Motivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem:


 Ursula Whitehead
 2 years ago
 Views:
Transcription
1 CDS270 Maryam Fazel Lecture 2 Topics from Optimization and Duality Motivation network utility maximization (NUM) problem: consider a network with S sources (users), each sending one flow at rate x s, through path L(s), with utility PSfrag replacements U s (x s ). there are L links, each one with capacity c l, shared by S(l) sources. motivation: Network Utility Maximization (NUM) problem application: Internet congestion control basic unconstrained minimization x 1 x 3 c 1 c 2 c 3 x 2 c 4 x 4 duality theory Lagrange dual of NUM, decentralization maximize S s=1 U s(x s ) subject to s S(l) x s c l, l = 1,..., L, optimization variable: x R S (domain of U s is x s 0)
2 when U s (x s ) are concave functions, NUM is a convex optimization problem TCP/AQM congestion control seminal paper [Kelly,Maulloo,Tan 98] important foundation of data networks: structure (APP/TCP/IP/MAC/PHY) layered different parts of NUM relate to different layers e.g., varying R, c would involve IP and PHY layers PSfrag replacements more later... a large class of congestion control mechanisms can be interpreted as distribured algorithms that solve NUM and its dual x s λ l AQM: RED, DropTail,... NUM useful as framework to understand network equilibrium and dynamics also a central problem in economics TCP: Reno, Vegas,... x s : source rate, updated by TCP (Transmission Control Protocol) λ l : link congestion measure, or price, updated by AQM (Active Queue Management) e.g., TCP Reno uses packet loss as congestion measure, TCP Vegas uses queueing delay
3 Topics we cover Unconstrained minimization basic unconstrained minimization, gradient descent duality theory Lagrange dual problem and its properties strong duality and Slater s condition interpretations of duality KKT optimality conditions theorems of alternatives Lagrange dual of NUM, decentralization minimize f(x) f : R n R, convex, differentiable minimizing sequence: x (k), k f(x (k) ) f optimality condition: f(x ) = 0 set of nonlinear equations; usually no analytical solution refs: Boyd & Vandenberghe, Convex Optimization, boyd/cvxbook. Chapters 5, and Bertsekas, Nonlinear Programming, Chapter 1. more generally, if 2 f(x) mi, m > 0, then f(x) f 1 2m f(x) 2... yields stopping criterion (if you know m)
4 Descent methods given starting point x dom f repeat 1. Compute a search direction v 2. Line search. Choose step size t > 0 3. Update. x := x + tv until stopping criterion is satisfied descent method: f(x (k+1) ) < f(x (k) ) since f convex, v must be a descent direction: f(x (k) ) T v (k) < 0 choose v (descent direction) v (k) = f(x (k) ) (gradient) v (k) = H (k) f(x (k) ), H (k) = H (k)t 0 v (k) = 2 f(x (k) ) 1 f(x (k) ) (Newton s) choose t (step size) exact line search: t = argmin s>0 f(x + sv) backtracking line search (picks step size that reduces f enough ) Sfrag replacements f(x) = c 2 c 1 f(x) = c 1 v (k) constant step size x x (k) f(x (k) ) (main idea extends to nondifferentiable f using subgradients)
5 Gradient descent method Example given starting point x dom f repeat 1. Compute search direction v = f(x) 2. Line search. Choose step size t 3. Update. x := x + tv until stopping criterion is satisfied converges with exact or backtracking line search. a simple (but conservative) convergence condition for t: suppose f has bounded curvature 2 f(x) MI; there exists z line segment [x, y] s.t. f(x + tv) f(x) = f(x) T (tv) (tv)t ( 2 f(z))(tv) t f t2 M f 2, then t t2 M < 0, or t < 2/M, guarantees descent. 2 9 minimize 1 2 (x2 1 + Mx 2 2) where M > 0, optimal point is x = 0. use exact line search; start at x (0) = (M, 1) (condition number κ = max{m, 1/M} determines convergence rate) 2 10
6 notes: convergence can be very slow, depends critically on condition number Lagrangian standard form optimization problem Newton s method (and variations) have much better convergence properties minimize subject to f 0 (x) f i (x) 0, i = 1,..., m however, in networking problems, gradient descent proves very useful, since main concern here is decentralization... (details later) optimal value p, domain D called primal problem (in context of duality) (for now) we don t assume convexity how about constrained optimization problems? next topic: Lagrangian duality Lagrangian L : R n+m R L(x, λ) = f 0 (x) + λ 1 f 1 (x) + + λ m f m (x) λ i called Lagrange multipliers or dual variables objective is augmented with weighted sum of constraint functions
