Types of Penalty Methods for Handling Constraints
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
PENALTY METHODS for the SOLUTION of GENERALIZED NASH EQUILIBRIUM PROBLEMS (WITH COMPLETE TEST PROBLEMS) Francisco Facchinei1
PENALTY METHODS FOR THE SOLUTION OF GENERALIZED NASH EQUILIBRIUM PROBLEMS (WITH COMPLETE TEST PROBLEMS) Francisco Facchinei1 and Christian Kanzow2 1 Sapienza University of Rome Department of Computer and System Sciences “A. Ruberti” Via Ariosto 25, 00185 Roma, Italy e-mail: [email protected] 2 University of W¨urzburg Institute of Mathematics Am Hubland, 97074 W¨urzburg, Germany e-mail: [email protected] February 11, 2009 Abstract The generalized Nash equilibrium problem (GNEP) is an extension of the classical Nash equilibrium problem where both the objective functions and the constraints of each player may depend on the rivals’ strategies. This class of problems has a multitude of important engineering applications and yet solution algorithms are extremely scarce. In this paper, we analyze in detail a globally convergent penalty method that has favorable theoretical properties. We also consider strengthened results for a particular subclass of problems very often considered in the literature. Basically our method reduces the GNEP to a single penalized (and nonsmooth) Nash equilibrium problem. We suggest a suitable method for the solution of the latter penalized problem and present extensive numerical results. Key Words: Nash equilibrium problem, Generalized Nash equilibrium problem, Jointly convex problem, Exact penalty function, Global convergence. 1 Introduction In this paper, we consider the Generalized Nash Equilibrium Problem (GNEP) and describe new globally convergent methods for its solution. The GNEP is an extension of the classical Nash Equilibrium Problem (NEP) where the feasible sets of each player may depend on ν n the rivals’ strategies. We assume there are N players and denote by x ∈ R ν the vector representing the ν-th player’s strategy. -
First-Order Methods for Semidefinite Programming
FIRST-ORDER METHODS FOR SEMIDEFINITE PROGRAMMING Zaiwen Wen Advisor: Professor Donald Goldfarb Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2009 c 2009 Zaiwen Wen All Rights Reserved ABSTRACT First-order Methods For Semidefinite Programming Zaiwen Wen Semidefinite programming (SDP) problems are concerned with minimizing a linear function of a symmetric positive definite matrix subject to linear equality constraints. These convex problems are solvable in polynomial time by interior point methods. However, if the number of constraints m in an SDP is of order O(n2) when the unknown positive semidefinite matrix is n n, interior point methods become impractical both in terms of the × time (O(n6)) and the amount of memory (O(m2)) required at each iteration to form the m × m positive definite Schur complement matrix M and compute the search direction by finding the Cholesky factorization of M. This significantly limits the application of interior-point methods. In comparison, the computational cost of each iteration of first-order optimization approaches is much cheaper, particularly, if any sparsity in the SDP constraints or other special structure is exploited. This dissertation is devoted to the development, analysis and evaluation of two first-order approaches that are able to solve large SDP problems which have been challenging for interior point methods. In chapter 2, we present a row-by-row (RBR) method based on solving a sequence of problems obtained by restricting the n-dimensional positive semidefinite constraint on the matrix X. By fixing any (n 1)-dimensional principal submatrix of X and using its − (generalized) Schur complement, the positive semidefinite constraint is reduced to a sim- ple second-order cone constraint. -
Constraint-Handling Techniques for Highly Constrained Optimization
