An Algorithm for Nonlinear Optimization Using Linear Programming and Equality Constrained Subproblems
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12. Coordinate Descent Methods
EE 546, Univ of Washington, Spring 2014 12. Coordinate descent methods theoretical justifications • randomized coordinate descent method • minimizing composite objectives • accelerated coordinate descent method • Coordinate descent methods 12–1 Notations consider smooth unconstrained minimization problem: minimize f(x) x RN ∈ n coordinate blocks: x =(x ,...,x ) with x RNi and N = N • 1 n i ∈ i=1 i more generally, partition with a permutation matrix: U =[PU1 Un] • ··· n T xi = Ui x, x = Uixi i=1 X blocks of gradient: • f(x) = U T f(x) ∇i i ∇ coordinate update: • x+ = x tU f(x) − i∇i Coordinate descent methods 12–2 (Block) coordinate descent choose x(0) Rn, and iterate for k =0, 1, 2,... ∈ 1. choose coordinate i(k) 2. update x(k+1) = x(k) t U f(x(k)) − k ik∇ik among the first schemes for solving smooth unconstrained problems • cyclic or round-Robin: difficult to analyze convergence • mostly local convergence results for particular classes of problems • does it really work (better than full gradient method)? • Coordinate descent methods 12–3 Steepest coordinate descent choose x(0) Rn, and iterate for k =0, 1, 2,... ∈ (k) 1. choose i(k) = argmax if(x ) 2 i 1,...,n k∇ k ∈{ } 2. update x(k+1) = x(k) t U f(x(k)) − k i(k)∇i(k) assumptions f(x) is block-wise Lipschitz continuous • ∇ f(x + U v) f(x) L v , i =1,...,n k∇i i −∇i k2 ≤ ik k2 f has bounded sub-level set, in particular, define • ⋆ R(x) = max max y x 2 : f(y) f(x) y x⋆ X⋆ k − k ≤ ∈ Coordinate descent methods 12–4 Analysis for constant step size quadratic upper bound due to block coordinate-wise -
Process Optimization
Process Optimization Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike Director, Minerals Processing Research Institute Horton Professor of Chemical Engineering Louisiana State University Department of Chemical Engineering, Lamar University, April, 10, 2007 Process Optimization • Typical Industrial Problems • Mathematical Programming Software • Mathematical Basis for Optimization • Lagrange Multipliers and the Simplex Algorithm • Generalized Reduced Gradient Algorithm • On-Line Optimization • Mixed Integer Programming and the Branch and Bound Algorithm • Chemical Production Complex Optimization New Results • Using one computer language to write and run a program in another language • Cumulative probability distribution instead of an optimal point using Monte Carlo simulation for a multi-criteria, mixed integer nonlinear programming problem • Global optimization Design vs. Operations • Optimal Design −Uses flowsheet simulators and SQP – Heuristics for a design, a superstructure, an optimal design • Optimal Operations – On-line optimization – Plant optimal scheduling – Corporate supply chain optimization Plant Problem Size Contact Alkylation Ethylene 3,200 TPD 15,000 BPD 200 million lb/yr Units 14 76 ~200 Streams 35 110 ~4,000 Constraints Equality 761 1,579 ~400,000 Inequality 28 50 ~10,000 Variables Measured 43 125 ~300 Unmeasured 732 1,509 ~10,000 Parameters 11 64 ~100 Optimization Programming Languages • GAMS - General Algebraic Modeling System • LINDO - Widely used in business applications -
The Gradient Sampling Methodology
The Gradient Sampling Methodology James V. Burke∗ Frank E. Curtisy Adrian S. Lewisz Michael L. Overtonx January 14, 2019 1 Introduction in particular, this leads to calling the negative gra- dient, namely, −∇f(x), the direction of steepest de- The principal methodology for minimizing a smooth scent for f at x. However, when f is not differentiable function is the steepest descent (gradient) method. near x, one finds that following the negative gradient One way to extend this methodology to the minimiza- direction might offer only a small amount of decrease tion of a nonsmooth function involves approximating in f; indeed, obtaining decrease from x along −∇f(x) subdifferentials through the random sampling of gra- may be possible only with a very small stepsize. The dients. This approach, known as gradient sampling GS methodology is based on