7 Lagrange dual function (Lagrange) dual function g : R m R { } g(λ) = inf x = inf x L(x, λ) (f 0(x) + λ 1 f 1 (x) + + λ m f m (x)) minimum of augmented cost as function of weights can be for some λ g is concave (even if f i not convex!) example: LP minimize subject to L(x, λ) = c T x + c T x a T i x b i 0, i = 1,..., m m λ i (a T i x b i ) i=1 = b T λ + (A T λ + c) T x { b hence g(λ) = T λ if A T λ + c = 0 otherwise 2 13 Lower bound property if λ 0 and x is primal feasible, then g(λ) f 0 (x) proof: if f i (x) 0 and λ i 0, f 0 (x) f 0 (x) + i ( inf z = g(λ) f 0 (z) + i λ i f i (x) λ i f i (z) f 0 (x) g(λ) is called the duality gap of (primal feasible) x and λ 0 minimize over primal feasible x to get, for any λ 0, g(λ) p λ R m is dual feasible if λ 0 and g(λ) > dual feasible points yield lower bounds on optimal value! 2 14 )
8 Lagrange dual problem Strong duality & constraint qualifications let s find best lower bound on p : maximize g(λ) subject to λ 0 called (Lagrange) dual problem (associated with primal problem) always a convex problem, even if primal isn t! optimal value denoted d we always have d p (called weak duality) p d is optimal duality gap for convex problems, we (usually) have strong duality: d = p (strong duality does not hold, in general, for nonconvex problems) when strong duality holds, dual optimal λ serves as certificate of optimality for primal optimal point x many conditions or constraint qualifications guarantee strong duality for convex problems Slater s condition: if primal problem is strictly feasible (and convex), i.e., there exists x relint D with then we have p = d f i (x) < 0, i = 1,..., m
9 Dual of linear program Dual of quadratic program (primal) LP minimize subject to c T x Ax b primal QP minimize x T P x subject to Ax b we assume P 0 for simplicity n variables, m inequality constraints dual of LP is (after making implicit equality constraints explicit) maximize b T λ subject to A T λ + c = 0 λ 0 dual of LP is also an LP (in std LP format) m variables, n equality constraints, m nonnegativity contraints for LP we have strong duality except in one (pathological) case: primal and dual both infeasible (p = +, d = ) 2 17 Lagrangian is L(x, λ) = x T P x + λ T (Ax b) x L(x, λ) = 0 yields x = (1/2)P 1 A T λ, hence dual function is g(λ) = (1/4)λ T AP 1 A T λ b T λ concave quadratic function all λ 0 are dual feasible dual of QP is maximize (1/4)λ T AP 1 A T λ b T λ subject to λ 0... another QP 2 18
10 Minmax & saddlepoint interpretation can express primal and dual problems in a more symmetric form: sup λ 0 ( f 0 (x) + ) m λ i f i (x) = i=1 so p = inf x sup λ 0 L(x, λ). { f0 (x) f i (x) 0, + otherwise, also by definition, d = sup λ 0 inf x L(x, λ). weak duality can be expressed as sup inf λ 0 x L(x, λ) inf x sup L(x, λ) λ 0 strong duality when equality holds. means can switch the order of minimization over x and maximization over λ 0. if x and λ are primal and dual optimal and strong duality holds, they form a saddlepoint for the Lagrangian (converse also true). Economic (price) interpretation minimize f 0 (x) subject to f i (x) 0, i = 1,..., m. f 0 (x) is cost of operating firm at operating condition x; constraints give resource limits. suppose: can violate f i (x) 0 by paying additional cost of λ i (in dollars per unit violation), i.e., incur cost λ i f i (x) if f i (x) > 0 can sell unused portion of ith constraint at same price, i.e., gain profit λ i f i (x) if f i (x) < 0 total cost to firm to operate at x at constraint prices λ i : m L(x, λ) = f 0 (x) + λ i f i (x) i=
11 interpretations: dual function: g(λ) = inf x L(x, λ) is optimal cost to firm at constraint prices λ i weak duality: cost can be lowered if firm allowed to pay for violated constraints (and get paid for nontight ones) duality gap: advantage to firm under this scenario strong duality: λ give prices for which firm has no advantage in being allowed to violate constraints... dual optimal λ problem called shadow prices for original Geometric interpretation of duality consider set A = { (u, t) R m+1 x f i (x) u i, f 0 (x) t } A is convex if f i are for λ 0, g(λ) = inf { [ λ 1 ] T [ u t ] [ u t ] } A PSfrag replacements t A t + λ T u = g(λ) g(λ)» λ 1 u
12 (Idea of) proof of Slater s theorem problem convex, strictly feasible = strong duality PSfrag replacements t Slater s condition: there exists (u, t) A with u 0; implies that all supporting hyperplanes at (0, p ) are nonvertical (µ 0 > 0) p A» 1 λ u (0, p ) A supporting hyperplane at (0, p ): (u, t) A = µ 0 (t p ) + µ T u 0 µ 0 0, µ 0, (µ, µ 0 ) 0 strong duality supp. hyperpl. with µ 0 > 0: for λ = µ/µ 0, we have p t + λ T u (t, u) A p g(λ )