Andreas Petrow Constraint-Handling Techniques for highly Constrained Optimization Intelligent Cooperative Systems Computational Intelligence Constraint-Handling Techniques for highly Constrained Optimization Master Thesis Andreas Petrow March 22, 2019 Supervisor: Prof. Dr.-Ing. habil. Sanaz Mostaghim Advisor: M.Sc. Heiner Zille Advisor: Dr.-Ing. Markus Schori Andreas Petrow: Constraint-Handling Techniques for highly Con- strained Optimization Otto-von-Guericke Universität Intelligent Cooperative Systems Computational Intelligence Magdeburg, 2019. Abstract This thesis relates to constraint handling techniques for highly constrained opti- mization. Many real world optimization problems are non-linear and restricted to constraints. To solve non-linear optimization problems population-based meta- heuristics are used commonly. To enable such optimization methods to fulfill the optimization with respect to the constraints modifications are required. Therefore a research field considering such constraint handling techniques establishes. How- ever the combination of large-scale optimization problems with constraint handling techniques is rare. Therefore this thesis involves with common constraint handling techniques and evaluates these on two benchmark problems. I Contents List of Figures VII List of TablesXI List of Acronyms XIII 1. Introduction and Motivation1 1.1. Aim of this thesis...........................2 1.2. Structure of this thesis........................3 2. Basics5 2.1. Constrained Optimization Problem..................5 2.2. Optimization Method.........................6 2.2.1. Classical optimization....................7 2.2.2. Population-based Optimization Methods...........7 3. Related Work 11 3.1. Penalty Methods........................... 12 3.1.1. Static Penalty Functions................... 12 3.1.2. Co-Evolution Penalty Methods................ 13 3.1.3. Dynamic Penalty Functions................. 13 3.1.4. Adaptive Penalty Functions................. 13 3.2. -
Matroidal Subdivisions, Dressians and Tropical Grassmannians
Matroidal subdivisions, Dressians and tropical Grassmannians vorgelegt von Diplom-Mathematiker Benjamin Frederik Schröter geboren in Frankfurt am Main Von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Wilhelm Stannat Gutachter: Prof. Dr. Michael Joswig Prof. Dr. Hannah Markwig Senior Lecturer Ph.D. Alex Fink Tag der wissenschaftlichen Aussprache: 17. November 2017 Berlin 2018 Zusammenfassung In dieser Arbeit untersuchen wir verschiedene Aspekte von tropischen linearen Räumen und deren Modulräumen, den tropischen Grassmannschen und Dressschen. Tropische lineare Räume sind dual zu Matroidunterteilungen. Motiviert durch das Konzept der Splits, dem einfachsten Fall einer polytopalen Unterteilung, wird eine neue Klasse von Matroiden eingeführt, die mit Techniken der polyedrischen Geometrie untersucht werden kann. Diese Klasse ist sehr groß, da sie alle Paving-Matroide und weitere Matroide enthält. Die strukturellen Eigenschaften von Split-Matroiden können genutzt werden, um neue Ergebnisse in der tropischen Geometrie zu erzielen. Vor allem verwenden wir diese, um Strahlen der tropischen Grassmannschen zu konstruieren und die Dimension der Dressschen zu bestimmen. Dazu wird die Beziehung zwischen der Realisierbarkeit von Matroiden und der von tropischen linearen Räumen weiter entwickelt. Die Strahlen einer Dressschen entsprechen den Facetten des Sekundärpolytops eines Hypersimplexes. Eine besondere Klasse von Facetten bildet die Verallgemeinerung von Splits, die wir Multi-Splits nennen und die Herrmann ursprünglich als k-Splits bezeichnet hat. Wir geben eine explizite kombinatorische Beschreibung aller Multi-Splits eines Hypersimplexes. Diese korrespondieren mit Nested-Matroiden. Über die tropische Stiefelabbildung erhalten wir eine Beschreibung aller Multi-Splits für Produkte von Simplexen. -
Numerical Solution of Saddle Point Problems
Acta Numerica (2005), pp. 1–137 c Cambridge University Press, 2005 DOI: 10.1017/S0962492904000212 Printed in the United Kingdom Numerical solution of saddle point problems Michele Benzi∗ Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA E-mail: [email protected] Gene H. Golub† Scientific Computing and Computational Mathematics Program, Stanford University, Stanford, California 94305-9025, USA E-mail: [email protected] J¨org Liesen‡ Institut f¨ur Mathematik, Technische Universit¨at Berlin, D-10623 Berlin, Germany E-mail: [email protected] We dedicate this paper to Gil Strang on the occasion of his 70th birthday Large linear systems of saddle point type arise in a wide variety of applica- tions throughout computational science and engineering. Due to their indef- initeness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems. ∗ Supported in part by the National Science Foundation grant DMS-0207599. † Supported in part by the Department of Energy of the United States Government. ‡ Supported in part by the Emmy Noether Programm of the Deutsche Forschungs- gemeinschaft. 2 M. Benzi, G. H. Golub and J. Liesen CONTENTS 1 Introduction 2 2 Applications leading to saddle point problems 5 3 Properties of saddle point matrices 14 4 Overview of solution algorithms 29 5 Schur complement reduction 30 6 Null space methods 32 7 Coupled direct solvers 40 8 Stationary iterations 43 9 Krylov subspace methods 49 10 Preconditioners 59 11 Multilevel methods 96 12 Available software 105 13 Concluding remarks 107 References 109 1. -
