12. Interior-point methods

Size: px
Start display at page:

Download "12. Interior-point methods"

Transcription

1 12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity analysis via self-concordance generalized inequalities 12 1 Inequality constrained minimization f (x) subject to f i (x), i = 1,..., m (1) f i convex, twice continuously differentiable A R p n with rank A = p we assume p is finite and attained we assume problem is strictly feasible: there exists x with x dom f, f i ( x) <, i = 1,..., m, A x = b hence, strong duality holds and dual optimum is attained Interior-point methods 12 2

2 Examples LP, QP, QCQP, GP entropy maximization with linear inequality constraints subject to n x i log x i F x g with dom f = R n ++ differentiability may require reformulating the problem, e.g., piecewise-linear minimization or l -norm approximation via LP SDPs and SOCPs are better handled as problems with generalized inequalities (see later) Interior-point methods 12 3 Logarithmic barrier reformulation of (1) via indicator function: subject to f (x) + m I (f i (x)) where I (u) = if u, I (u) = otherwise (indicator function of R ) approximation via logarithmic barrier f (x) (1/t) m log( f i(x)) subject to an equality constrained problem for t >, (1/t) log( u) is a smooth approximation of I approximation improves as t u Interior-point methods 12 4

3 logarithmic barrier function φ(x) = log( f i (x)), dom φ = {x f 1 (x) <,..., f m (x) < } convex (follows from composition rules) twice continuously differentiable, with derivatives φ(x) = 2 φ(x) = 1 f i (x) f i(x) 1 f i (x) 2 f i(x) f i (x) T + 1 f i (x) 2 f i (x) Interior-point methods 12 5 Central path for t >, define x (t) as the solution of subject to tf (x) + φ(x) (for now, assume x (t) exists and is unique for each t > ) central path is {x (t) t > } example: central path for an LP c T x subject to a T i x b i, i = 1,..., 6 hyperplane c T x = c T x (t) is tangent to level curve of φ through x (t) c x x (1) Interior-point methods 12 6

4 Dual points on central path x = x (t) if there exists a w such that t f (x) + 1 f i (x) f i(x) + A T w =, therefore, x (t) s the Lagrangian L(x, λ (t), ν (t)) = f (x) + λ i (t)f i (x) + ν (t) T (Ax b) where we define λ i (t) = 1/( tf i(x (t)) and ν (t) = w/t this confirms the intuitive idea that f (x (t)) p if t : p g(λ (t), ν (t)) = L(x (t), λ (t), ν (t)) = f (x (t)) m/t Interior-point methods 12 7 Interpretation via KKT conditions x = x (t), λ = λ (t), ν = ν (t) satisfy 1. primal constraints: f i (x), i = 1,..., m, 2. dual constraints: λ 3. approximate complementary slackness: λ i f i (x) = 1/t, i = 1,..., m 4. gradient of Lagrangian with respect to x vanishes: f (x) + λ i f i (x) + A T ν = difference with KKT is that condition 3 replaces λ i f i (x) = Interior-point methods 12 8

5 Force field interpretation centering problem (for problem with no equality constraints) tf (x) m log( f i(x)) force field interpretation tf (x) is potential of force field F (x) = t f (x) log( f i (x)) is potential of force field F i (x) = (1/f i (x)) f i (x) the forces balance at x (t): F (x (t)) + F i (x (t)) = Interior-point methods 12 9 example c T x subject to a T i x b i, i = 1,..., m objective force field is constant: F (x) = tc constraint force field decays as inverse distance to constraint hyperplane: F i (x) = where H i = {x a T i x = b i} a i b i a T i x, F i(x) 2 = 1 dist(x, H i ) c 3c t = 1 t = 3 Interior-point methods 12 1

