications, maximum likelihood (ML) joint channel and data estimation, infinite-impulse-response (IIR) filter design and evaluation of minimum symbol-error-rate (MSER) decision feedback equalizer (DFE) are used to demonstrate the effectiveness of the ASA.
TL;DR: A new nonlinear algorithm is described for the solution of the probabilistic non linear programming model that exploits the structure of the optimization problem and is more efficient than other standard algorithms for nonlinear programming problems.
Abstract: Railway passenger transportation plays a fundamental role in Europe, particularly in view of the growing number of trains offering valuable services such as high speed travel, high comfort, etc. Hence, it is advantageous to submit seat inventories to a Yield Management system to get the maximum revenue. We consider a deterministic linear programming model and a probabilistic nonlinear programming model for the network problem with non-nested seat allocation. A first comparative analysis of the computational results obtained by the two models, both in terms of the overall expected revenue and in terms of CPU time, is carried out. Furthermore, we describe a new nonlinear algorithm for the solution of the probabilistic nonlinear programming model that exploits the structure of the optimization problem. The numerical results obtained on a set of real data show that, for this class of problems, this algorithm is more efficient than other standard algorithms for nonlinear programming problems.
TL;DR: In this article, a projective approach is used to define necessary and sufficient conditions for optimization problems with explicit or implicit constraints, and a particular emphasis is given to mathematical programming problems with non-polyhedral constraints.
Abstract: Using a projective approach, new necessary conditions and new sufficient conditions for optimization problems with explicit or implicit constraints are examined. They are compared to previous ones. A particular emphasis is given to mathematical programming problems with non-polyhedral constraints. This case occurs in particular when the constraints are defined in functional spaces.
TL;DR: In this article, an optimization strategy was developed to determine the optimum cutting parameters for multipass milling operations like plain milling and face milling, based on the "maximum production rate" criterion and incorporating eight technological constraints.
Abstract: This paper outlines the development of an optimization strategy to determine the optimum cutting parameters for multipass milling operations like plain milling and face milling. The developed strategy is based on the “maximum production rate” criterion and incorporates eight technological constraints. The optimum number of passes is determined via dynamic programming, and the optimal values of the cutting conditions are found based on the objective function developed for the typified criterion by using a non-linear programming technique called “geometric programming”. This paper also underlies the importance of using optimization strategies rather than handbook recommendations as well as pointing out the superiority of the multipass over the single-pass optimization approach.
TL;DR: A methodology based on genetic algorithm has been developed for lower cost design of new, and augmentation of existing water distribution networks through application to several case studies and results results in a lower cost solution.
Abstract: A methodology based on genetic algorithm has been developed for lower cost design of new, and augmentation of existing water distribution networks. The results have been compared with those of non-linear programming technique through application to several case studies. The genetic algorithm results in a lower cost solution. Parameters governing the convergence of the solutions in non-linear and genetic algorithms are also discussed.
01 Jul 1999
TL;DR: Criteria by which one can classify, analyze, and evaluate approaches to solving multidisciplinary design optimization (MDO) problems are discussed.
Abstract: We discuss criteria by which one can classify, analyze, and evaluate approaches to solving multidisciplinary design optimization (MDO) problems. Central to our discussion is the often overlooked distinction between questions of formulating MDO problems and solving the resulting computational problem. We illustrate our general remarks by comparing several approaches to MDO that have been proposed.
01 Jun 1999
TL;DR: Experimental results demonstrate that the sequence-of-linear-programming method is orders of magnitude faster than the best-known method based on conjugate gradients, with constantly better optimization solutions.
Abstract: This paper presents a new method for determining the widths of the power and ground routes in integrated circuits so that the area required by the routes is minimized subject to the reliability constraints The basic idea is to transform the resulting constrained nonlinear programming problem into a sequence of linear programs Theoretically, we show that the sequence of linear programs always converges to the optimum solution of the relaxed convex problem Experimental results demonstrate that the sequence-of-linear-programming method is orders of magnitude faster than the best-known method based on conjugate gradients, with constantly better optimization solutions
Book•
08 Nov 1999
TL;DR: This paper presents a meta-modelling procedure called "Constrained Maximization and Minimization of Linear Equations", which automates the very labor-intensive and therefore time-heavy process of solving nonlinear Equations problems.