13 Sensitivity analysis via duality define p (u) as the optimal value of PSfrag replacements minimize subject to p (u) 0 f 0 (x) f i (x) u i, i = 1,..., m 0 u epi p p (0) λ T u λ gives lower bound on p (u): p (u) p m i=1 λ i u i if λ i large: u i < 0 greatly increases p if λ i small: u i > 0 does not decrease p too much if p (u) is differentiable, λ i = p (0) u i λ i is sensitivity of p w.r.t. ith constraint 2 25 Complementary slackness suppose x, λ are primal, dual feasible with zero duality gap (hence, they are primal, dual optimal) f 0 (x ) = g(λ ) ( = inf x f 0 (x ) + f 0 (x) + ) m λ i f i (x) i=1 m λ i f i (x ) i=1 hence we have m i=1 λ i f i(x ) = 0, and so λ i f i (x ) = 0, i = 1,..., m called complementary slackness condition ith constraint inactive at optimum = λ i = 0 λ i > 0 at optimum = ith constraint active at optimum 2 26
14 suppose KKT optimality conditions f i are differentiable x, λ are (primal, dual) optimal, with zero duality gap so if x, λ are (primal, dual) optimal, with zero duality gap, they satisfy f i (x ) 0 λ i 0 λ i f i(x ) = 0 f 0 (x ) + i λ i f i(x ) = 0 the KarushKuhnTucker (KKT) optimality conditions by complementary slackness we have f 0 (x ) + i λ i f i (x ) = inf x ( f 0 (x) + i λ i f i (x) ) conversely, if the problem is convex and x, λ satisfy KKT, then they are (primal, dual) optimal i.e., x minimizes L(x, λ ) therefore f 0 (x ) + i λ i f i (x ) =
15 Generalized inequalities dual problem minimize subject to f 0 (x) f i (x) Ki 0, i = 1,..., L maximize g(λ 1,..., λ L ) subject to λ i K i 0, i = 1,..., L Ki are generalized inequalities on R m i f i : R n R m i are K i convex Lagrangian L : R n R m 1 R m L R, L(x, λ 1,..., λ L ) = f 0 (x) + λ T 1 f 1 (x) + + λ T Lf L (x) dual function g(λ 1,..., λ L ) = inf x ( f0 (x) + λ T 1 f 1 (x) + + λ T Lf L (x) ) weak duality: d p always strong duality: d = p usually Slater condition: if primal is strictly feasible, i.e., x relint D : f i (x) Ki 0, i = 1,..., L then d = p λ i dual feasible if λ i K i 0, g(λ 1,..., λ L ) > lower bound property: if x primal feasible and (λ 1,..., λ L ) is dual feasible, then g(λ 1,..., λ L ) f 0 (x) (hence, g(λ 1,..., λ L ) p )
16 Example: semidefinite programming minimize c T x subject to F 0 + x 1 F x n F n 0 Lagrangian (multiplier Z = Z T R m m ) L(x, Z) = c T x + Tr Z(F 0 + x 1 F x n F n ) dual function g(z) = ( inf c T x + Tr Z(F 0 + x 1 F x n F n ) ) x = { Tr F0 Z if Tr F i Z + c i = 0, i = 1,..., n otherwise dual problem maximize subject to Tr F 0 Z Tr F i Z + c i = 0, i = 1,..., n Z = Z T 0 strong duality holds if there exists x with F 0 + x 1 F x n F n Theorem of alternatives apply Lagrange duality theory to the feasibility problem: f 1,..., f m convex with dom f i = R n exactly one of the following is true: 1. there exists x with f i (x) < 0, i = 1,..., m 2. there exists λ 0 with λ 0, g(λ) = inf x (λ 1f 1 (x) + + λ m f m (x)) 0 called alternatives use in practice: λ that satisfies 2nd condition proves f i (x) < 0 is infeasible example: f i (x) = a T i x b i 1. there exists x with Ax b 2. there exists λ 0, λ 0, b T λ 0, A T λ =
17 proof. from convex duality: primal problem minimize subject to (variables x, t) dual problem t f i (x) t, i = 1,..., m maximize g(λ) subject to λ 0 1 T λ = 1 Slater s condition is satisfied, hence p = d 1st alternative: p < 0 2nd alternative: p Dual of NUM back to the Network Utility Maximization problem we saw before, primal: maximize Us (x s ) subject to Rx c, where R is the routing matrix. Lagrangian: L(x, λ) = s = s D(λ) = max L(x, λ) = x s U s (x s ) + ( λ l c l ) R ls x s l s [ ( ) ] U s (x s ) R ls λ l x s + c l λ l l l max L s (x s, λ s ) + x s l c l λ l where λ s = l R lsλ l is endtoend path price for source s. 2 34
18 dual: minimize subject to s max x s 0 (U s (x s ) λ s x s ) + l c lλ l λ 0, dualbased distributed algorithm one way to solve iteratively: use (a variation on) gradient method for dual additivity of total utility and flow constraints lead to decomposition into individual source terms: a form of dual decomposition this decomposition is called horizontal : users in network each source x s can keep its utility private across each source only needs to know λ s to solve subproblem max xs 0(U s (x s ) λ s x s ) optimal prices λ serve as a coordinating signal that align individual (selfish) optimality with social optimality source algorithm: singlevariable optimization x s(λ s ) = argmax [ U s (x s ) ( l R ls λ l (t))x s ], s. if U s are strictly concave, twice continously differentiable, x (λ s ) = U 1 (λ s ) called demand function in microeconomics more later
19 link algorithm: if U s strictly concave, twice continously differentiable, D λ l (λ(k)) = c l s R ls x s (λ(k)) use gradient projection method: [ ( λ l (k+1) = λ l (k) α(k) c l s R ls x s (λ(k)))] +, l α(k) is step size, certain choices guarantee λ(k) λ and x (λ(k)) x. interpretation: balancing supply and demand through pricing link update is decentralized, each link using only local information a model for sourcelink dynamics 2 37
5. Duality. Lagrangian
5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized
