A Survey of Random Methods for Parameter Optimization
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Direct Search Methods for Nonsmooth Problems Using Global Optimization Techniques
Direct Search Methods for Nonsmooth Problems using Global Optimization Techniques A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS by Blair Lennon Robertson University of Canterbury 2010 To Peter i Abstract This thesis considers the practical problem of constrained and unconstrained local optimization. This subject has been well studied when the objective function f is assumed to smooth. However, nonsmooth problems occur naturally and frequently in practice. Here f is assumed to be nonsmooth or discontinuous without forcing smoothness assumptions near, or at, a potential solution. Various methods have been presented by others to solve nonsmooth optimization problems, however only partial convergence results are possible for these methods. In this thesis, an optimization method which use a series of local and localized global optimization phases is proposed. The local phase searches for a local minimum and gives the methods numerical performance on parts of f which are smooth. The localized global phase exhaustively searches for points of descent in a neighborhood of cluster points. It is the localized global phase which provides strong theoretical convergence results on nonsmooth problems. Algorithms are presented for solving bound constrained, unconstrained and con- strained nonlinear nonsmooth optimization problems. These algorithms use direct search methods in the local phase as they can be applied directly to nonsmooth prob- lems because gradients are not explicitly required. The localized global optimization phase uses a new partitioning random search algorithm to direct random sampling into promising subsets of Rn. The partition is formed using classification and regression trees (CART) from statistical pattern recognition. -
Algorithm Selection for Black-Box Continuous Optimization Problems: a Survey on Methods and Challenges
Accepted Manuscript Algorithm selection for black-box continuous optimization problems: a survey on methods and challenges Mario A. Muñoz, Yuan Sun, Michael Kirley, Saman K. Halgamuge PII: S0020-0255(15)00368-0 DOI: http://dx.doi.org/10.1016/j.ins.2015.05.010 Reference: INS 11563 To appear in: Information Sciences Received Date: 18 November 2014 Revised Date: 17 March 2015 Accepted Date: 1 May 2015 Please cite this article as: M.A. Muñoz, Y. Sun, M. Kirley, S.K. Halgamuge, Algorithm selection for black-box continuous optimization problems: a survey on methods and challenges, Information Sciences (2015), doi: http:// dx.doi.org/10.1016/j.ins.2015.05.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Algorithm selection for black-box continuous optimization problems: a survey on methods and challenges Mario A. Munoz˜ a,b,∗, Yuan Sunb, Michael Kirleyc, Saman K. Halgamugeb aSchool of Mathematical Sciences, Monash University, Clayton, Victoria 3800 Australia bDepartment of Mechanical Engineering, The University of Melbourne, Parkville, Victoria 3010 Australia cDepartment of Computer and Information Systems, The University of Melbourne, Parkville, Victoria 3010 Australia Abstract Selecting the most appropriate algorithm to use when attempting to solve a black-box continuous optimization problem is a challenging task. -
Stochastic Optimization,” in Handbook of Computational Statistics: Concepts and Methods (2Nd Ed.) (J
Citation: Spall, J. C. (2012), “Stochastic Optimization,” in Handbook of Computational Statistics: Concepts and Methods (2nd ed.) (J. Gentle, W. Härdle, and Y. Mori, eds.), Springer−Verlag, Heidelberg, Chapter 7, pp. 173–201. dx.doi.org/10.1007/978-3-642-21551-3_7 STOCHASTIC OPTIMIZATION James C. Spall The Johns Hopkins University Applied Physics Laboratory 11100 Johns Hopkins Road Laurel, Maryland 20723-6099 U.S.A. [email protected] Stochastic optimization algorithms have been growing rapidly in popularity over the last decade or two, with a number of methods now becoming “industry standard” approaches for solving challenging optimization problems. This chapter provides a synopsis of some of the critical issues associated with stochastic optimization and a gives a summary of several popular algorithms. Much more complete discussions are available in the indicated references. To help constrain the scope of this article, we restrict our attention to methods using only measurements of the criterion (loss function). Hence, we do not cover the many stochastic methods using information such as gradients of the loss function. Section 1 discusses some general issues in stochastic optimization. Section 2 discusses random search methods, which are simple and surprisingly powerful in many applications. Section 3 discusses stochastic approximation, which is a foundational approach in stochastic optimization. Section 4 discusses a popular method that is based on connections to natural evolution—genetic algorithms. Finally, Section 5 offers some concluding remarks. 1 Introduction 1.1 General Background Stochastic optimization plays a significant role in the analysis, design, and operation of modern systems. Methods for stochastic optimization provide a means of coping with inherent system noise and coping with models or systems that are highly nonlinear, high dimensional, or otherwise inappropriate for classical deterministic methods of optimization. -
