DOCSLIB.ORG

Problem Set 7 Convex Optimization (Winter 2018)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Problem Set 7 Convex Optimization (Winter 2018) Problem set 7 Convex Optimization (Winter 2018) Due Friday, February 23rd at 5pm • Upload both a pdf writeup of your work and a zip file containing any supplemental files to Canvas by the due date. Submissions without a non-zipped pdf writeup won’t be graded. • Late homeworks will be penalized 25% per day and will not be accepted after the next Monday at 5pm unless you have made arrangements with us well before the due date. We strongly encourage you to typeset your solutions. • You may collaborate with other students to complete this assignment, but all submitted work should be yours. When you go to write down your solutions, please do so on your own. Description In this assignment, you will experiment with the log-barrier method. Partiallly completed Python code is included with the assignment. Areas that you need to fill in are marked with “TODO” comments. You should turn in an archive containing all of your source code, and a separate pdf document containing all plots and answers to underlined questions. 1 Log-Barrier Function Most of the functions in the file “algorithms.py” are the same as in previous programming as- signments (but filled-in this time). The objective log barrier function in “algorithms.py” takes as parameters an objective function f0, a constraint function φ (which has the same calling convention as the objective function), a vector x, and a scaling parameter t, and returns the value, gradient and Hessian of the log-barrier objective: f(x; t) = tf0(x) + φ(x) 1 1.1 Implementation The function quadratic log barrier in “functions.py” is supposed to compute the value, gradient, and Hessian of: 1 X f(x; t) = t x>Qx + v>x − log (a x + b ) 2 i;: i i This is the log-barrier objective function f(x; t) = tf0(x) + φ(x) for a quadratic objective f0, log-barrier function φ, and linear constraints Ax + b ≥ 0. Fill in this function. 1.2 Experiments The function main quadratic in “functions.py” minimizes a quadratic objective subject to three linear constraints, with t = 3, (note: this is an extremely small value of t), and plots the iterates of gradient descent and Newtons method. What do you observe? Remember that you can zoom in on the plots! The function main quadratic2 in “functions.py” plots the number of iterations required by gradient 1 descent and Newton’s method for t between 8 and 2 (once more, these are extremely small values of t, but the performance of gradient descent should give you a clue as to why we didn’t use larger t). What do you observe? 2 Log Barrier Method 2.1 Implementation–Scalar Constraints One of the most appealing properties of the log-barrier method is that, given oracle access to each constraint function fi and its first and second derivatives, for constraints of the form fi(x) ≥ 0, we may calculate the value, gradient, and Hessian of the log-barrier function gi(x) = − log fi(x). The function objective scalar constraints in “algorithms.py” implements this. It takes as param- P P eters a list of constraint functions fi and a point x, and should return i gi(x), i rgi(x) and P 2 i r gi(x). Each constraint function takes x as a parameter, and returns fi(x), rfi(x), and 2 r fi(x). Fill in the function objective scalar constraints. 2 2.2 Implementation–Log Barrier The function quadratic in “functions.py” implements a constraint function which returns the value, gradient, and Hessian of: 1 f (x) = x>Qx + v>x + c i 2 which will be used to implement constraints fi(x) ≥ 0 (in this case, we will need Q 0). The function log barrier in “algorithms.py” contains a mostly-complete implementation of the log-barrier method (Algorithm 11.1 in Boyd and Vandenberghe). This function takes (among other things) a list of constraint functions as a parameter, and constructs the log-barrier objective P f(x; t) = tf0(x) − i log fi(x) using objective log barrier and your filled-in version of objec- tive scalar constraints. Fill in the function log barrier. 2.3 Experiments The function main linear in “functions.py” optimizes a linear program: min v>x x > s.t. ai x ≥ bi 8i The parameter µ controls how quickly the log-barrier scaling parameter t changes between inner iterations of Newton. The generated plot shows the total number of Newton iterations performed for a run of the log-barrier method for various values of µ (with t0 = 1). Before running the code, describe the dependence on µ which you expect to see here. Does the plot meet your expectations? What solution is found for this LP? 3.