More informationEE/AA 578, Univ of Washington, Fall Duality
7. Duality EE/AA 578, Univ of Washington, Fall 2016 Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized
More informationConvex Optimization & Lagrange Duality
Convex Optimization & Lagrange Duality Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Convex optimization Optimality condition Lagrange duality KKT
More informationConvex Optimization Boyd & Vandenberghe. 5. Duality
5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized
More informationConvex Optimization M2
Convex Optimization M2 Lecture 3 A. d Aspremont. Convex Optimization M2. 1/49 Duality A. d Aspremont. Convex Optimization M2. 2/49 DMs DM par email: dm.daspremont@gmail.com A. d Aspremont. Convex Optimization
More informationLecture: Duality.
Lecture: Duality http://bicmr.pku.edu.cn/~wenzw/opt2016fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/35 Lagrange dual problem weak and strong
More informationExtreme Abridgment of Boyd and Vandenberghe s Convex Optimization
Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization Compiled by David Rosenberg Abstract Boyd and Vandenberghe s Convex Optimization book is very wellwritten and a pleasure to read. The
More informationI.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec  Spring 2010
I.3. LMI DUALITY Didier HENRION henrion@laas.fr EECI Graduate School on Control Supélec  Spring 2010 Primal and dual For primal problem p = inf x g 0 (x) s.t. g i (x) 0 define Lagrangian L(x, z) = g 0
More informationLecture: Duality of LP, SOCP and SDP
1/33 Lecture: Duality of LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2017.html wenzw@pku.edu.cn Acknowledgement:
More informationICSE4030 Kernel Methods in Machine Learning
ICSE4030 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 28. September, 2016 Juho Rousu 28. September, 2016 1 / 38 Convex optimization Convex optimisation This
More information9. Dual decomposition and dual algorithms
EE 546, Univ of Washington, Spring 2016 9. Dual decomposition and dual algorithms dual gradient ascent example: network rate control dual decomposition and the proximal gradient method examples with simple
More informationCSE4830 Kernel Methods in Machine Learning
CSE4830 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 27. September, 2017 Juho Rousu 27. September, 2017 1 / 45 Convex optimization Convex optimisation This
More informationSolving Dual Problems
Lecture 20 Solving Dual Problems We consider a constrained problem where, in addition to the constraint set X, there are also inequality and linear equality constraints. Specifically the minimization problem
More informationPrimal/Dual Decomposition Methods
Primal/Dual Decomposition Methods Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470  Convex Optimization Fall 201819, HKUST, Hong Kong Outline of Lecture Subgradients
More informationLecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem
Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 00 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R
More informationLagrange Duality. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)
Lagrange Duality Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470  Convex Optimization Fall 201718, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual
More informationDuality. Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities
Duality Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities Lagrangian Consider the optimization problem in standard form
More informationCSCI : Optimization and Control of Networks. Review on Convex Optimization
CSCI7000016: Optimization and Control of Networks Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one
More informationsubject to (x 2)(x 4) u,
Exercises Basic definitions 5.1 A simple example. Consider the optimization problem with variable x R. minimize x 2 + 1 subject to (x 2)(x 4) 0, (a) Analysis of primal problem. Give the feasible set, the
More informationA Brief Review on Convex Optimization
A Brief Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one convex, two nonconvex sets): A Brief Review
More informationConstrained Optimization and Lagrangian Duality
CIS 520: Machine Learning Oct 02, 2017 Constrained Optimization and Lagrangian Duality Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may
More informationRate Control in Communication Networks
From Models to Algorithms Department of Computer Science & Engineering The Chinese University of Hong Kong February 29, 2008 Outline Preliminaries 1 Preliminaries Convex Optimization TCP Congestion Control
More informationConvex Optimization and Modeling
Convex Optimization and Modeling Duality Theory and Optimality Conditions 5th lecture, 12.05.2010 Jun.Prof. Matthias Hein Program of today/next lecture Lagrangian and duality: the Lagrangian the dual