the idea of stabilizing (GS), gained a solid theoretical foundation about a this definition of steepest descent by instead finding decade ago [BLO05, Kiw07], and has developed into a a direction to approximately solve comprehensive methodology for handling nonsmooth, potentially nonconvex functions in the context of op- min max gT d; (2) kdk2≤1 g2@¯ f(x) timization algorithms. In this article, we summarize the foundations of the GS methodology, provide an ¯ where @f(x) is the -subdifferential of f at x [Gol77]. overview of the enhancements and extensions to it To understand the context of this idea, recall that that have developed over the past decade, and high- the (Clarke) subdifferential of a locally Lipschitz f light some interesting open questions related to GS. -
A New Algorithm of Nonlinear Conjugate Gradient Method with Strong Convergence*
Volume 27, N. 1, pp. 93–106, 2008 Copyright © 2008 SBMAC ISSN 0101-8205 www.scielo.br/cam A new algorithm of nonlinear conjugate gradient method with strong convergence* ZHEN-JUN SHI1,2 and JINHUA GUO2 1College of Operations Research and Management, Qufu Normal University Rizhao, Sahndong 276826, P.R. China 2Department of Computer and Information Science, University of Michigan Dearborn, Michigan 48128-1491, USA E-mails: [email protected]; [email protected] / [email protected] Abstract. The nonlinear conjugate gradient method is a very useful technique for solving large scale minimization problems and has wide applications in many fields. In this paper, we present a new algorithm of nonlinear conjugate gradient method with strong convergence for unconstrained minimization problems. The new algorithm can generate an adequate trust region radius automatically at each iteration and has global convergence and linear convergence rate under some mild conditions. Numerical results show that the new algorithm is efficient in practical computation and superior to other similar methods in many situations. Mathematical subject classification: 90C30, 65K05, 49M37. Key words: unconstrained optimization, nonlinear conjugate gradient method, global conver- gence, linear convergence rate. 1 Introduction Consider an unconstrained minimization problem min f (x), x ∈ Rn, (1) where Rn is an n-dimensional Euclidean space and f : Rn −→ R is a continu- ously differentiable function. #724/07. Received: 10/V/07. Accepted: 24/IX/07. *The work was supported in part by NSF CNS-0521142, USA. 94 A NEW ALGORITHM OF NONLINEAR CONJUGATE GRADIENT METHOD When n is very large (for example, n > 106) the related problem is called large scale minimization problem. -
Using a New Nonlinear Gradient Method for Solving Large Scale Convex Optimization Problems with an Application on Arabic Medical Text
Using a New Nonlinear Gradient Method for Solving Large Scale Convex Optimization Problems with an Application on Arabic Medical Text Jaafar Hammouda*, Ali Eisab, Natalia Dobrenkoa, Natalia Gusarovaa aITMO University, St. Petersburg, Russia *[email protected] bAleppo University, Aleppo, Syria Abstract Gradient methods have applications in multiple fields, including signal processing, image processing, and dynamic systems. In this paper, we present a nonlinear gradient method for solving convex supra-quadratic functions by developing the search direction, that done by hybridizing between the two conjugate coefficients HRM [2] and NHS [1]. The numerical results proved the effectiveness of the presented method by applying it to solve standard problems and reaching the exact solution if the objective function is quadratic convex. Also presented in this article, an application to the problem of named entities in the Arabic medical language, as it proved the stability of the proposed method and its efficiency in terms of execution time. Keywords: Convex optimization; Gradient methods; Named entity recognition; Arabic; e-Health. 1. Introduction Several nonlinear conjugate gradient methods have been presented for solving high-dimensional unconstrained optimization problems which are given as [3]: min f(x) ; x∈Rn (1) To solve problem (1), we start with the following iterative relationship: 푥(푘+1) = 푥푘 + 훼푘 푑푘, 푘 = 0,1,2, (2) where α푘 > 0 is a step size, that is calculated by strong Wolfe-Powell’s conditions [4] 푇 푓(푥푘 + 훼푘푑푘) ≤ 푓(푥푘) + 훿훼푘푔푘 푑푘, (3) 푇 푇 |푔(푥푘 + 훼푘푑푘) 푑푘| ≤ 휎|푔푘 푑푘| where 0 < 훿 < 휎 < 1 푑푘 is the search direction that is computed as follow [3] −푔푘 , if 푘 = 0 (4) 푑푘+1 = { −푔푘+1 + 훽푘푑푘 , if 푘 ≥ 1 Where 푔푘 = 푔(푥푘) = ∇푓(푥푘) represent the gradient vector for 푓(푥) at the point 푥푘 , 훽푘 ∈ ℝ is known as CG coefficient that characterizes different CG methods. -