A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems
' : .v ' : V. • ' '^!V'> r" '%': ' W ' ::r'' NISTIR 4900 Computing and Applied Mathematics Laboratory A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems P.T. Boggs, J.W. Tolle, A.J. Kearsley August 1992 Technology Administration U.S. DEPARTMENT OF COMMERCE National Institute of Standards and Technology Gaithersburg, MD 20899 “QC— 100 .U56 4900 1992 Jc. NISTIR 4900 A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems P. T. Boggs J. W. Tone A. J. Kearsley U.S. DEPARTMENT OF COMMERCE Technology Administration National Institute of Standards and Technology Computing and Applied Mathematics Laboratory Applied and Computational Mathematics Division Gaithersburg, MD 20899 August 1992 U.S. DEPARTMENT OF COMMERCE Barbara Hackman Franklin, Secretary TECHNOLOGY ADMINISTRATION Robert M. White, Under Secretary for Technology NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY John W. Lyons, Director si.- MT-- ’•,•• /'k /-.<• -Vj* #' ^ ( "> M !f>' J ' • ''f,i sgsHv.w,^. ^ , .!n'><^.'". tr 'V 10 i^ailCin'i.-'J/ > ' it i>1l • •»' W " %'. ' ' -fft ?" . , - '>! .-' .S A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems * Paul T. Boggs ^ Jon W. ToUe ^ Anthony J. Kearsley § August 4, 1992 Abstract We consider the inequality constrained nonlinear programming problem and an SQP algorithm for its solution. We are primarily concerned with two aspects of the general procedure, namely, the approximate solution of the quadratic program, and the need for an appropriate merit function. We first describe an (iterative) interior-point method for the quadratic programming subproblem that, no matter when it its terminated, yields a descent direction for a suggested new merit function. An algorithm based on ideas from trust-region and truncated Newton methods, is suggested and some of our preliminary numerical results are discussed. -
A Bound Strengthening Method for Optimal Transmission Switching in Power Systems
1 A Bound Strengthening Method for Optimal Transmission Switching in Power Systems Salar Fattahi∗, Javad Lavaei∗;+, and Alper Atamturk¨ ∗ ∗ Industrial Engineering and Operations Research, University of California, Berkeley + Tsinghua-Berkeley Shenzhen Institute, University of California, Berkeley. Abstract—This paper studies the optimal transmission switch- transmission switches) in the system. Furthermore, The Energy ing (OTS) problem for power systems, where certain lines are Policy Act of 2005 explicitly addresses the “difficulties of fixed (uncontrollable) and the remaining ones are controllable siting major new transmission facilities” and calls for the via on/off switches. The goal is to identify a topology of the power grid that minimizes the cost of the system operation while utilization of better transmission technologies [2]. satisfying the physical and operational constraints. Most of the Unlike in the classical network flows, removing a line from existing methods for the problem are based on first converting a power network may improve the efficiency of the network the OTS into a mixed-integer linear program (MILP) or mixed- due to physical laws. This phenomenon has been observed integer quadratic program (MIQP), and then iteratively solving and harnessed to improve the power system performance by a series of its convex relaxations. The performance of these methods depends heavily on the strength of the MILP or MIQP many authors. The notion of optimally switching the lines of formulations. In this paper, it is shown that finding the strongest a transmission network was introduced by O’Neill et al. [3]. variable upper and lower bounds to be used in an MILP or Later on, it has been shown in a series of papers that the MIQP formulation of the OTS based on the big-M or McCormick incorporation of controllable transmission switches in a grid inequalities is NP-hard. -