More informationLecture 18: Optimization Programming
Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equalityconstrained Optimization Inequalityconstrained Optimization Mixtureconstrained Optimization 3 Quadratic Programming
More informationLagrange duality. The Lagrangian. We consider an optimization program of the form
Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization
More information12. Interiorpoint methods
12. Interiorpoint methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More information14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.
CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity
More informationDual Decomposition.
1/34 Dual Decomposition http://bicmr.pku.edu.cn/~wenzw/opt2017fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghes lecture notes Outline 2/34 1 Conjugate function 2 introduction:
More informationIntroduction to Machine Learning Lecture 7. Mehryar Mohri Courant Institute and Google Research
Introduction to Machine Learning Lecture 7 Mehryar Mohri Courant Institute and Google Research mohri@cims.nyu.edu Convex Optimization Differentiation Definition: let f : X R N R be a differentiable function,
More informationIntroduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.14.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following
More informationConvex Optimization. Dani Yogatama. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. February 12, 2014
Convex Optimization Dani Yogatama School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA February 12, 2014 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12,
More informationOptimization for Machine Learning
Optimization for Machine Learning (Problems; Algorithms  A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html
More informationSubgradients. subgradients and quasigradients. subgradient calculus. optimality conditions via subgradients. directional derivatives
Subgradients subgradients and quasigradients subgradient calculus optimality conditions via subgradients directional derivatives Prof. S. Boyd, EE392o, Stanford University Basic inequality recall basic
More informationOn the Method of Lagrange Multipliers
On the Method of Lagrange Multipliers Reza Nasiri Mahalati November 6, 2016 Most of what is in this note is taken from the Convex Optimization book by Stephen Boyd and Lieven Vandenberghe. This should
More informationLagrangian Duality and Convex Optimization
Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DSGA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?
More informationConvex Optimization. Newton s method. ENSAE: Optimisation 1/44
Convex Optimization Newton s method ENSAE: Optimisation 1/44 Unconstrained minimization minimize f(x) f convex, twice continuously differentiable (hence dom f open) we assume optimal value p = inf x f(x)
More informationLecture 7: Convex Optimizations
Lecture 7: Convex Optimizations Radu Balan, David Levermore March 29, 2018 Convex Sets. Convex Functions A set S R n is called a convex set if for any points x, y S the line segment [x, y] := {tx + (1
More information4TE3/6TE3. Algorithms for. Continuous Optimization
4TE3/6TE3 Algorithms for Continuous Optimization (Duality in Nonlinear Optimization ) Tamás TERLAKY Computing and Software McMaster University Hamilton, January 2004 terlaky@mcmaster.ca Tel: 27780 Optimality
More informationGeneralization to inequality constrained problem. Maximize
Lecture 11. 26 September 2006 Review of Lecture #10: Second order optimality conditions necessary condition, sufficient condition. If the necessary condition is violated the point cannot be a local minimum
More informationInterior Point Algorithms for Constrained Convex Optimization
Interior Point Algorithms for Constrained Convex Optimization Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Inequality constrained minimization problems
More informationISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints
ISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints Instructor: Prof. Kevin Ross Scribe: Nitish John October 18, 2011 1 The Basic Goal The main idea is to transform a given constrained
More informationKarushKuhnTucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36725
KarushKuhnTucker Conditions Lecturer: Ryan Tibshirani Convex Optimization 10725/36725 1 Given a minimization problem Last time: duality min x subject to f(x) h i (x) 0, i = 1,... m l j (x) = 0, j =
More informationNonlinear Programming and the KuhnTucker Conditions
Nonlinear Programming and the KuhnTucker Conditions The KuhnTucker (KT) conditions are firstorder conditions for constrained optimization problems, a generalization of the firstorder conditions we
More informationNonlinear Optimization: What s important?