6 Barrier method given strictly feasible x, t := t () >, µ > 1, tolerance ɛ >. repeat 1. Centering step. Compute x (t) by minimizing tf + φ, subject to. 2. Update. x := x (t). 3. Stopping criterion. quit if m/t < ɛ. 4. Increase t. t := µt. terminates with f (x) p ɛ (stopping criterion follows from f (x (t)) p m/t) centering usually done using Newton s method, starting at current x choice of µ involves a trade-off: large µ means fewer outer iterations, more inner (Newton) iterations; typical values: µ = 1 2 several heuristics for choice of t () Interior-point methods Convergence analysis number of outer (centering) iterations: exactly log(m/(ɛt () )) log µ plus the initial centering step (to compute x (t () )) centering problem tf (x) + φ(x) see convergence analysis of Newton s method tf + φ must have closed sublevel sets for t t () classical analysis requires strong convexity, Lipschitz condition analysis via self-concordance requires self-concordance of tf + φ Interior-point methods 12 12

7 duality gap Examples inequality form LP (m = 1 inequalities, n = 5 variables) µ = 5 µ = 15 µ = µ starts with x on central path (t () = 1, duality gap 1) terminates when t = 1 8 (gap 1 6 ) centering uses Newton s method with backtracking total number of not very sensitive for µ 1 Interior-point methods geometric program (m = 1 inequalities and n = 5 variables) subject to ( 5 ) log k=1 exp(at k x + b k) ( 5 ) log k=1 exp(at ik x + b ik), i = 1,..., m 1 2 duality gap µ = 15 µ = 5 µ = Interior-point methods 12 14

8 family of standard LPs (A R m 2m ) c T x subject to, x m = 1,..., 1; for each m, solve 1 randomly generated instances m number of iterations grows very slowly as m ranges over a 1 : 1 ratio Interior-point methods Feasibility and phase I methods feasibility problem: find x such that f i (x), i = 1,..., m, (2) phase I: computes strictly feasible starting point for barrier method basic phase I method (over x, s) s subject to f i (x) s, i = 1,..., m (3) if x, s feasible, with s <, then x is strictly feasible for (2) if optimal value p of (3) is positive, then problem (2) is infeasible if p = and attained, then problem (2) is feasible (but not strictly); if p = and not attained, then problem (2) is infeasible Interior-point methods 12 16

9 sum of infeasibilities phase I method 1 T s b i a T i x max subject to s, f i (x) s i, i = 1,..., m for infeasible problems, produces a solution that satisfies many more inequalities than basic phase I method example (infeasible set of 1 linear inequalities in 5 variables) 6 6 number 4 2 number b i a T i x max b i a T i x sum left: basic γ phase I solution; satisfies 39 inequalities γ right: sum of infeasibilities Newton phase I iterations solution; satisfies 79 solutions Infeasible Feasible 1 Interior-point methods example: family of linear inequalities Ax b + γ b 1 data chosen to be strictly feasible for γ >, infeasible for γ 2 use basic phase I, terminate when s < or dual objective is positive Infeasible γ Infeasible Feasible Feasible γ γ 2 1 number of iterations roughly proportional to log(1/ γ ) Interior-point methods 12 18

10 Complexity analysis via self-concordance same assumptions as on page 12 2, plus: sublevel sets (of f, on the feasible set) are bounded tf + φ is self-concordant with closed sublevel sets second condition holds for LP, QP, QCQP may require reformulating the problem, e.g., subject to n x i log x i F x g n x i log x i subject to F x g, x needed for complexity analysis; barrier method works even when self-concordance assumption does not apply Interior-point methods per centering step: from self-concordance theory # µtf (x) + φ(x) µtf (x + ) φ(x + ) γ bound on effort of computing x + = x (µt) starting at x = x (t) γ, c are constants (depend only on Newton algorithm parameters) from duality (with λ = λ (t), ν = ν (t)): + c µtf (x) + φ(x) µtf (x + ) φ(x + ) = µtf (x) µtf (x + ) + log( µtλ i f i (x + )) m log µ µtf (x) µtf (x + ) µt λ i f i (x + ) m m log µ µtf (x) µtg(λ, ν) m m log µ = m(µ 1 log µ) Interior-point methods 12 2