Abstract: FOUNDATIONS: LINEAR METHODS. Matrix Algebra. Systems of Linear Equations. FOUNDATIONS: NONLINEAR METHODS. Unconstrained Maximization and Minimization. Constrained Maximization and Minimization. APPLICATIONS: ITERATIVE METHODS FOR NONLINEAR PROBLEMS. Solving Nonlinear Equations. Solving Unconstrained Maximization and Minimization Problems. APPLICATIONS: CONSTRAINED OPTIMIZATION IN LINEAR MODELS. Linear Programming: Fundamentals. Linear Programming: Extensions. Linear Programming: Interior Point Methods. APPLICATIONS: CONSTRAINED OPTIMIZATION IN NONLINEAR MODELS. Nonlinear Programming: Fundamentals. Nonlinear Programming: Duality and Computational Methods. Problems. References.
IBM1
TL;DR: This paper describes a method of optimally sizing digital circuits on a static-timing basis based on gradient-based, nonlinear optimization and can accommodate transistor-level schematics without the need for pre-characterization.
Abstract: This paper describes a method of optimally sizing digital circuits on a static-timing basis. All paths through the logic are considered simultaneously and no input patterns need be specified by the user. The method is unique in that it is based on gradient-based, nonlinear optimization and can accommodate transistor-level schematics without the need for pre-characterization. It employs efficient time-domain simulation and gradient computation for each channel-connected component. A large-scale, general-purpose, nonlinear optimization package is used to solve the tuning problem. A prototype tuner has been developed that accommodates combinational circuits consisting of parameterized library cells. Numerical results are presented.
TL;DR: In this article, the incorporation of a genetic algorithm methodology for determining the critical slip surface in multiple-wedge stability analysis was described, which was found to be sufficiently robust to handle layered soils with weak, thin layers, and as efficient and accurate as the convention.
Abstract: Most procedures for determining the critical slip surface in slope-stability analysis rely on traditional nonlinear optimization techniques. The main shortcoming of these techniques is the uncertainty as to robustness of the algorithms to locate the global minimum factor of safety rather than the local minimum factor of safety for complicated and nonhomogeneous geological subsoil conditions. This paper describes the incorporation of a genetic algorithm methodology for determining the critical slip surface in multiple-wedge stability analysis. This search strategy is becoming increasingly popular in engineering optimization problems because it has been shown in a wide variety of problems to be suitably robust for the search not to become trapped in local optima. Three examples are presented to demonstrate the effectiveness of the genetic algorithm approach. The search strategy was found to be sufficiently robust to handle layered soils with weak, thin layers, and as efficient and accurate as the convention...
TL;DR: It is shown that the quadratic growth condition and the Mangasarian--Fromovitz constraint qualification (MFCQ) imply that local minima of nonlinear programs are isolated stationary points, and an exact penalty sequential quadRatic programming algorithm will induce at least R-linear convergence of the iterates to such a local minimum.
Abstract: We show that the quadratic growth condition and the Mangasarian--Fromovitz constraint qualification (MFCQ) imply that local minima of nonlinear programs are isolated stationary points. As a result, when started sufficiently close to such points, an $L_\infty$ exact penalty sequential quadratic programming algorithm will induce at least R-linear convergence of the iterates to such a local minimum. We construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the MFCQ but for which no positive semidefinite augmented Lagrangian exists. We present numerical results obtained using several nonlinear programming packages on this example and discuss its implications for some algorithms.
TL;DR: Novel theoretical and algorithmic developments for the solution of mixed- integer optimization problems involving uncertainty are presented, which can be posed as multiparametric mixed-integer optimization models, where uncertainty is described by a set of parameters bounded between lower and upper bounds.
Abstract: In this paper we present novel theoretical and algorithmic developments for the solution of mixed-integer optimization problems involving uncertainty, which can be posed as multiparametric mixed-integer optimization models, where uncertainty is described by a set of parameters bounded between lower and upper bounds. In particular, we address convex nonlinear formulations involving (i) 0−1 integer variables and (ii) uncertain parameters appearing linearly and separately and present on the right-hand side of the constraints. The developments reported in this work are based upon decomposition principles where the problem is decomposed into two iteratively converging subproblems: (i) a primal and (ii) a master subproblem, representing valid parametric upper and lower bounds on the final solution, respectively. The primal subproblem is formulated by fixing the integer variables which results in a multiparametric nonlinear programming (mp-NLP) problem, which is solved by outer-approximating the nonlinear funct...