Nonlinear Optimization: What s important? Julian Hall 10th May 2012 Convexity: convex problems A local minimizer is a global minimizer A solution of f (x) = 0 (stationary point) is a minimizer A global
More informationOptimality, Duality, Complementarity for Constrained Optimization
Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of WisconsinMadison May 2014 Wright (UWMadison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear
More informationNonlinear Programming (Hillier, Lieberman Chapter 13) CHEME7155 Production Planning and Control
Nonlinear Programming (Hillier, Lieberman Chapter 13) CHEME7155 Production Planning and Control 19/4/2012 Lecture content Problem formulation and sample examples (ch 13.1) Theoretical background Graphical
More informationA Tutorial on Convex Optimization II: Duality and Interior Point Methods
A Tutorial on Convex Optimization II: Duality and Interior Point Methods Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California 94304 email: hhindi@parc.com Abstract In recent years, convex
More informationConvex Optimization and SVM
Convex Optimization and SVM Problem 0. Cf lecture notes pages 12 to 18. Problem 1. (i) A slab is an intersection of two half spaces, hence convex. (ii) A wedge is an intersection of two half spaces, hence
More informationPrimaldual Subgradient Method for Convex Problems with Functional Constraints
Primaldual Subgradient Method for Convex Problems with Functional Constraints Yurii Nesterov, CORE/INMA (UCL) Workshop on embedded optimization EMBOPT2014 September 9, 2014 (Lucca) Yu. Nesterov Primaldual
More informationLecture 6: Conic Optimization September 8
IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions
More informationThe Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem:
HT05: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford Convex Optimization and slides based on Arthur Gretton s Advanced Topics in Machine Learning course
More informationLecture 9 Sequential unconstrained minimization
S. Boyd EE364 Lecture 9 Sequential unconstrained minimization brief history of SUMT & IP methods logarithmic barrier function central path UMT & SUMT complexity analysis feasibility phase generalized inequalities
More information1. f(β) 0 (that is, β is a feasible point for the constraints)
xvi 2. The lasso for linear models 2.10 Bibliographic notes Appendix Convex optimization with constraints In this Appendix we present an overview of convex optimization concepts that are particularly useful
More informationLagrangian Duality Theory
Lagrangian Duality Theory Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapter 14.14 1 Recall Primal and Dual
More informationPart IB Optimisation
Part IB Optimisation Theorems Based on lectures by F. A. Fischer Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationSupport Vector Machines
Support Vector Machines Support vector machines (SVMs) are one of the central concepts in all of machine learning. They are simply a combination of two ideas: linear classification via maximum (or optimal
More informationMore on Lagrange multipliers
More on Lagrange multipliers CE 377K April 21, 2015 REVIEW The standard form for a nonlinear optimization problem is min x f (x) s.t. g 1 (x) 0. g l (x) 0 h 1 (x) = 0. h m (x) = 0 The objective function
More informationTutorial on Convex Optimization for Engineers Part II
Tutorial on Convex Optimization for Engineers Part II M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box 100565 D98684 Ilmenau, Germany jens.steinwandt@tuilmenau.de
More information12. Interiorpoint methods
12. Interiorpoint methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationNetwork Utility Maximization With Nonconcave Utilities Using SumofSquares Method
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 25, 2005 MoC4.6 Network Utility Maximization With Nonconcave Utilities
More informationSupport Vector Machines: Maximum Margin Classifiers
Support Vector Machines: Maximum Margin Classifiers Machine Learning and Pattern Recognition: September 16, 2008 Piotr Mirowski Based on slides by Sumit Chopra and FuJie Huang 1 Outline What is behind
More informationApplications of Linear Programming
Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 9 Nonlinear programming In case of LP, the goal
More informationHomework Set #6  Solutions
EE 15  Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 111 Homework Set #6  Solutions 1 a The feasible set is the interval [ 4] The unique
More informationAlternative Decompositions for Distributed Maximization of Network Utility: Framework and Applications
Alternative Decompositions for Distributed Maximization of Network Utility: Framework and Applications Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470  Convex Optimization
More informationLecture 10: Linear programming duality and sensitivity 00
Lecture 10: Linear programming duality and sensitivity 00 The canonical primal dual pair 1 A R m n, b R m, and c R n maximize z = c T x (1) subject to Ax b, x 0 n and minimize w = b T y (2) subject to