11 total number of (excluding first centering step) log(m/(t () ( ɛ)) m(µ 1 log µ) # N = log µ γ ) + c N figure shows N for typical values of γ, c, m = 1, m t () ɛ = µ confirms trade-off in choice of µ in practice, #iterations is in the tens; not very sensitive for µ 1 Interior-point methods polynomial-time complexity of barrier method for µ = 1 + 1/ m: ( ( )) m m/t () N = O log ɛ number of for fixed gap reduction is O( m) multiply with cost of one Newton iteration (a polynomial function of problem dimensions), to get bound on number of flops this choice of µ optimizes worst-case complexity; in practice we choose µ fixed (µ = 1,..., 2) Interior-point methods 12 22

12 Generalized inequalities f (x) subject to f i (x) Ki, i = 1,..., m f convex, f i : R n R k i, i = 1,..., m, convex with respect to proper cones K i R k i f i twice continuously differentiable A R p n with rank A = p we assume p is finite and attained we assume problem is strictly feasible; hence strong duality holds and dual optimum is attained examples of greatest interest: SOCP, SDP Interior-point methods Generalized logarithm for proper cone ψ : R q R is generalized logarithm for proper cone K R q if: dom ψ = int K and 2 ψ(y) for y K ψ(sy) = ψ(y) + θ log s for y K, s > (θ is the degree of ψ) examples nonnegative orthant K = R n +: ψ(y) = n log y i, with degree θ = n positive semidefinite cone K = S n +: ψ(y ) = log det Y (θ = n) second-order cone K = {y R n+1 (y y 2 n) 1/2 y n+1 }: ψ(y) = log(y 2 n+1 y 2 1 y 2 n) (θ = 2) Interior-point methods 12 24

13 properties (without proof): for y K, ψ(y) K, y T ψ(y) = θ nonnegative orthant R n +: ψ(y) = n log y i ψ(y) = (1/y 1,..., 1/y n ), y T ψ(y) = n positive semidefinite cone S n +: ψ(y ) = log det Y ψ(y ) = Y 1, tr(y ψ(y )) = n second-order cone K = {y R n+1 (y yn) 2 1/2 y n+1 }: y 1 2 ψ(y) = yn+1 2 y2 1. y2 n y n, yt ψ(y) = 2 y n+1 Interior-point methods Logarithmic barrier and central path logarithmic barrier for f 1 (x) K1,..., f m (x) Km : φ(x) = ψ i ( f i (x)), dom φ = {x f i (x) Ki, i = 1,..., m} ψ i is generalized logarithm for K i, with degree θ i φ is convex, twice continuously differentiable central path: {x (t) t > } where x (t) solves subject to tf (x) + φ(x) Interior-point methods 12 26

14 Dual points on central path x = x (t) if there exists w R p, t f (x) + Df i (x) T ψ i ( f i (x)) + A T w = (Df i (x) R ki n is derivative matrix of f i ) therefore, x (t) s Lagrangian L(x, λ (t), ν (t)), where λ i (t) = 1 t ψ i( f i (x (t))), ν (t) = w t from properties of ψ i : λ i (t) K i, with duality gap f (x (t)) g(λ (t), ν (t)) = (1/t) θ i Interior-point methods example: semidefinite programming (with F i S p ) subject to c T x F (x) = n x if i + G logarithmic barrier: φ(x) = log det( F (x) 1 ) central path: x (t) s tc T x log det( F (x)); hence tc i tr(f i F (x (t)) 1 ) =, i = 1,..., n dual point on central path: Z (t) = (1/t)F (x (t)) 1 is feasible for maximize tr(gz) subject to tr(f i Z) + c i =, i = 1,..., n Z duality gap on central path: c T x (t) tr(gz (t)) = p/t Interior-point methods 12 28

15 Barrier method given strictly feasible x, t := t () >, µ > 1, tolerance ɛ >. repeat 1. Centering step. Compute x (t) by minimizing tf + φ, subject to. 2. Update. x := x (t). 3. Stopping criterion. quit if ( P i θ i)/t < ɛ. 4. Increase t. t := µt. only difference is duality gap m/t on central path is replaced by i θ i/t number of outer iterations: log(( i θ i)/(ɛt () )) log µ complexity analysis via self-concordance applies to SDP, SOCP Interior-point methods duality gap 1 2 Examples second-order cone program (5 variables, 5 SOC constraints in R 6 ) duality gap duality gap µ = 5 µ = 2 µ = µ semidefinite program (1 variables, LMI constraint in S 1 ) µ = 15 µ = 5 µ = µ Interior-point methods 12 3