STATE TRANSITION ALGORITHM Xiaojun Zhou, Chunhua Yang And
JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2012.8.1039 MANAGEMENT OPTIMIZATION Volume 8, Number 4, November 2012 pp. 1039{1056 STATE TRANSITION ALGORITHM Xiaojun Zhou, Chunhua Yang and Weihua Gui School of Information Science and Engineering Central South University, Changsha, 410083, China (Communicated by Hussein A. Abbass) Abstract. In terms of the concepts of state and state transition, a new heuris- tic random search algorithm named state transition algorithm is proposed. For continuous function optimization problems, four special transformation opera- tors called rotation, translation, expansion and axesion are designed. Adjusting measures of the transformations are mainly studied to keep the balance of ex- ploration and exploitation. Convergence analysis is also discussed about the algorithm based on random search theory. In the meanwhile, to strengthen the search ability in high dimensional space, communication strategy is in- troduced into the basic algorithm and intermittent exchange is presented to prevent premature convergence. Finally, experiments are carried out for the algorithms. With 10 common benchmark unconstrained continuous functions used to test the performance, the results show that state transition algorithms are promising algorithms due to their good global search capability and con- vergence property when compared with some popular algorithms. 1. Introduction. The concept of state means to a situation which a material sys- tem maintains, and it is characterized by a group of physical qualities. The process of a system turning from a state to another is called state transition, which can be described by a state transition matrix. The idea of state transition was created by a Russian mathematician named Markov when he expected to represent a specific sto- chastic process (known as Markov process)[22]. -
Optimization in Diamond
1 Faculty of Behavioural, Management & Social Sciences Optimization in Diamond Joris van der Meulen M.Sc. Thesis September 2016 Supervisors: dr. ir. M. R. K. Mes dr. ir. J. M. J. Schutten ir. N. Nijenmanting Production & Logistics Management (Industrial Engineering & Management) Faculty of Behavioural, Management & Social Sciences University of Twente P.O. Box 217 7500 AE Enschede The Netherlands Management Summary Diamond is a decision making tool that enables users to construct models of the processes that take place in dairy factories and optimize these processes by varying the product mix and technology settings in these models. Differential Evolution (DE), a stochastic optimization method, is implemented in Diamond to perform these optimizations. DE's effectiveness and efficiency depend onthe values of several auxiliary optimization parameters. In the current situation, a user of Diamond has to set values for those parameters before performing an optimization. The goal of this research is to find an approach for determining the values of DE's auxiliary optimization parameters in Diamond so that they do not have to be set by the user anymore. We apply the approaches of parameter selection and meta-optimization to tune the auxiliary optimization parameters of DE in Diamond. Parameter se- lection comes down to selecting values for the auxiliary optimization parameters relying on conventions and default values. Meta-optimization involves treating the search for good auxiliary parameter values as an optimization problem in its own right. It hence requires the implementation of an optimization method on the meta-level. The meta-level optimizer aims to find good values for the auxiliary parameters of DE, which in turn aims to find good values for the optimization variables of the actual problem in Diamond. -
Stochastic Derivative Free Optimization Methodwith Momentum
Published as a conference paper at ICLR 2020 ASTOCHASTIC DERIVATIVE FREE OPTIMIZATION METHOD WITH MOMENTUM Eduard Gorbunov∗ Adel Bibi MIPT, Russia and IITP RAS, Russia and RANEPA, Russia KAUST, Saudi Arabia [email protected] [email protected] Ozan Sener El Houcine Bergou Intel Labs KAUST, Saudi Arabia and MaIAGE, INRA, France [email protected] [email protected] Peter Richtárik KAUST, Saudi Arabia and MIPT, Russia [email protected] ABSTRACT We consider the problem of unconstrained minimization of a smooth objective function in Rd in setting where only function evaluations are possible. We propose and analyze stochastic zeroth-order method with heavy ball momentum. In particu- lar, we propose, SMTP, a momentum version of the stochastic three-point method (STP) Bergou et al. (2019). We show new complexity results for non-convex, convex and strongly convex functions. We test our method on a collection of learning to continuous control tasks on several MuJoCo Todorov et al. (2012) envi- ronments with varying difficulty and compare against STP, other state-of-the-art derivative-free optimization algorithms and against policy gradient methods. SMTP significantly outperforms STP and all other methods that we considered in our numerical experiments. Our second contribution is SMTP with importance sam- pling which we call SMTP_IS. We provide convergence analysis of this method for non-convex, convex and strongly convex objectives. 1 INTRODUCTION In this paper, we consider the following minimization problem min f(x); (1) d x2R where f : Rd ! R is "smooth" but not necessarily a convex function in a Derivative-Free Opti- mization (DFO) setting where only function evaluations are possible. -