Load more

Recommended publications
  • Advances in Interior Point Methods and Column Generation
    Advances in Interior Point Methods and Column Generation Pablo Gonz´alezBrevis Doctor of Philosophy University of Edinburgh 2013 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Pablo Gonz´alezBrevis) ii To my endless inspiration and angels on earth: Paulina, Crist´obal and Esteban iii Abstract In this thesis we study how to efficiently combine the column generation technique (CG) and interior point methods (IPMs) for solving the relaxation of a selection of integer programming problems. In order to obtain an efficient method a change in the column generation technique and a new reoptimization strategy for a primal-dual interior point method are proposed. It is well-known that the standard column generation technique suffers from un- stable behaviour due to the use of optimal dual solutions that are extreme points of the restricted master problem (RMP). This unstable behaviour slows down column generation so variations of the standard technique which rely on interior points of the dual feasible set of the RMP have been proposed in the literature. Among these tech- niques, there is the primal-dual column generation method (PDCGM) which relies on sub-optimal and well-centred dual solutions. This technique dynamically adjusts the column generation tolerance as the method approaches optimality. Also, it relies on the notion of the symmetric neighbourhood of the central path so sub-optimal and well-centred solutions are obtained. We provide a thorough theoretical analysis that guarantees the convergence of the primal-dual approach even though sub-optimal solu- tions are used in the course of the algorithm.
    [Show full text]
  • Chapter 8 Constrained Optimization 2: Sequential Quadratic Programming, Interior Point and Generalized Reduced Gradient Methods
    Chapter 8: Constrained Optimization 2 CHAPTER 8 CONSTRAINED OPTIMIZATION 2: SEQUENTIAL QUADRATIC PROGRAMMING, INTERIOR POINT AND GENERALIZED REDUCED GRADIENT METHODS 8.1 Introduction In the previous chapter we examined the necessary and sufficient conditions for a constrained optimum. We did not, however, discuss any algorithms for constrained optimization. That is the purpose of this chapter. The three algorithms we will study are three of the most common. Sequential Quadratic Programming (SQP) is a very popular algorithm because of its fast convergence properties. It is available in MATLAB and is widely used. The Interior Point (IP) algorithm has grown in popularity the past 15 years and recently became the default algorithm in MATLAB. It is particularly useful for solving large-scale problems. The Generalized Reduced Gradient method (GRG) has been shown to be effective on highly nonlinear engineering problems and is the algorithm used in Excel. SQP and IP share a common background. Both of these algorithms apply the Newton- Raphson (NR) technique for solving nonlinear equations to the KKT equations for a modified version of the problem. Thus we will begin with a review of the NR method. 8.2 The Newton-Raphson Method for Solving Nonlinear Equations Before we get to the algorithms, there is some background we need to cover first. This includes reviewing the Newton-Raphson (NR) method for solving sets of nonlinear equations. 8.2.1 One equation with One Unknown The NR method is used to find the solution to sets of nonlinear equations. For example, suppose we wish to find the solution to the equation: xe+=2 x We cannot solve for x directly.
    [Show full text]
  • Cancer Treatment Optimization
    CANCER TREATMENT OPTIMIZATION A Thesis Presented to The Academic Faculty by Kyungduck Cha In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology April 2008 CANCER TREATMENT OPTIMIZATION Approved by: Professor Eva K. Lee, Advisor Professor Renato D.C. Monteiro H. Milton Stewart School of Industrial H. Milton Stewart School of Industrial and Systems Engineering and Systems Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Earl R. Barnes Professor Nolan E. Hertel H. Milton Stewart School of Industrial G. W. Woodruff School of Mechanical and Systems Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Ellis L. Johnson Date Approved: March 28, 2008 H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology To my parents, wife, and son. iii ACKNOWLEDGEMENTS First of all, I would like to express my sincere appreciation to my advisor, Dr. Eva Lee for her invaluable guidance, financial support, and understanding throughout my Ph.D program of study. Her inspiration provided me many ideas in tackling with my thesis problems and her enthusiasm encouraged me to produce better works. I would also like to thank Dr. Earl Barnes, Dr. Ellis Johnson, Dr. Renato D.C. Monteiro, and Dr. Nolan Hertel for their willingness to serve on my dissertation committee and for their valuable feedbacks. I also thank my friends, faculty, and staff, at Georgia Tech, who have contributed to this thesis in various ways. Specially, thank you to the members at the Center for Operations Research in Medicine and HealthCare for their friendship.