16 family of SDPs (A S n, x R n ) 1 T x subject to A + diag(x) n = 1,..., 1, for each n solve 1 randomly generated instances n 1 3 Interior-point methods Primal-dual interior-point methods more efficient than barrier method when high accuracy is needed update primal and dual variables at each iteration; no distinction between inner and outer iterations often exhibit superlinear asymptotic convergence search directions can be interpreted as Newton directions for modified KKT conditions can start at infeasible points cost per iteration same as barrier method Interior-point methods 12 32

12. Interior-point methods

12. Interior-point methods 12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity

More information

Convex Optimization. Newton s method. ENSAE: Optimisation 1/44

Convex 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 information

Lecture 9 Sequential unconstrained minimization

Lecture 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 information

Interior Point Algorithms for Constrained Convex Optimization

Interior 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 information

Barrier Method. Javier Peña Convex Optimization /36-725

Barrier Method. Javier Peña Convex Optimization /36-725 Barrier Method Javier Peña Convex Optimization 10-725/36-725 1 Last time: Newton s method For root-finding 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 information

5. Duality. Lagrangian

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 information

Convex Optimization M2

Convex 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 information

Convex Optimization and l 1 -minimization

Convex Optimization and l 1 -minimization Convex Optimization and l 1 -minimization Sangwoon Yun Computational Sciences Korea Institute for Advanced Study December 11, 2009 2009 NIMS Thematic Winter School Outline I. Convex Optimization II. l

More information

Convex Optimization Boyd & Vandenberghe. 5. Duality

Convex 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 information

4. Convex optimization problems

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

More information

CSCI 1951-G Optimization Methods in Finance Part 09: Interior Point Methods

CSCI 1951-G Optimization Methods in Finance Part 09: Interior Point Methods CSCI 1951-G Optimization Methods in Finance Part 09: Interior Point Methods March 23, 2018 1 / 35 This material is covered in S. Boyd, L. Vandenberge s book Convex Optimization https://web.stanford.edu/~boyd/cvxbook/.

More information

Lecture: Convex Optimization Problems

Lecture: Convex Optimization Problems 1/36 Lecture: Convex Optimization Problems http://bicmr.pku.edu.cn/~wenzw/opt-2015-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/36 optimization

More information

Advances in Convex Optimization: Theory, Algorithms, and Applications

Advances in Convex Optimization: Theory, Algorithms, and Applications Advances in Convex Optimization: Theory, Algorithms, and Applications Stephen Boyd Electrical Engineering Department Stanford University (joint work with Lieven Vandenberghe, UCLA) ISIT 02 ISIT 02 Lausanne

More information

Lecture: Duality of LP, SOCP and SDP

Lecture: 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 information

Lecture: Duality.

Lecture: Duality. Lecture: Duality http://bicmr.pku.edu.cn/~wenzw/opt-2016-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/35 Lagrange dual problem weak and strong

More information

ICS-E4030 Kernel Methods in Machine Learning

ICS-E4030 Kernel Methods in Machine Learning ICS-E4030 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 information

10 Numerical methods for constrained problems

10 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 information

Lagrange 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) Lagrange Duality Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual

More information

CS-E4830 Kernel Methods in Machine Learning

CS-E4830 Kernel Methods in Machine Learning CS-E4830 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 information

Convex optimization problems. Optimization problem in standard form

Convex optimization problems. Optimization problem in standard form Convex optimization problems optimization problem in standard form convex optimization problems linear optimization quadratic optimization geometric programming quasiconvex optimization generalized inequality

More information

A Brief Review on Convex Optimization

A 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 information

CS711008Z Algorithm Design and Analysis

CS711008Z Algorithm Design and Analysis CS711008Z Algorithm Design and Analysis Lecture 8 Linear programming: interior point method Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 31 Outline Brief

More information

Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization

Extreme 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 well-written and a pleasure to read. The

