A Tutorial on Convex Optimization
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4. Convex Optimization Problems
Convex Optimization — Boyd & Vandenberghe 4. Convex optimization problems optimization problem in standard form • convex optimization problems • quasiconvex optimization • linear optimization • quadratic optimization • geometric programming • generalized inequality constraints • semidefinite programming • vector optimization • 4–1 Optimization problem in standard form minimize f0(x) subject to f (x) 0, i =1,...,m i ≤ hi(x)=0, i =1,...,p x Rn is the optimization variable • ∈ f : Rn R is the objective or cost function • 0 → f : Rn R, i =1,...,m, are the inequality constraint functions • i → h : Rn R are the equality constraint functions • i → optimal value: p⋆ = inf f (x) f (x) 0, i =1,...,m, h (x)=0, i =1,...,p { 0 | i ≤ i } p⋆ = if problem is infeasible (no x satisfies the constraints) • ∞ p⋆ = if problem is unbounded below • −∞ Convex optimization problems 4–2 Optimal and locally optimal points x is feasible if x dom f and it satisfies the constraints ∈ 0 ⋆ a feasible x is optimal if f0(x)= p ; Xopt is the set of optimal points x is locally optimal if there is an R> 0 such that x is optimal for minimize (over z) f0(z) subject to fi(z) 0, i =1,...,m, hi(z)=0, i =1,...,p z x≤ R k − k2 ≤ examples (with n =1, m = p =0) f (x)=1/x, dom f = R : p⋆ =0, no optimal point • 0 0 ++ f (x)= log x, dom f = R : p⋆ = • 0 − 0 ++ −∞ f (x)= x log x, dom f = R : p⋆ = 1/e, x =1/e is optimal • 0 0 ++ − f (x)= x3 3x, p⋆ = , local optimum at x =1 • 0 − −∞ Convex optimization problems 4–3 Implicit constraints the standard form optimization problem has an implicit -
Convex Optimisation Matlab Toolbox
UNLOCBOX: USER’S GUIDE MATLAB CONVEX OPTIMIZATION TOOLBOX Lausanne - February 2014 PERRAUDIN Nathanaël, KALOFOLIAS Vassilis LTS2 - EPFL Abstract Nowadays the trend to solve optimization problems is to use specific algorithms rather than very gen- eral ones. The UNLocBoX provides a general framework allowing the user to design his own algorithms. To do so, the framework try to stay as close from the mathematical problem as possible. More precisely, the UNLocBoX is a Matlab toolbox designed to solve convex optimization problem of the form K min ∑ fn(x); x2C n=1 using proximal splitting techniques. It is mainly composed of solvers, proximal operators and demonstra- tion files allowing the user to quickly implement a problem. Contents 1 Introduction 1 2 Literature 2 3 Installation and initialization2 3.1 Dependencies.........................................2 3.2 GPU computation.......................................2 4 Structure of the UNLocBoX2 5 Problems of interest4 6 Solvers 4 6.1 Defining functions......................................5 6.2 Selecting a solver.......................................5 6.3 Optional parameters......................................6 6.4 Plug-ins: how to tune your algorithm.............................6 6.5 Returned arguments......................................8 7 Proximal operators9 7.1 Constraints..........................................9 8 Example 10 References 16 LTS2 - EPFL 1 Introduction During your research, you have most likely faced the situation where you need to solve a convex optimiza- tion problem. Maybe you were trying to find an optimal combination of parameters, you were transforming some process into something automatic or, even more simply, your algorithm was equivalent to solving convex problem. In this situation, you usually face a dilemma: you need a specific algorithm to solve your problem, but you do not want to spend hours writing it. -
Conditional Gradient Sliding for Convex Optimization ∗
CONDITIONAL GRADIENT SLIDING FOR CONVEX OPTIMIZATION ∗ GUANGHUI LAN y AND YI ZHOU z Abstract. In this paper, we present a new conditional gradient type method for convex optimization by calling a linear optimization (LO) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time, and as a result, can achieve the optimal complexity bounds in terms of not only the number of callsp to the LO oracle, but also the number of gradient evaluations. More specifically, we show that the CGS method requires O(1= ) and O(log(1/)) gradient evaluations, respectively, for solving smooth and strongly convex problems, while still maintaining the optimal O(1/) bound on the number of calls to LO oracle. We also develop variants of the CGS method which can achieve the optimal complexity bounds for solving stochastic optimization problems and an important class of saddle point optimization problems. To the best of our knowledge, this is the first time that these types of projection-free optimal first-order methods have been developed in the literature. Some preliminary numerical results have also been provided to demonstrate the advantages of the CGS method. Keywords: convex programming, complexity, conditional gradient method, Frank-Wolfe method, Nesterov's method AMS 2000 subject classification: 90C25, 90C06, 90C22, 49M37 1. Introduction. The conditional gradient (CndG) method, which was initially developed by Frank and Wolfe in 1956 [13] (see also [11, 12]), has been considered one of the earliest first-order methods for solving general convex programming (CP) problems. -