Constrained Cohort Intelligence Using Static and Dynamic Penalty Function Approach for Mechanical Components Design
Constrained Cohort Intelligence using Static and Dynamic Penalty Function Approach for Mechanical Components Design Omkar Kulkarni1, Ninad Kulkarni1, Anand J Kulkarni2,3*, Ganesh Kakandikar1 1 Department of Mechanical Engineering, Faculty of Engineering, Zeal College of Engineering and Research, Pune 2 Odette School of Business, University of Windsor, 401 Sunset Avenue, Windsor, Ontario N9B 3P4, Canada Email: [email protected]; Ph: 1 519 253 3000 x4939 3Department of Mechanical Engineering, Symbiosis Institute of Technology, Symbiosis International University, Pune MH 412 115 India Email: [email protected]; [email protected], Ph: 91 20 39116468 Abstract Most of the metaheuristics can efficiently solve unconstrained problems; however, their performance may degenerate if the constraints are involved. This paper proposes two constraint handling approaches for an emerging metaheuristic of Cohort Intelligence (CI). More specifically CI with static penalty function approach (SCI) and CI with dynamic penalty function approach (DCI) are proposed. The approaches have been tested by solving several constrained test problems. The performance of the SCI and DCI have been compared with algorithms like GA, PSO, ABC, d-Ds. In addition, as well as three real world problems from mechanical engineering domain with improved solutions. The results were satisfactory and validated the applicability of CI methodology for solving real world problems. Keywords: Cohort Intelligence, Static Penalty Function Approach, Dynamic Penalty Function Approach, Constrained Optimization 1. Introduction In past few years, several metaheuristics have been proposed in the field of optimization. They include number of nature/bio-inspired optimization techniques such as Evolutionary Algorithms (EAs), Swarm Intelligence (SI), etc. Some of the SI techniques include Particle swarm optimization (PSO), Cuckoo Search Algorithm (CS) (Yang et al. -
A Penalty Method for PDE-Constrained Optimization in Inverse Problems
A penalty method for PDE-constrained optimization in inverse problems T. van Leeuwen1 and F.J. Herrmann2 1Mathematical Institute, Utrecht University, Utrecht, the Netherlands. 2 Dept. of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Vancouver (BC), Canada. E-mail: [email protected] Abstract. Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand-sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the parameters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the non-linearity of the problem and is less sensitive to the initial iterate. arXiv:1504.02249v2 [math.OC] 23 Sep 2015 A penalty method for inverse problems 2 1. -
A Double-Multiplicative Dynamic Penalty Approach for Constrained Evolutionary Optimization
Struct Multidisc Optim (2008) 35:431–445 DOI 10.1007/s00158-007-0143-1 RESEARCH PAPER A double-multiplicative dynamic penalty approach for constrained evolutionary optimization Simone Puzzi & Alberto Carpinteri Received: 13 July 2006 /Revised: 5 February 2007 /Accepted: 25 March 2007 / Published online: 2 June 2007 # Springer-Verlag 2007 Abstract The present paper proposes a double-multiplica- include genetic algorithms (GAs; Holland 1975; Goldberg tive penalty strategy for constrained optimization by means 1989), evolutionary programming (Fogel 1966), evolution of genetic algorithms (GAs). The aim of this research is to strategies (Rechenberg 1973; Schwefel 1981), and biolog- provide a simple and efficient way of handling constrained ical evolution methods (Arutyunyan and Drozdov 1988; optimization problems in the GA framework without the Azegami 1990; Mattheck and Burkhardt 1990). The present need for tuning the values of penalty factors for any given trend is to decrease the differences among these paradigms optimization problem. After a short review on the most and refer, in generic terms, to evolutionary algorithms popular and effective exterior penalty formulations, the (EAs) or evolution programs (EPs) when talking about any proposed penalty strategy is presented and tested on five of them (Michalewicz 1996). different benchmark problems. The obtained results are The reason behind the success of these approaches is that compared with the best solutions provided in the literature, traditional nonlinear programming methods are local search showing the effectiveness of the proposed approach. techniques and are, thus, prone to converge toward local optima. This is a serious drawback in many practical Keywords Genetic algorithms . Exterior penalty function . structural optimization problems where the design space Constraint handling . -