Constrained Optimization 5
Constrained Optimization 5 Most problems in structural optimization must be formulated as constrained min- imization problems. In a typical structural design problem the objective function is a fairly simple function of the design variables (e.g., weight), but the design has to satisfy a host of stress, displacement, buckling, and frequency constraints. These constraints are usually complex functions of the design variables available only from an analysis of a finite element model of the structure. This chapter offers a review of methods that are commonly used to solve such constrained problems. The methods described in this chapter are for use when the computational cost of evaluating the objective function and constraints is small or moderate. In these meth- ods the objective function or constraints these are calculated exactly (e.g., by a finite element program) whenever they are required by the optimization algorithm. This approach can require hundreds of evaluations of objective function and constraints, and is not practical for problems where a single evaluation is computationally ex- pensive. For these more expensive problems we go through an intermediate stage of constructing approximations for the objective function and constraints, or at least for the more expensive functions. The optimization is then performed on the approx- imate problem. This approximation process is described in the next chapter. The basic problem that we consider in this chapter is the minimization of a function subject to equality and inequality constraints minimize f(x) such that hi(x) = 0, i = 1, . , ne , (5.1) gj(x) ≥ 0, j = 1, . , ng . The constraints divide the design space into two domains, the feasible domain where the constraints are satisfied, and the infeasible domain where at least one of the constraints is violated. -
Tabu Search for Constraint Satisfaction
Tabu Search for Constraint Solving and Its Applications Jin-Kao Hao LERIA University of Angers 2 Boulevard Lavoisier 49045 Angers Cedex 01 - France 1. Introduction The Constraint Satisfaction Problem (CSP) is a very general problem able to conveniently formulate a wide range of important applications in combinatorial optimization. These include well-known classical problems such as graph k-coloring and satisfiability, as well as many practical applications related to resource assignments, planning and timetabling. The feasibility problem of Integer Programming can also be easily represented by the CSP formalism. As elaborated below, we provide case studies for specific CSP applications, including the important realm of frequency assignment, and establish that tabu search is the best performing approach for these applications. The Constraint Satisfaction Problem is defined by a triplet (X, D, C) where: – X = {x1,· · ·,xn} is a finite set of n variables. – D = {Dx1 ,· · ·,Dxn} is a set of associated domains. Each domain Dxi specifies the finite set of possible values of the variable xi. – C = {C1,· · ·,Cp} is a finite set of p constraints. Each constraint is defined on a set of variables and specifies which combinations of values are compatible for these variables. Given such a triplet, the basic problem consists in finding a complete assignment of the values to the variables such that that all the constraints are satisfied. Such an assignment is called a solution to the given CSP instance. Since the set of all assignments is given by the Cartesian product Dx1 ×· · ·×Dxn, solving a CSP means to determine a particular assignment among a potentially huge search space. -
Conditional Gradient Method for Multiobjective Optimization