More information

CSCI : Optimization and Control of Networks. Review on Convex Optimization

CSCI : Optimization and Control of Networks. Review on Convex Optimization CSCI7000-016: 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 information

I.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010

I.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 information

Sparse Optimization Lecture: Basic Sparse Optimization Models

Sparse Optimization Lecture: Basic Sparse Optimization Models Sparse Optimization Lecture: Basic Sparse Optimization Models Instructor: Wotao Yin July 2013 online discussions on piazza.com Those who complete this lecture will know basic l 1, l 2,1, and nuclear-norm

More information

4. Convex optimization problems

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

More information

EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 17

EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 17 EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 17 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory May 29, 2012 Andre Tkacenko

More information

Lecture 16: October 22

Lecture 16: October 22 0-725/36-725: Conve Optimization Fall 208 Lecturer: Ryan Tibshirani Lecture 6: October 22 Scribes: Nic Dalmasso, Alan Mishler, Benja LeRoy Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:

More information

A Tutorial on Convex Optimization II: Duality and Interior Point Methods

A 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 information

Duality. 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 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 information

EE/AA 578, Univ of Washington, Fall Duality

EE/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 information

Primal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization /36-725

Primal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization /36-725 Primal-Dual Interior-Point Methods Ryan Tibshirani Convex Optimization 10-725/36-725 Given the problem Last time: barrier method min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h i, i = 1,...

More information

Primal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization

Primal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization Primal-Dual Interior-Point Methods Ryan Tibshirani Convex Optimization 10-725 Given the problem Last time: barrier method min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h i, i = 1,... m are

More information

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

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

More information

Lagrange duality. The Lagrangian. We consider an optimization program of the form

Lagrange 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 information

Convex Optimization & Lagrange Duality

Convex 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 information

Primal-Dual Interior-Point Methods

Primal-Dual Interior-Point Methods Primal-Dual Interior-Point Methods Lecturer: Aarti Singh Co-instructor: Pradeep Ravikumar Convex Optimization 10-725/36-725 Outline Today: Primal-dual interior-point method Special case: linear programming

More information

Inequality constrained minimization: log-barrier method

Inequality constrained minimization: log-barrier method Inequality constrained minimization: log-barrier method We wish to solve min c T x subject to Ax b with n = 50 and m = 100. We use the barrier method with logarithmic barrier function m ϕ(x) = log( (a

More information

Constrained Optimization and Lagrangian Duality

Constrained 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 information

ELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications

ELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications ELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications Professor M. Chiang Electrical Engineering Department, Princeton University March

More information

Introduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs

Introduction 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 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following

More information

Applications of Linear Programming

Applications of Linear Programming Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 9 Non-linear programming In case of LP, the goal

More information

Lecture: Introduction to LP, SDP and SOCP

Lecture: Introduction to LP, SDP and SOCP Lecture: Introduction to LP, SDP and SOCP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2015.html wenzw@pku.edu.cn Acknowledgement:

More information

10. Unconstrained minimization

10. Unconstrained minimization Convex Optimization Boyd & Vandenberghe 10. Unconstrained minimization terminology and assumptions gradient descent method steepest descent method Newton s method self-concordant functions implementation

More information

Lecture 7: Convex Optimizations

Lecture 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 information

Lecture 14: Optimality Conditions for Conic Problems

Lecture 14: Optimality Conditions for Conic Problems EE 227A: Conve Optimization and Applications March 6, 2012 Lecture 14: Optimality Conditions for Conic Problems Lecturer: Laurent El Ghaoui Reading assignment: 5.5 of BV. 14.1 Optimality for Conic Problems

More information

Convex Optimization. Prof. Nati Srebro. Lecture 12: Infeasible-Start Newton s Method Interior Point Methods

Convex Optimization. Prof. Nati Srebro. Lecture 12: Infeasible-Start Newton s Method Interior Point Methods Convex Optimization Prof. Nati Srebro Lecture 12: Infeasible-Start 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 information

subject to (x 2)(x 4) u,

subject 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 information

Lecture 18: Optimization Programming

Lecture 18: Optimization Programming Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming

More information

Determinant maximization with linear. S. Boyd, L. Vandenberghe, S.-P. Wu. Information Systems Laboratory. Stanford University

Determinant maximization with linear. S. Boyd, L. Vandenberghe, S.-P. Wu. Information Systems Laboratory. Stanford University Determinant maximization with linear matrix inequality constraints S. Boyd, L. Vandenberghe, S.-P. Wu Information Systems Laboratory Stanford University SCCM Seminar 5 February 1996 1 MAXDET problem denition

More information

Lagrangian Duality and Convex Optimization

Lagrangian Duality and Convex Optimization Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DS-GA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?