Techniques of Variational Analysis
J. M. Borwein and Q. J. Zhu Techniques of Variational Analysis An Introduction October 8, 2004 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo To Tova, Naomi, Rachel and Judith. To Charles and Lilly. And in fond and respectful memory of Simon Fitzpatrick (1953-2004). Preface Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse ¯elds. The discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques. The motivation to write this book came from a desire to share our pleasure in applying such variational techniques and promoting these powerful tools. Potential readers of this book will be researchers and graduate students who might bene¯t from using variational methods. The only broad prerequisite we anticipate is a working knowledge of un- dergraduate analysis and of the basic principles of functional analysis (e.g., those encountered in a typical introductory functional analysis course). We hope to attract researchers from diverse areas { who may fruitfully use varia- tional techniques { by providing them with a relatively systematical account of the principles of variational analysis. We also hope to give further insight to graduate students whose research already concentrates on variational analysis. Keeping these two di®erent reader groups in mind we arrange the material into relatively independent blocks. We discuss various forms of variational princi- ples early in Chapter 2. We then discuss applications of variational techniques in di®erent areas in Chapters 3{7. -
Additional Exercises for Convex Optimization
Additional Exercises for Convex Optimization Stephen Boyd Lieven Vandenberghe January 4, 2020 This is a collection of additional exercises, meant to supplement those found in the book Convex Optimization, by Stephen Boyd and Lieven Vandenberghe. These exercises were used in several courses on convex optimization, EE364a (Stanford), EE236b (UCLA), or 6.975 (MIT), usually for homework, but sometimes as exam questions. Some of the exercises were originally written for the book, but were removed at some point. Many of them include a computational component using one of the software packages for convex optimization: CVX (Matlab), CVXPY (Python), or Convex.jl (Julia). We refer to these collectively as CVX*. (Some problems have not yet been updated for all three languages.) The files required for these exercises can be found at the book web site www.stanford.edu/~boyd/cvxbook/. You are free to use these exercises any way you like (for example in a course you teach), provided you acknowledge the source. In turn, we gratefully acknowledge the teaching assistants (and in some cases, students) who have helped us develop and debug these exercises. Pablo Parrilo helped develop some of the exercises that were originally used in MIT 6.975, Sanjay Lall developed some other problems when he taught EE364a, and the instructors of EE364a during summer quarters developed others. We’ll update this document as new exercises become available, so the exercise numbers and sections will occasionally change. We have categorized the exercises into sections that follow the book chapters, as well as various additional application areas. Some exercises fit into more than one section, or don’t fit well into any section, so we have just arbitrarily assigned these. -
Efficient Convex Quadratic Optimization Solver for Embedded MPC Applications
DEGREE PROJECT IN ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2018 Efficient Convex Quadratic Optimization Solver for Embedded MPC Applications ALBERTO DÍAZ DORADO KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE Efficient Convex Quadratic Optimization Solver for Embedded MPC Applications ALBERTO DÍAZ DORADO Master in Electrical Engineering Date: October 2, 2018 Supervisor: Arda Aytekin , Martin Biel Examiner: Mikael Johansson Swedish title: Effektiv Konvex Kvadratisk Optimeringslösare för Inbäddade MPC-applikationer School of Electrical Engineering and Computer Science iii Abstract Model predictive control (MPC) is an advanced control technique that requires solving an optimization problem at each sampling instant. Sev- eral emerging applications require the use of short sampling times to cope with the fast dynamics of the underlying process. In many cases, these applications also need to be implemented on embedded hard- ware