More informationECE Optimization for wireless networks Final. minimize f o (x) s.t. Ax = b,
ECE 788  Optimization for wireless networks Final Please provide clear and complete answers. PART I: Questions  Q.. Discuss an iterative algorithm that converges to the solution of the problem minimize
More informationQuiz Discussion. IE417: Nonlinear Programming: Lecture 12. Motivation. Why do we care? Jeff Linderoth. 16th March 2006
Quiz Discussion IE417: Nonlinear Programming: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th March 2006 Motivation Why do we care? We are interested in
More informationConstrained Optimization
1 / 22 Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University March 30, 2015 2 / 22 1. Equality constraints only 1.1 Reduced gradient 1.2 Lagrange
More informationLectures 9 and 10: Constrained optimization problems and their optimality conditions
Lectures 9 and 10: Constrained optimization problems and their optimality conditions Coralia Cartis, Mathematical Institute, University of Oxford C6.2/B2: Continuous Optimization Lectures 9 and 10: Constrained
More informationMath 273a: Optimization Subgradients of convex functions
Math 273a: Optimization Subgradients of convex functions Made by: Damek Davis Edited by Wotao Yin Department of Mathematics, UCLA Fall 2015 online discussions on piazza.com 1 / 42 Subgradients Assumptions
More information10 Numerical methods for constrained problems
10 Numerical methods for constrained problems min s.t. f(x) h(x) = 0 (l), g(x) 0 (m), x X The algorithms can be roughly divided the following way: ˆ primal methods: find descent direction keeping inside
More informationUtility, Fairness and Rate Allocation
Utility, Fairness and Rate Allocation Laila Daniel and Krishnan Narayanan 11th March 2013 Outline of the talk A rate allocation example Fairness criteria and their formulation as utilities Convex optimization
More informationOptimality Conditions for Constrained Optimization
72 CHAPTER 7 Optimality Conditions for Constrained Optimization 1. First Order Conditions In this section we consider first order optimality conditions for the constrained problem P : minimize f 0 (x)
More informationDuality. Geoff Gordon & Ryan Tibshirani Optimization /
Duality Geoff Gordon & Ryan Tibshirani Optimization 10725 / 36725 1 Duality in linear programs Suppose we want to find lower bound on the optimal value in our convex problem, B min x C f(x) E.g., consider
More informationLECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE
LECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE CONVEX ANALYSIS AND DUALITY Basic concepts of convex analysis Basic concepts of convex optimization Geometric duality framework  MC/MC Constrained optimization
More information2.098/6.255/ Optimization Methods Practice True/False Questions
2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 13 line supporting sentence
More informationNumerical optimization
Numerical optimization Lecture 4 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of nonrigid shapes Stanford University, Winter 2009 2 Longest Slowest Shortest Minimal Maximal
More informationMidterm Review. Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.
Midterm Review Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapter 14, Appendices) 1 Separating hyperplane
More informationSubgradients. subgradients. strong and weak subgradient calculus. optimality conditions via subgradients. directional derivatives
Subgradients subgradients strong and weak subgradient calculus optimality conditions via subgradients directional derivatives Prof. S. Boyd, EE364b, Stanford University Basic inequality recall basic inequality
More informationNumerical optimization. Numerical optimization. Longest Shortest where Maximal Minimal. Fastest. Largest. Optimization problems
1 Numerical optimization Alexander & Michael Bronstein, 20062009 Michael Bronstein, 2010 tosca.cs.technion.ac.il/book Numerical optimization 048921 Advanced topics in vision Processing and Analysis of
More informationCONSTRAINED NONLINEAR PROGRAMMING
149 CONSTRAINED NONLINEAR PROGRAMMING We now turn to methods for general constrained nonlinear programming. These may be broadly classified into two categories: 1. TRANSFORMATION METHODS: In this approach
More informationFinite Dimensional Optimization Part III: Convex Optimization 1
John Nachbar Washington University March 21, 2017 Finite Dimensional Optimization Part III: Convex Optimization 1 1 Saddle points and KKT. These notes cover another important approach to optimization,
More informationShiqian Ma, MAT258A: Numerical Optimization 1. Chapter 4. Subgradient
Shiqian Ma, MAT258A: Numerical Optimization 1 Chapter 4 Subgradient Shiqian Ma, MAT258A: Numerical Optimization 2 4.1. Subgradients definition subgradient calculus duality and optimality conditions Shiqian
More informationLinear and Combinatorial Optimization
Linear and Combinatorial Optimization The dual of an LPproblem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality
More informationConstrained optimization: direct methods (cont.)