More information

4. Algebra and Duality

4. Algebra and Duality 4-1 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone

More information

Primal-Dual Interior-Point Methods. Javier Peña Convex Optimization /36-725

Primal-Dual Interior-Point Methods. Javier Peña Convex Optimization /36-725 Primal-Dual Interior-Point Methods Javier Peña Convex Optimization 10-725/36-725 Last time: duality revisited Consider the problem min x subject to f(x) Ax = b h(x) 0 Lagrangian L(x, u, v) = f(x) + u T

More information

14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.

14. 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 information

Lecture 6: Conic Optimization September 8

Lecture 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 information

Convex Optimization and Support Vector Machine

Convex Optimization and Support Vector Machine Convex Optimization and Support Vector Machine Problem 0. Consider a two-class classification problem. The training data is L n = {(x 1, t 1 ),..., (x n, t n )}, where each t i { 1, 1} and x i R p. We

More information

Primal-dual Subgradient Method for Convex Problems with Functional Constraints

Primal-dual Subgradient Method for Convex Problems with Functional Constraints Primal-dual Subgradient Method for Convex Problems with Functional Constraints Yurii Nesterov, CORE/INMA (UCL) Workshop on embedded optimization EMBOPT2014 September 9, 2014 (Lucca) Yu. Nesterov Primal-dual

More information

8. Geometric problems

8. Geometric problems 8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 Minimum volume ellipsoid around a set Löwner-John ellipsoid

More information

On the Method of Lagrange Multipliers

On 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 information

Agenda. Interior Point Methods. 1 Barrier functions. 2 Analytic center. 3 Central path. 4 Barrier method. 5 Primal-dual path following algorithms

Agenda. Interior Point Methods. 1 Barrier functions. 2 Analytic center. 3 Central path. 4 Barrier method. 5 Primal-dual path following algorithms Agenda Interior Point Methods 1 Barrier functions 2 Analytic center 3 Central path 4 Barrier method 5 Primal-dual path following algorithms 6 Nesterov Todd scaling 7 Complexity analysis Interior point

More information

Tutorial on Convex Optimization for Engineers Part II

Tutorial 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 D-98684 Ilmenau, Germany jens.steinwandt@tu-ilmenau.de

More information

EE364a Review Session 5

EE364a 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 phone-in office hours: tuesdays 6-7pm (650-723-1156) 1 Complementary

More information

Homework 4. Convex Optimization /36-725

Homework 4. Convex Optimization /36-725 Homework 4 Convex Optimization 10-725/36-725 Due Friday November 4 at 5:30pm submitted to Christoph Dann in Gates 8013 (Remember to a submit separate writeup for each problem, with your name at the top)

More information

Convex Optimization and Modeling

Convex 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 information

Convex Optimization and SVM

Convex 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 information

15. Conic optimization

15. Conic optimization L. Vandenberghe EE236C (Spring 216) 15. Conic optimization conic linear program examples modeling duality 15-1 Generalized (conic) inequalities Conic inequality: a constraint x K where K is a convex cone

More information

Lecture: Algorithms for LP, SOCP and SDP

Lecture: Algorithms for LP, SOCP and SDP 1/53 Lecture: Algorithms for LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2018.html wenzw@pku.edu.cn Acknowledgement:

More information

Equality constrained minimization

Equality constrained minimization Chapter 10 Equality constrained minimization 10.1 Equality constrained minimization problems In this chapter we describe methods for solving a convex optimization problem with equality constraints, minimize