with limited resources. As a result, the use of model predictive controllers in these application domains remains challenging. This work deals with the implementation of an interior point algo- rithm for use in embedded MPC applications. We propose a modular software design that allows for high solver customization, while still producing compact and fast code. Our interior point method includes an efficient implementation of a novel approach to constraint softening, which has only been tested in high-level languages before. We show that a well conceived low-level implementation of integrated constraint softening adds no significant overhead to the solution time, and hence, constitutes an attractive alternative in embedded MPC solvers. iv Sammanfattning Modell prediktiv reglering (MPC) är en avancerad regler-teknik som involverar att lösa ett optimeringsproblem vid varje sampeltillfälle. -
1 Linear Programming
ORF 523 Lecture 9 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a [email protected]. In this lecture, we see some of the most well-known classes of convex optimization problems and some of their applications. These include: • Linear Programming (LP) • (Convex) Quadratic Programming (QP) • (Convex) Quadratically Constrained Quadratic Programming (QCQP) • Second Order Cone Programming (SOCP) • Semidefinite Programming (SDP) 1 Linear Programming Definition 1. A linear program (LP) is the problem of optimizing a linear function over a polyhedron: min cT x T s.t. ai x ≤ bi; i = 1; : : : ; m; or written more compactly as min cT x s.t. Ax ≤ b; for some A 2 Rm×n; b 2 Rm: We'll be very brief on our discussion of LPs since this is the central topic of ORF 522. It suffices to say that LPs probably still take the top spot in terms of ubiquity of applications. Here are a few examples: • A variety of problems in production planning and scheduling 1 • Exact formulation of several important combinatorial optimization problems (e.g., min-cut, shortest path, bipartite matching) • Relaxations for all 0/1 combinatorial programs • Subroutines of branch-and-bound algorithms for integer programming • Relaxations for cardinality constrained (compressed sensing type) optimization prob- lems • Computing Nash equilibria in zero-sum games • ::: 2 Quadratic Programming Definition 2. A quadratic program (QP) is an optimization problem with a quadratic ob- jective and linear constraints min xT Qx + qT x + c x s.t. Ax ≤ b: Here, we have Q 2 Sn×n, q 2 Rn; c 2 R;A 2 Rm×n; b 2 Rm: The difficulty of this problem changes drastically depending on whether Q is positive semidef- inite (psd) or not. -
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] April 20, 2017 2 3 Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier Abstract: Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, com- puter vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincar´eformula (Poincar´e,1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a \correct" account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space. -
Linear Programming and Convex Optimization
Lecture 2 : Linear programming and convex optimization Rajat Mittal ? IIT Kanpur We talked about optimization problems and why they are important. We also looked at one of the class of problems called least square problems. Next we look at another class, 1 Linear Programming 1.1 Definition Linear programming is one of the well studied classes of optimization problem. We already discussed that a linear program is one which has linear objective and constraint functions. This implies that a standard linear program looks like P T min i cixi = c x T subject to ai xi ≤ bi 8i 2 f1; ··· ; mg n Here the vectors c; a1; ··· ; am 2 R and scalars bi 2 R are the problem parameters. 1.2 Examples { Max flow: Given a graph, start(s) and end node (t), capacities on every edge; find out the maximum flow possible through edges. s t Fig. 1. Max flow problem: there will be capacities for every edge in the problem statement The linear program looks like: P max fs;ug f(s; u) P P s.t. fu;vg f(u; v) = fv;ug f(v; u) 8v 6= s; t; ∼ 0 ≤ f(u; v) ≤ c(u; v) Note: There is another one which can be made using the flow through paths. ? Thanks to books from Boyd and Vandenberghe, Dantzig and Thapa, Papadimitriou and Steiglitz { Another example: This time, I will give the linear program and you will tell me what real world situation models it :). max 2x1 + 4x2 s.t. x1 + x2 ≤ 10; x1 ≤ 4 1.3 Solving linear programs You might have already had a course on linear optimization. -
Convex Optimization: Algorithms and Complexity