Conditional gradient method for multiobjective optimization P. B. Assun¸c~ao ∗ O. P. Ferreira ∗ L. F. Prudente ∗ April 7, 2020 Abstract: We analyze the conditional gradient method, also known as Frank-Wolfe method, for constrained multiobjective optimization. The constraint set is assumed to be convex and com- pact, and the objectives functions are assumed to be continuously differentiable. The method is considered with different strategies for obtaining the step sizes. Asymptotic convergence proper- ties and iteration-complexity bounds with and without convexity assumptions on the objective functions are stablished. Numerical experiments are provided to illustrate the effectiveness of the method and certify the obtained theoretical results. Keywords: Conditional gradient method; multiobjective optimization; Pareto optimality; con- strained optimization problem 1 Introduction In the constrained multiobjective optimization, we seek to simultaneously minimize several objec- tive functions on a set C. One strategy for solving multiobjective problems that has become very popular consists in the extension of methods for scalar-valued to multiobjective-valued optimiza- tion, instead of using scalarization approaches [23]. As far as we know, this strategy originated from the work [14] in which the authors proposed the steepest descent methods for multiobjec- tive optimization. Since of then, new properties related to this method have been discovered and several variants of it have been considered, see for example [4,5,16,19,20,26,27]. In recent years, there has been a significant increase in the number of papers addressing concepts, techniques, and methods for multiobjective optimization, see for example [6, 9, 13, 15, 25, 41, 42, 45, 46, 48, 53, 54]. -
Successive Linear Programming to The
THE UNIVERSITY OF MANITOBA AN APPLICATION OF SUCCESSIVE LINEAR PROGRAMMING TO THE OPTIMIZATION OF AN INTERCONNECTED HYDRO UTILMY OPERATION by @ KARoLI REZNICEK A Thesis Submitted to the Faculfy of Graduate Studies in Partial Fulfillment of the Requirements for the Degree of Master of Science DEPARTMENT OF CTVIL ENGINEERING Winnipeg, Manitoba DECEMBER 1988 Per¡niesion trae been granted L'autorleatlon a êtê accordêe to the Natlonal Llbrary of å la Blbllothèque natlonale Canada to nicrofiln thlg du Canada de nlcrofllner thesls and to lend or eell cette thèse et de prêter ou copies of the filn. de vendre dee exenplairee du f llrn. The author (copyright owner) Lrauteur (tltulaire du drolt has reaerved other d'auteur) ae rêeerve lee pubtication ríghts, and autres droits de publlcatlon; neither the thesie nor nl ta thèee nl de longe extensive extracts from it extraite de celle-cl ne Day be printed or otherwiee doivent être irnprlnês ou reproduced wlthout his/her autrement reprodults BanB aon written perurieeion. autorieatlon écrite. rsBN 0-3L5-5i.576-7 AN APPLICATION OF SUCCESSIVE LINEAR PROGRAMMING TO THE OPTIMIZATION OF AN INTERCONNECTED HYDRO UTILITY OPERATION BY KAROLJ REZNICEK A thesis submirted to the Faculty of craduate studies of the university of Manitoba in partiar furfilrment of the requirenrents of the degree of I'fASTER OF SCIENCE o 1989 Permission has beerr grarìred ro the LIBRARy oF THE UNIVER- S¡TY OF MANITOBA to lend or sell copies of this thesis. ro the NATIONAL LIBRARy oF CANADA to microfìrnr this thesis and to lend or seil copies of the firm, and uNIVERsrry llf ICROFILMS to publish an absrract of this thesis. -
Metaheuristic Techniques in Enhancing the Efficiency and Performance of Thermo-Electric Cooling Devices
Review Metaheuristic Techniques in Enhancing the Efficiency and Performance of Thermo-Electric Cooling Devices Pandian Vasant 1,2,*, Utku Kose 3 and Junzo Watada 4 1 Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 2 Department of Fundamental and Applied Sciences, Universiti Teknologi Petronas, Seri Iskandar, Perak 32610, Malaysia 3 Computer Sciences Application& Research Center, Usak University, Usak 64000, Turkey; [email protected] 4 Department of Computer & Information System, Universiti Teknologi Petronas, Seri Iskandar, Perak 32610, Malaysia; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +60-16-489-2932 Received: 11 August 2017; Accepted: 2 September 