More information

minimize x subject to (x 2)(x 4) u,

minimize 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 information

Lecture 15 Newton Method and Self-Concordance. October 23, 2008

Lecture 15 Newton Method and Self-Concordance. October 23, 2008 Newton Method and Self-Concordance October 23, 2008 Outline Lecture 15 Self-concordance Notion Self-concordant Functions Operations Preserving Self-concordance Properties of Self-concordant Functions Implications

More information

E5295/5B5749 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 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 information

The Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System

The Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System The Q-parametrization (Youla) Lecture 3: Synthesis by Convex Optimization controlled variables z Plant distubances w Example: Spring-mass system measurements y Controller control inputs u Idea for lecture

More information

10-725/ Optimization Midterm Exam

10-725/ Optimization Midterm Exam 10-725/36-725 Optimization Midterm Exam November 6, 2012 NAME: ANDREW ID: Instructions: This exam is 1hr 20mins long Except for a single two-sided sheet of notes, no other material or discussion is permitted

More information

8. Geometric problems

8. Geometric problems 8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 1 Minimum volume ellipsoid around a set Löwner-John ellipsoid

More information

Lagrangian Duality Theory

Lagrangian 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 14.1-4 1 Recall Primal and Dual

More information

Convex Optimization Lecture 13

Convex Optimization Lecture 13 Convex Optimization Lecture 13 Today: Interior-Point (continued) Central Path method for SDP Feasibility and Phase I Methods From Central Path to Primal/Dual Central'Path'Log'Barrier'Method Init: Feasible&#

More information

1. Gradient method. gradient method, first-order methods. quadratic bounds on convex functions. analysis of gradient method

1. Gradient method. gradient method, first-order methods. quadratic bounds on convex functions. analysis of gradient method L. Vandenberghe EE236C (Spring 2016) 1. Gradient method gradient method, first-order methods quadratic bounds on convex functions analysis of gradient method 1-1 Approximate course outline First-order

More information

Convex Optimization M2

Convex Optimization M2 Convex Optimization M2 Lecture 8 A. d Aspremont. Convex Optimization M2. 1/57 Applications A. d Aspremont. Convex Optimization M2. 2/57 Outline Geometrical problems Approximation problems Combinatorial

More information

Duality revisited. Javier Peña Convex Optimization /36-725

Duality revisited. Javier Peña Convex Optimization /36-725 Duality revisited Javier Peña Conve Optimization 10-725/36-725 1 Last time: barrier method Main idea: approimate the problem f() + I C () with the barrier problem f() + 1 t φ() tf() + φ() where t > 0 and

More information

FRTN10 Multivariable Control, Lecture 13. Course outline. The Q-parametrization (Youla) Example: Spring-mass System

FRTN10 Multivariable Control, Lecture 13. Course outline. The Q-parametrization (Youla) Example: Spring-mass System FRTN Multivariable Control, Lecture 3 Anders Robertsson Automatic Control LTH, Lund University Course outline The Q-parametrization (Youla) L-L5 Purpose, models and loop-shaping by hand L6-L8 Limitations

More information

Analytic Center Cutting-Plane Method

Analytic Center Cutting-Plane Method Analytic Center Cutting-Plane Method S. Boyd, L. Vandenberghe, and J. Skaf April 14, 2011 Contents 1 Analytic center cutting-plane method 2 2 Computing the analytic center 3 3 Pruning constraints 5 4 Lower

More information

Interior Point Methods. We ll discuss linear programming first, followed by three nonlinear problems. Algorithms for Linear Programming Problems

Interior Point Methods. We ll discuss linear programming first, followed by three nonlinear problems. Algorithms for Linear Programming Problems AMSC 607 / CMSC 764 Advanced Numerical Optimization Fall 2008 UNIT 3: Constrained Optimization PART 4: Introduction to Interior Point Methods Dianne P. O Leary c 2008 Interior Point Methods We ll discuss

More information

Optimisation convexe: performance, complexité et applications.