Foundations and Trends R in Machine Learning Vol. 8, No. 3-4 (2015) 231–357 c 2015 S. Bubeck DOI: 10.1561/2200000050 Convex Optimization: Algorithms and Complexity Sébastien Bubeck Theory Group, Microsoft Research [email protected] Contents 1 Introduction 232 1.1 Some convex optimization problems in machine learning . 233 1.2 Basic properties of convexity . 234 1.3 Why convexity? . 237 1.4 Black-box model . 238 1.5 Structured optimization . 240 1.6 Overview of the results and disclaimer . 240 2 Convex optimization in finite dimension 244 2.1 The center of gravity method . 245 2.2 The ellipsoid method . 247 2.3 Vaidya’s cutting plane method . 250 2.4 Conjugate gradient . 258 3 Dimension-free convex optimization 262 3.1 Projected subgradient descent for Lipschitz functions . 263 3.2 Gradient descent for smooth functions . 266 3.3 Conditional gradient descent, aka Frank-Wolfe . 271 3.4 Strong convexity . 276 3.5 Lower bounds . 279 3.6 Geometric descent . 284 ii iii 3.7 Nesterov’s accelerated gradient descent . 289 4 Almost dimension-free convex optimization in non-Euclidean spaces 296 4.1 Mirror maps . 298 4.2 Mirror descent . 299 4.3 Standard setups for mirror descent . 301 4.4 Lazy mirror descent, aka Nesterov’s dual averaging . 303 4.5 Mirror prox . 305 4.6 The vector field point of view on MD, DA, and MP . 307 5 Beyond the black-box model 309 5.1 Sum of a smooth and a simple non-smooth term . 310 5.2 Smooth saddle-point representation of a non-smooth function312 5.3 Interior point methods . -
Efficient Convex Optimization for Linear
Efficient Convex Optimization for Linear MPC Stephen J. Wright Abstract MPC formulations with linear dynamics and quadratic objectives can be solved efficiently by using a primal-dual interior-point framework, with complexity proportional to the length of the horizon. An alternative, which is more able to ex- ploit the similarity of the problems that are solved at each decision point of linear MPC, is to use an active-set approach, in which the MPC problem is viewed as a convex quadratic program that is parametrized by the initial state x0. Another alter- native is to identify explicitly polyhedral regions of the space occupied by x0 within which the set of active constraints remains constant, and to pre-calculate solution operators on each of these regions. All these approaches are discussed here. 1 Introduction In linear model predictive control (linear MPC), the problem to be solved at each decision point has linear dynamics and a quadratic objective. This a classic problem in optimization — quadratic programming (QP) — which is convex when (as is usu- ally true) the quadratic objective is convex. It remains a convex QP even when linear constraints on the states and controls are allowed at each stage, or when only linear functions of the state can be observed. Moreover, this quadratic program has special structure that can be exploited by algorithms that solve it, particularly interior-point algorithms. In deployment of linear MPC, unless there is an unanticipated upset, the quadratic program to be solved differs only slightly from one decision point to the next. The question arises of whether the solution at one decision point can be used to “warm- start” the algorithm at the next decision point. -
A Tutorial on Convex Optimization
A Tutorial on Convex Optimization Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California email: [email protected] Abstract— In recent years, convex optimization has be- quickly deduce the convexity (and hence tractabil- come a computational tool of central importance in engi- ity) of many problems, often by inspection; neering, thanks to it’s ability to solve very large, practical 2) to review and introduce some canonical opti- engineering problems reliably and efficiently. The goal of this tutorial is to give an overview of the basic concepts of mization problems, which can be used to model convex sets, functions and convex optimization problems, so problems and for which reliable optimization code that the reader can more readily recognize and formulate can be readily obtained; engineering problems using modern convex optimization. 3) to emphasize modeling and formulation; we do This tutorial coincides with the publication of the new book not discuss topics like duality or writing custom on convex optimization, by Boyd and Vandenberghe [7], who have made available a large amount of free course codes. material and links to freely available code. These can be We assume that the reader has a working knowledge of downloaded and used immediately by the audience both linear algebra and vector calculus, and some (minimal) for self-study and to solve real problems. exposure to optimization. Our presentation is quite informal. Rather than pro- I. INTRODUCTION vide details for all the facts and claims presented, our Convex optimization can be described as a fusion goal is instead to give the reader a flavor for what is of three disciplines: optimization [22], [20], [1], [3], possible with convex optimization.