Constrained optimization: direct methods (cont.) Jussi Hakanen Postdoctoral researcher jussi.hakanen@jyu.fi Direct methods Also known as methods of feasible directions Idea in a point x h, generate a
More informationRouting. Topics: 6.976/ESD.937 1
Routing Topics: Definition Architecture for routing data plane algorithm Current routing algorithm control plane algorithm Optimal routing algorithm known algorithms and implementation issues new solution
More informationminimize x subject to (x 2)(x 4) u,
Math 6366/6367: Optimization and Variational Methods Sample Preliminary Exam Questions 1. Suppose that f : [, L] R is a C 2 function with f () on (, L) and that you have explicit formulae for
More informationOptimization Theory. Lectures 46
Optimization Theory Lectures 46 Unconstrained Maximization Problem: Maximize a function f:ú n 6 ú within a set A f ú n. Typically, A is ú n, or the nonnegative orthant {x0ú n x$0} Existence of a maximum:
More informationBarrier Method. Javier Peña Convex Optimization /36725
Barrier Method Javier Peña Convex Optimization 10725/36725 1 Last time: Newton s method For rootfinding F (x) = 0 x + = x F (x) 1 F (x) For optimization x f(x) x + = x 2 f(x) 1 f(x) Assume f strongly
More informationUses of duality. Geoff Gordon & Ryan Tibshirani Optimization /
Uses of duality Geoff Gordon & Ryan Tibshirani Optimization 10725 / 36725 1 Remember conjugate functions Given f : R n R, the function is called its conjugate f (y) = max x R n yt x f(x) Conjugates appear
More informationMachine Learning. Lecture 6: Support Vector Machine. Feng Li.
Machine Learning Lecture 6: Support Vector Machine Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 2018 Warm Up 2 / 80 Warm Up (Contd.)
More informationDual methods and ADMM. Barnabas Poczos & Ryan Tibshirani Convex Optimization /36725
Dual methods and ADMM Barnabas Poczos & Ryan Tibshirani Convex Optimization 10725/36725 1 Given f : R n R, the function is called its conjugate Recall conjugate functions f (y) = max x R n yt x f(x)
More information5 Handling Constraints
5 Handling Constraints Engineering design optimization problems are very rarely unconstrained. Moreover, the constraints that appear in these problems are typically nonlinear. This motivates our interest
More informationE5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming
E5295/5B5749 Convex optimization with engineering applications Lecture 5 Convex programming and semidefinite programming A. Forsgren, KTH 1 Lecture 5 Convex optimization 2006/2007 Convex quadratic program
More informationEE364a Review Session 5
EE364a Review Session 5 EE364a Review announcements: homeworks 1 and 2 graded homework 4 solutions (check solution to additional problem 1) scpd phonein office hours: tuesdays 67pm (6507231156) 1 Complementary
More informationDuality Theory of Constrained Optimization
Duality Theory of Constrained Optimization Robert M. Freund April, 2014 c 2014 Massachusetts Institute of Technology. All rights reserved. 1 2 1 The Practical Importance of Duality Duality is pervasive
More informationOptimization. A first course on mathematics for economists
Optimization. A first course on mathematics for economists Xavier MartinezGiralt Universitat Autònoma de Barcelona xavier.martinez.giralt@uab.eu II.3 Static optimization  NonLinear programming OPT p.1/45
More informationMathematical Foundations 1 Constrained Optimization. Constrained Optimization. An intuitive approach 2. First Order Conditions (FOC) 7
Mathematical Foundations  Constrained Optimization Constrained Optimization An intuitive approach First Order Conditions (FOC) 7 Constraint qualifications 9 Formal statement of the FOC for a maximum
More informationConvex Optimization. Prof. Nati Srebro. Lecture 12: InfeasibleStart Newton s Method Interior Point Methods
Convex Optimization Prof. Nati Srebro Lecture 12: InfeasibleStart Newton s Method Interior Point Methods Equality Constrained Optimization f 0 (x) s. t. A R p n, b R p Using access to: 2 nd order oracle
More information4. Algebra and Duality
41 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: nonconvex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone
More information