2017; Published: 25 October 2017 Abstract: The objective of this paper is to focus on the technical issues of single-stage thermo-electric coolers (TECs) and two-stage TECs and then apply new methods in optimizing the dimensions of TECs. In detail, some metaheuristics—simulated annealing (SA) and differential evolution (DE)—are applied to search the optimal design parameters of both types of TEC, which yielded cooling rates and coefficients of performance (COPs) individually and simultaneously. The optimization findings obtained by using SA and DE are validated by applying them in some defined test cases taking into consideration non-linear inequality and non-linear equality constraint conditions. The performance of SA and DE are verified after comparing the findings with the ones obtained applying the genetic algorithm (GA) and hybridization technique (HSAGA and HSADE). Mathematical modelling and parameter setting of TEC is combined with SA and DE to find better optimal findings. -
Constrained Quadratic Optimization: Theory and Application for Wireless Communication Systems
Constrained Quadratic Optimization: Theory and Application for Wireless Communication Systems by Erik Hons A thesis presented to the University of Waterloo in ful¯lment of the thesis requirement for the degree of Master of Applied Science in Electrical and Computer Engineering Waterloo, Ontario, Canada, 2001 c Erik Hons 2001 ° I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by pho- tocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or pho- tocopying this thesis. Please sign below, and give address and date. iii Abstract The suitability of general constrained quadratic optimization as a modeling and solution tool for wireless communication is discussed. It is found to be appropriate due to the many quadratic entities present in wireless communication models. It is posited that decisions which a®ect the minimum mean squared error at the receiver may achieve improved performance if those decisions are energy constrained. That theory is tested by applying a speci¯c type of constrained quadratic optimization problem to the forward link of a cellular DS-CDMA system. Through simulation it is shown that when channel coding is used, energy constrained methods typi- cally outperform unconstrained methods. Furthermore, a new energy-constrained method is presented which signi¯cantly outperforms a similar published method in several di®erent environments. -
Branch and Bound Algorithm for Dependency Parsing with Non-Local Features
Branch and Bound Algorithm for Dependency Parsing with Non-local Features Xian Qian and Yang Liu Computer Science Department The University of Texas at Dallas qx,yangl @hlt.utdallas.edu { } Abstract Charniak and Johnson, 2005; Hall, 2007). It typi- cally has two steps: the low level classifier gener- Graph based dependency parsing is inefficient ates the top k hypotheses using local features, then when handling non-local features due to high the high level classifier reranks these candidates us- computational complexity of inference. In ing global features. Since the reranking quality is this paper, we proposed an exact and effi- bounded by the oracle performance of candidates, cient decoding algorithm based on the Branch some work has combined candidate generation and and Bound (B&B) framework where non- local features are bounded by a linear combi- reranking steps using cube pruning (Huang, 2008; nation of local features. Dynamic program- Zhang and McDonald, 2012) to achieve higher or- ming is used to search the upper bound. Ex- acle performance. They parse a sentence in bottom periments are conducted on English PTB and up order and keep the top k derivations for each s- Chinese CTB datasets. We achieved competi- pan using k best parsing (Huang and Chiang, 2005). tive Unlabeled Attachment Score (UAS) when After merging the two spans, non-local features are 93.17% no additional resources are available: used to rerank top k combinations. This approach for English and 87.25% for Chinese. Parsing speed is 177 words per second for English and is very efficient and flexible to handle various non- 97 words per second for Chinese.