    [Show full text]
  • The Central Curve in Linear Programming
    THE CENTRAL CURVE IN LINEAR PROGRAMMING JESUS´ A. DE LOERA, BERND STURMFELS, AND CYNTHIA VINZANT Abstract. The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instance-specific bound on the total curvature of the central path, a quantity relevant for interior point methods. The global geometry of central curves is studied in detail. 1. Introduction We consider the standard linear programming problem in its primal and dual formulation: (1) Maximize cT x subject to Ax = b and x ≥ 0; (2) Minimize bT y subject to AT y − s = c and s ≥ 0: Here A is a fixed matrix of rank d having n columns. The vectors c 2 Rn and b 2 image(A) may vary. In most of our results we assume that b and c are generic. This implies that both the primal optimal solution and the dual optimal solution are unique. We also assume that the problem (1) is bounded and both problems (1) and (2) are strictly feasible. Before describing our contributions, we review some basics from the theory of linear pro- gramming [26,33].
    [Show full text]
  • IMPROVED PARTICLE SWARM OPTIMIZATION for NON-LINEAR PROGRAMMING PROBLEM with BARRIER METHOD Raju Prajapati1 1University of Engineering & Management, Kolkata, W.B
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Gyandhara International Academic Publication (GIAP): Journals International Journal of Students’ Research in Technology & Management eISSN: 2321-2543, Vol 5, No 4, 2017, pp 72-80 https://doi.org/10.18510/ijsrtm.2017.5410 IMPROVED PARTICLE SWARM OPTIMIZATION FOR NON-LINEAR PROGRAMMING PROBLEM WITH BARRIER METHOD Raju Prajapati1 1University Of Engineering & Management, Kolkata, W.B. India Om Prakash Dubey2, Randhir Kumar3 2,3Bengal College Of Engineering & Technology, Durgapur, W.B. India email: [email protected] Article History: Submitted on 10th September, Revised on 07th November, Published on 30th November 2017 Abstract: The Non-Linear Programming Problems (NLPP) are computationally hard to solve as compared to the Linear Programming Problems (LPP). To solve NLPP, the available methods are Lagrangian Multipliers, Sub gradient method, Karush-Kuhn-Tucker conditions, Penalty and Barrier method etc. In this paper, we are applying Barrier method to convert the NLPP with equality constraint to an NLPP without constraint. We use the improved version of famous Particle Swarm Optimization (PSO) method to obtain the solution of NLPP without constraint. SCILAB programming language is used to evaluate the solution on sample problems. The results of sample problems are compared on Improved PSO and general PSO. Keywords. Non-Linear Programming Problem, Barrier Method, Particle Swarm Optimization, Equality and Inequality Constraints. INTRODUCTION A large number of algorithms are available to solve optimization problems, inspired by evolutionary process in nature. e.g. Genetic Algorithm, Particle Swarm Optimization, Ant Colony Optimization, Bee Colony Optimization, Memetic Algorithm, etc.
    [Show full text]
  • Barrier Method
    Barrier Method Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: Newton's method Consider the problem min f(x) x n for f convex, twice differentiable, with dom(f) = R . Newton's (0) n method: choose initial x R , repeat 2 (k) (k−1) 2 (k−1) −1 (k−1) x = x tk f(x ) f(x ); k = 1; 2; 3;::: − r r Step sizes tk chosen by backtracking line search If f Lipschitz, f strongly convex, 2f Lipschitz, then Newton's r r method has a local convergence rate O(log log(1/)) Downsides: Requires solving systems in Hessian quasi-Newton • Can only handle equality constraints this lecture • 2 Hierarchy of second-order methods Assuming all problems are convex, you can think of the following hierarchy that we've worked through: Quadratic problems are the easiest: closed-form solution • Equality-constrained quadratic problems are still easy: we use • KKT conditions to derive closed-form solution Equality-constrained smooth problems are next: use Newton's • method to reduce this to a sequence of equality-constrained quadratic problems Inequality- and equality-constrained smooth problems are • what we cover now: use interior-point methods to reduce this to a sequence of equality-constrained smooth problems 3 Log barrier function Consider the convex optimization problem min f(x) x subject to hi(x) 0; i = 1; : : : m ≤ Ax = b We will assume that f, h1; : : : hm are convex, twice differentiable, n each with domain R . The function m φ(x) = log( hi(x)) − − i=1 X is called the log barrier for the above problem.