A LEVEL SET GLOBAL OPTIMIZATION METHOD for NONLINEAR ENGINEERING PROBLEMS by Hassen Au Yassien
A LEVEL SET GLOBAL OPTIMIZATION METHOD FOR NONLINEAR ENGINEERING PROBLEMS By Hassen Au Yassien B. Sc. in Civil Engineering, Addis Ababa University, Ethiopia, 1979 Diploma in Hydrology, Free University of Brussels, Belgium 194 Masters in Hydrology, Free University of Brussels, Belgium 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE. OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE COLUMBIA December 1993 © Hassen Au Yassien, 1993 __________________________________ In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of C The University of British Columbia Vancouver, Canada Date kcvL to DE-6 (2)88) Abstract The mathematical equations used in civil engineering design procedures are predomi nantly nonlinear. Most civil engineering design optimization problems would therefore require the use of nonlinear programming (NLP) techniques for their solution. Those NLP packages with the ability to handle practical sizes of problems, and have been available on mainframe computers for many years, are only now becoming available on microcomputers. On top of this, these existing NLP techniques, which are dominated by the gradient methods, do not guarantee global solutions. -
Randomized Derivative-Free Optimization of Noisy Convex Functions∗
Randomized Derivative-Free Optimization of Noisy Convex Functions∗ Ruobing Cheny Stefan M. Wildz July 12, 2015 Abstract We propose STARS, a randomized derivative-free algorithm for unconstrained opti- mization when the function evaluations are contaminated with random noise. STARS takes dynamic, noise-adjusted smoothing stepsizes that minimize the least-squares er- ror between the true directional derivative of a noisy function and its finite difference approximation. We provide a convergence rate analysis of STARS for solving convex problems with additive or multiplicative noise. Experimental results show that (1) STARS exhibits noise-invariant behavior with respect to different levels of stochastic noise; (2) the practical performance of STARS in terms of solution accuracy and con- vergence rate is significantly better than that indicated by the theoretical result; and (3) STARS outperforms a selection of randomized zero-order methods on both additive- and multiplicative-noisy functions. 1 Introduction We propose STARS, a randomized derivative-free algorithm for unconstrained optimization when the function evaluations are contaminated with random noise. Formally, we address the stochastic optimization problem h i min f(x) = f~(x; ξ) ; (1.1) n Eξ x2R where the objective f(x) is assumed to be differentiable but is available only through noisy realizations f~(x; ξ). In particular, although our analysis will at times assume that the gradi- ent of the objective function f(x) exist and be Lipschitz continuous, we assume that direct evaluation of these derivatives is impossible. Of special interest to this work are situations when derivatives are unavailable or unreliable because of stochastic noise in the objective function evaluations. -
Learning to Guide Random Search
Published as a conference paper at ICLR 2020 LEARNING TO GUIDE RANDOM SEARCH Ozan Sener Vladlen Koltun Intel Labs Intel Labs ABSTRACT We are interested in derivative-free optimization of high-dimensional functions. The sample complexity of existing methods is high and depends on problem dimensionality, unlike the dimensionality-independent rates of first-order meth- ods. The recent success of deep learning suggests that many datasets lie on low-dimensional manifolds that can be represented by deep nonlinear models. We therefore consider derivative-free optimization of a high-dimensional func- tion that lies on a latent low-dimensional manifold. We develop an online learn- ing approach that learns this manifold while performing the optimization. In other words, we jointly learn the manifold and optimize the function. Our anal- ysis suggests that the presented method significantly reduces sample complex- ity. We empirically evaluate the method on continuous optimization benchmarks and high-dimensional continuous control problems. Our method achieves sig- nificantly lower sample complexity than Augmented Random Search, Bayesian optimization, covariance matrix adaptation (CMA-ES), and other derivative-free optimization algorithms. 1 INTRODUCTION A typical approach to machine learning problems is to define an objective function and optimize it over a dataset. First-order optimization methods are widely used for this purpose since they can scale to high-dimensional problems and their convergence rates are independent of problem dimensionality in most cases. However, gradients are not available in many important settings such as control, black-box optimization, and interactive learning with humans in the loop. Derivative- free optimization (DFO) can be used to tackle such problems. -