A Generalized Newton-Penalty Algorithm for Large Scale Ill-Conditioned Quadratic Problems Abstract MSC (2000) No.: 90C06, 90C20
Available at Applications and Applied http://pvamu.edu/aam Mathematics: Appl. Appl. Math. ISSN: 1932-9466 An International Journal (AAM) Vol. 4, Issue 2 (December 2009), pp. 273 – 278 (Previously, Vol. 4, No. 2) A Generalized Newton-Penalty Algorithm for Large Scale Ill-Conditioned Quadratic Problems Maziar Salahi and Moslem Ganji Department of Mathematics Faculty of Sciences, University of Guilan P.O. Box 1914, Rasht, Iran Received: July 11, 2009; Accepted: October 27, 2009 Abstract Large scale quadratic problems arise in many real world applications. It is quite often that the coefficient matrices in these problems are ill-conditioned. Thus, if the problem data are available even with small error, then solving them using classical algorithms might result to meaningless solutions. In this short paper, we propose an efficient generalized Newton-penalty algorithm for solving these problems. Our computational results show that our new simple algorithm is much faster and better than the approach of Rojas et al. (2000), which requires parameter tuning for different problems. Keywords: Large Scale Ill-Quadratic Problems, Penalty Method MSC (2000) No.: 90C06, 90C20 1. Introduction Large scale quadratic problems arise in many disciplines like image restoration [ Rojas and Steihaug (2002)]. This is a well studied problem and efficient algorithms have been developed to solve various forms of , for example see Salahi (2009 a, b). It is often the case that the problem is ill- conditioned. Thus, even a small error in problem data might significantly change the solution. 273 274 Maziar Salahi and Moslem Ganji Regularization is a technique to deal with such situation and the well known one is the Tikhonov regularization [Tikhonov (1963)], which considers the following problem instead of where is the so called regularization parameter [Hansen ]. -
An Augmented Lagrangian Method for Optimization Problems in Banach Spaces Arxiv:1807.04467V1 [Math.OC] 12 Jul 2018
An Augmented Lagrangian Method for Optimization Problems in Banach Spaces∗ Christian Kanzowy Daniel Stecky Daniel Wachsmuthy October 23, 2017 Abstract. We propose a variant of the classical augmented Lagrangian method for con- strained optimization problems in Banach spaces. Our theoretical framework does not require any convexity or second-order assumptions and allows the treatment of inequality constraints with infinite-dimensional image space. Moreover, we discuss the convergence properties of our algorithm with regard to feasibility, global optimality, and KKT conditions. Some numerical results are given to illustrate the practical viability of the method. Keywords. Constrained optimization, augmented Lagrangian method, Banach space, inequality constraints, global convergence. 1 Introduction Let X, Y be (real) Banach spaces and let f : X ! R, g : X ! Y be given mappings. The aim of this paper is to describe an augmented Lagrangian method for the solution of the constrained optimization problem min f(x) subject to (s.t.) g(x) ≤ 0: (P ) We assume that Y,! L2(Ω) densely for some measure space Ω, where the natural order on L2(Ω) induces the order on Y . A detailed description together with some remarks about this setting is given in Section 2. Augmented Lagrangian methods for the solution of optimization problems belong to the most famous and successful algorithms for the solution of finite-dimensional problems and are described in almost all text books on continuous optimization, see, e.g. [8, 29]. Their generalization to infinite-dimensional problems has received considerable attention throughout the last decades [6, 7, 16, 18, 20, 21, 22, 24, 25]. However, most existing arXiv:1807.04467v1 [math.OC] 12 Jul 2018 approaches either assume a very specific problem structure [6, 7], require strong convexity assumptions [18] or consider only the case where Y is finite-dimensional [20, 24].