    [Show full text]
  • Electrical Networks, Hyperplane Arrangements and Matroids By
    Electrical Networks, Hyperplane Arrangements and Matroids by Robert P. Lutz A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2019 Doctoral Committee: Professor Jeffrey Lagarias, Chair Professor Alexander Barvinok Associate Professor Chris Peikert Professor Karen Smith Professor David Speyer Robert P. Lutz a.k.a. Bob Lutz [email protected] orcid.org/0000-0002-5407-7041 c Robert P. Lutz 2019 Dedication This dissertation is dedicated to my parents. ii Acknowledgments I thank Trevor Hyde, Thomas Zaslavsky, and anonymous referees for helpful com- ments on individual chapters of this thesis. I thank Alexander Barvinok for his careful reading of an early version of the thesis, and many useful comments. Finally I thank my advisor, Jeffrey Lagarias, for his extensive comments on the thesis, for his keen advice and moral support, and for many helpful conversations throughout my experience as a graduate student. This work was partially supported by NSF grants DMS-1401224 and DMS-1701576. iii Table of Contents Dedication ii Acknowledgments iii List of Figures vii Abstract ix Chapter 1: Introduction 1 Chapter 2: Background 4 2.1 Electrical networks . 4 2.2 Hyperplane arrangements . 10 2.2.1 Supersolvable arrangements . 13 2.2.2 Graphic arrangements . 14 2.3 Matroids . 15 2.3.1 Graphic matroids . 17 2.3.2 Complete principal truncations . 17 Chapter 3: Electrical networks and hyperplane arrangements 19 3.1 Main definition and examples . 19 3.2 Main results . 24 3.3 Combinatorics of Dirichlet arrangements . 26 3.3.1 Intersection poset and connected partitions .
    [Show full text]
  • Price-Based Distributed Optimization in Large-Scale Networked Systems
    Price-Based Distributed Optimization in Large-Scale Networked Systems Dissertation document submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements of degree of Doctor of Philosophy in the School of Dynamic Systems College of Engineering and Applied Sciences by Baisravan HomChaudhuri Summer 2013 MS, University of Cincinnati, 2010 BE, Jadavpur University, 2007 Committee Chair: Dr. Manish Kumar c 2013 Baisravan HomChaudhuri, All Rights Reserved Abstract This work is intended towards the development of distributed optimization methods for large-scale complex networked systems. The advancement in tech- nological fields such as networking, communication and computing has facili- tated the development of networks which are massively large-scale in nature. The examples of such systems include power grid, communication network, Internet, large manufacturing plants, and cloud computing clusters. One of the important challenges in these networked systems is the evaluation of the optimal point of operation of the system. Most of the centralized optimiza- tion algorithms suffer from the curse of dimensionality that raises the issue of scalability of algorithms for large-scale systems. The problem is essentially challenging not only due to high-dimensionality of the problem, but also due to distributed nature of resources, lack of global information and uncertain and dynamic nature of operation of most of these systems. The inadequacies of the traditional centralized optimization
    [Show full text]
  • 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.