Derivative-Free Optimization Methods
Derivative-free optimization methods Jeffrey Larson, Matt Menickelly and Stefan M. Wild Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL 60439, USA [email protected], [email protected], [email protected] 4th April 2019 Dedicated to the memory of Andrew R. Conn for his inspiring enthusiasm and his many contribu- tions to the renaissance of derivative-free optimization methods. Abstract In many optimization problems arising from scientific, engineering and artificial intelligence applications, objective and constraint functions are available only as the output of a black-box or simulation oracle that does not provide derivative information. Such settings necessitate the use of methods for derivative-free, or zeroth-order, optimization. We provide a review and perspectives on developments in these methods, with an emphasis on highlighting recent developments and on unifying treatment of such problems in the non-linear optimization and machine learning literature. We categorize methods based on assumed properties of the black-box functions, as well as features of the methods. We first overview the primary setting of deterministic methods applied to unconstrained, non-convex optimization problems where the objective function is defined by a deterministic black-box oracle. We then discuss developments in randomized methods, methods that assume some additional structure about the objective (including convexity, separability and general non-smooth compositions), methods for problems where the output of the black-box oracle -
Solution of Stochastic Quadratic Programming with Imperfect Probability Distribution Using Nelder-Mead Simplex Method
Journal of Applied Mathematics and Physics, 2018, 6, 1111-1119 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 Solution of Stochastic Quadratic Programming with Imperfect Probability Distribution Using Nelder-Mead Simplex Method Xinshun Ma*, Xin Liu Department of Mathematics and Physics, North China Electric Power University, Baoding, China How to cite this paper: Ma, X.S. and Liu, Abstract X. (2018) Solution of Stochastic Quadratic Programming with Imperfect Probability Stochastic quadratic programming with recourse is one of the most important Distribution Using Nelder-Mead Simplex topics in the field of optimization. It is usually assumed that the probability Method. Journal of Applied Mathematics distribution of random variables has complete information, but only part of and Physics, 6, 1111-1119. https://doi.org/10.4236/jamp.2018.65095 the information can be obtained in practical situation. In this paper, we pro- pose a stochastic quadratic programming with imperfect probability distribu- Received: April 16, 2018 tion based on the linear partial information (LPI) theory. A direct optimizing Accepted: May 28, 2018 Published: May 31, 2018 algorithm based on Nelder-Mead simplex method is proposed for solving the problem. Finally, a numerical example is given to demonstrate the efficiency Copyright © 2018 by authors and of the algorithm. Scientific Research Publishing Inc. This work is licensed under the Creative Keywords Commons Attribution International License (CC BY 4.0). Stochastic Quadratic Programming, LPI, Nelder-Mead http://creativecommons.org/licenses/by/4.0/ Open Access 1. Introduction Stochastic programming is an important method to solve decision problems in random environment. -
Arxiv:1205.6548V4 [Math.OC] 9 Dec 2013
JOURNAL OF INDUSTRIAL AND doi: MANAGEMENT OPTIMIZATION Volume 00, Number 0, Xxxx XXXX pp. 000{000 STATE TRANSITION ALGORITHM Xiaojun Zhou, Chunhua Yang and Weihua Gui School of Information Science and Engineering Central South University, Changsha, 410083, China Abstract. In terms of the concepts of state and state transition, a new heuris- tic random search algorithm named state transition algorithm is proposed. For continuous function optimization problems, four special transformation opera- tors called rotation, translation, expansion and axesion are designed. Adjusting measures of the transformations are mainly studied to keep the balance of ex- ploration and exploitation. Convergence analysis is also discussed about the algorithm based on random search theory. In the meanwhile, to strengthen the search ability in high dimensional space, communication strategy is in- troduced into the basic algorithm and intermittent exchange is presented to prevent premature convergence. Finally, experiments are carried out for the algorithms. With 10 common benchmark unconstrained continuous functions used to test the performance, the results show that state transition algorithms are promising algorithms due to their good global search capability and con- vergence property when compared with some popular algorithms. 1. Introduction. The concept of state means to a situation which a material sys- tem maintains, and it is characterized by a group of physical qualities. The process of a system turning from a state to another is called state transition, which can be described by a state transition matrix. The idea of state transition was created by a Russian mathematician named Markov when he expected to represent a specific sto- chastic process (known as Markov process)[22].