Optimisation convexe: performance, complexité et applications. Optimisation convexe: performance, complexité et applications. Introduction, convexité, dualité. A. d Aspremont. M2 OJME: Optimisation convexe. 1/128 Today Convex optimization: introduction Course organization

More information

Lecture 8 Plus properties, merit functions and gap functions. September 28, 2008

Lecture 8 Plus properties, merit functions and gap functions. September 28, 2008 Lecture 8 Plus properties, merit functions and gap functions September 28, 2008 Outline Plus-properties and F-uniqueness Equation reformulations of VI/CPs Merit functions Gap merit functions FP-I book:

More information

Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem

Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 0-0 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R

More information

Example: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma

Example: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma 4-1 Algebra and Duality P. Parrilo and S. Lall 2006.06.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone of valid

More information

Convex 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 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 information

More First-Order Optimization Algorithms

More First-Order Optimization Algorithms More First-Order Optimization Algorithms Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapters 3, 8, 3 The SDM

More information

Machine Learning. Support Vector Machines. Manfred Huber

Machine Learning. Support Vector Machines. Manfred Huber Machine Learning Support Vector Machines Manfred Huber 2015 1 Support Vector Machines Both logistic regression and linear discriminant analysis learn a linear discriminant function to separate the data

More information

11. Equality constrained minimization

11. Equality constrained minimization Convex Optimization Boyd & Vandenberghe 11. Equality constrained minimization equality constrained minimization eliminating equality constraints Newton s method with equality constraints infeasible start

More information

Lecture 14 Barrier method

Lecture 14 Barrier method L. Vandenberghe EE236A (Fall 2013-14) Lecture 14 Barrier method centering problem Newton decrement local convergence of Newton method short-step barrier method global convergence of Newton method predictor-corrector

More information

Tutorial on Convex Optimization: Part II

Tutorial on Convex Optimization: Part II Tutorial on Convex Optimization: Part II Dr. Khaled Ardah Communications Research Laboratory TU Ilmenau Dec. 18, 2018 Outline Convex Optimization Review Lagrangian Duality Applications Optimal Power Allocation

More information

18. Primal-dual interior-point methods

18. Primal-dual interior-point methods L. Vandenberghe EE236C (Spring 213-14) 18. Primal-dual interior-point methods primal-dual central path equations infeasible primal-dual method primal-dual method for self-dual embedding 18-1 Symmetric

More information

Lecture 24: August 28

Lecture 24: August 28 10-725: Optimization Fall 2012 Lecture 24: August 28 Lecturer: Geoff Gordon/Ryan Tibshirani Scribes: Jiaji Zhou,Tinghui Zhou,Kawa Cheung Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:

More information

Geometric problems. Chapter Projection on a set. The distance of a point x 0 R n to a closed set C R n, in the norm, is defined as

Geometric problems. Chapter Projection on a set. The distance of a point x 0 R n to a closed set C R n, in the norm, is defined as Chapter 8 Geometric problems 8.1 Projection on a set The distance of a point x 0 R n to a closed set C R n, in the norm, is defined as dist(x 0,C) = inf{ x 0 x x C}. The infimum here is always achieved.

More information

EE 227A: Convex Optimization and Applications October 14, 2008

EE 227A: Convex Optimization and Applications October 14, 2008 EE 227A: Convex Optimization and Applications October 14, 2008 Lecture 13: SDP Duality Lecturer: Laurent El Ghaoui Reading assignment: Chapter 5 of BV. 13.1 Direct approach 13.1.1 Primal problem Consider

More information

Homework Set #6 - Solutions

Homework Set #6 - Solutions EE 15 - Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 11-1 Homework Set #6 - Solutions 1 a The feasible set is the interval [ 4] The unique

More information

Written Examination

Written Examination Division of Scientific Computing Department of Information Technology Uppsala University Optimization Written Examination 202-2-20 Time: 4:00-9:00 Allowed Tools: Pocket Calculator, one A4 paper with notes

More information

Solution to EE 617 Mid-Term Exam, Fall November 2, 2017

Solution to EE 617 Mid-Term Exam, Fall November 2, 2017 Solution to EE 67 Mid-erm Exam, Fall 207 November 2, 207 EE 67 Solution to Mid-erm Exam - Page 2 of 2 November 2, 207 (4 points) Convex sets (a) (2 points) Consider the set { } a R k p(0) =, p(t) for t

More information