    [Show full text]
  • Interior Point Method
    LECTURE 6: INTERIOR POINT METHOD 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm Motivation • Simplex method works well in general, but suffers from exponential-time computational complexity. • Klee-Minty example shows simplex method may have to visit every vertex to reach the optimal one. • Total complexity of an iterative algorithm = # of iterations x # of operations in each iteration • Simplex method - Simple operations: Only check adjacent extreme points - May take many iterations: Klee-Minty example Question: any fix? • Complexity of the simplex method Worst case performance of the simplex method Klee-Minty Example: •Victor Klee, George J. Minty, “How good is the simplex algorithm?’’ in (O. Shisha edited) Inequalities, Vol. III (1972), pp. 159-175. Klee-Minty Example Klee-Minty Example Karmarkar’s (interior point) approach • Basic idea: approach optimal solutions from the interior of the feasible domain • Take more complicated operations in each iteration to find a better moving direction • Require much fewer iterations General scheme of an interior point method • An iterative method that moves in the interior of the feasible domain Interior movement (iteration) • Given a current interior feasible solution , we have > 0 An interior movement has a general format Key knowledge • 1. Who is in the interior? - Initial solution • 2. How do we know a current solution is optimal? - Optimality condition • 3. How to move to a new solution? - Which direction to move? (good feasible direction) - How far to go? (step-length) Q1 - Who is in the interior? • Standard for LP • Who is at the vertex? • Who is on the edge? • Who is on the boundary? • Who is in the interior? What have learned before Who is in the interior? • Two criteria for a point x to be an interior feasible solution: 1.
    [Show full text]
  • Efficiency Improvements in Meta-Heuristic Algorithms to Solve
    Electrical Power and Energy Systems 82 (2016) 288–302 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes Efficiency improvements in meta-heuristic algorithms to solve the optimal power flow problem ⇑ S. Surender Reddy a, , P.R. Bijwe b a Department of Railroad and Electrical Engineering, Woosong University, South Korea b Department of Electrical Engineering, Indian Institute of Technology Delhi, India article info abstract Article history: This paper presents three efficient approaches for solving the Optimal Power Flow (OPF) problem using Received 24 September 2015 the meta-heuristic algorithms. Mathematically, OPF is formulated as non-linear equality and inequality Received in revised form 15 March 2016 constrained optimization problem. The main drawback of meta-heuristic algorithm based OPF is the Accepted 16 March 2016 excessive execution time required due to the large number of load flows/power flows needed in the solu- Available online 29 March 2016 tion process. The proposed efficient approaches uses the concept of incremental power flow model based on sensitivities, and lower, upper bounds of objective function values. By using these approaches, the Keywords: number of load flows/power flows to be performed are substantially, resulting in the solution speed Differential evolution up. The original advantages of meta-heuristic algorithms, such as ability to handle complex non- Enhanced genetic algorithm Meta-heuristic algorithms linearities, discontinuities in the objective function, discrete variables handling, and multi-objective opti- Power system optimization mization, are still available in the proposed efficient approaches. The proposed OPF formulation includes Optimal power flow the active and reactive power generation limits, Valve Point Loading (VPL) effects and Prohibited Linear programming Operating Zones (POZs) of generating units.
    [Show full text]
  • 9 Self-Concordant Barriers
    Efficient Methods in Optimization Fall 2020 9 Self-concordant barriers n o A barrier function on a set X ⊂ R is a function F : X ! R such that limx2@X F (x) = +1. For convenience we may define F (x) = +1 for x 62 int K. Adding a barrier to the objective function of a minimization problem over X prevents the iterates of an iterative algorithm to step out of the interior of X. In contrast to barriers, a penalty function on X is a function F : Rn ! R which is zero on X, positive outside of X and increases with the distance from X. It hence penalizes iterates which move too far away from X. From one iterate to the next the weight of a barrier function is decreased, while the weight of a penalty function is increased. Both the weighted barrier function and the weighted penalty function thus converge point-wise to the function 0; x 2 X; F ∗(x) = +1; x 62 X outside of @X. In the case of conic programs min hc; xi : Ax = b x2K the difficulty is hidden in the conic constraint x 2 K. Here K ⊂ Rn is a regular convex cone. The basic idea of interior-point methods for solving conic programs is to eliminate the conic constraint by the addition of a barrier function F : Ko ! R to the linear objective. The barrier function should satisfy the following requirements: • F (x) is convex (the problem should remain convex), • F (x) is sufficiently smooth (to be able to use second-order methods), • limx!@K F (x) = +1 (acts as a barrier for K), • F behaves well with the Newton method.
    [Show full text]