Tutorial on Convex Optimization for Engineers Part II

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Tutorial on Convex Optimization for Engineers Part II"

Transcription

1 Tutorial on Convex Optimization for Engineers Part II M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box D Ilmenau, Germany January 2014

2 Outline 1. Introduction 2. Convex sets 3. Convex functions 4. Convex optimization problems 5. Lagrangian duality theory 6. Disciplined convex programming and CVX

3 4. Convex optimization problems

4 Optimization problem in standard form minimize f 0 (x) subject to f i (x) 0, i = 1,...,m h i (x) = 0, i = 1,...,p x R n is the optimization variable f 0 : R n R is the objective or cost function f i : R n R, i = 1,...,m, are the inequality constraint functions h i : R n R are the equality constraint functions optimal value: p = inf{f 0 (x) f i (x) 0, i = 1,...,m, h i (x) = 0, i = 1,...,p} p = if problem is infeasible (no x satisfies the constraints) p = if problem is unbounded below Convex optimization problems 4 2

5 Feasibility problem find x subject to f i (x) 0, i = 1,...,m h i (x) = 0, i = 1,...,p can be considered a special case of the general problem with f 0 (x) = 0: minimize 0 subject to f i (x) 0, i = 1,...,m h i (x) = 0, i = 1,...,p p = 0 if constraints are feasible; any feasible x is optimal p = if constraints are infeasible Convex optimization problems 4 5

6 Convex optimization problem standard form convex optimization problem minimize f 0 (x) subject to f i (x) 0, i = 1,...,m a T i x = b i, i = 1,...,p f 0, f 1,..., f m are convex; equality constraints are affine problem is quasiconvex if f 0 is quasiconvex (and f 1,..., f m convex) often written as minimize f 0 (x) subject to f i (x) 0, i = 1,...,m Ax = b important property: feasible set of a convex optimization problem is convex Convex optimization problems 4 6

7 Equivalent convex problems two problems are (informally) equivalent if the solution of one is readily obtained from the solution of the other, and vice-versa some common transformations that preserve convexity: eliminating equality constraints is equivalent to minimize f 0 (x) subject to f i (x) 0, i = 1,...,m Ax = b minimize (over z) f 0 (Fz +x 0 ) subject to f i (Fz +x 0 ) 0, i = 1,...,m where F and x 0 are such that Ax = b x = Fz +x 0 for some z Convex optimization problems 4 11

8 introducing equality constraints is equivalent to minimize f 0 (A 0 x+b 0 ) subject to f i (A i x+b i ) 0, i = 1,...,m minimize (over x, y i ) f 0 (y 0 ) subject to f i (y i ) 0, i = 1,...,m y i = A i x+b i, i = 0,1,...,m introducing slack variables for linear inequalities is equivalent to minimize f 0 (x) subject to a T i x b i, i = 1,...,m minimize (over x, s) f 0 (x) subject to a T i x+s i = b i, i = 1,...,m s i 0, i = 1,...m Convex optimization problems 4 12

9 epigraph form: standard form convex problem is equivalent to minimize (over x, t) t subject to f 0 (x) t 0 f i (x) 0, i = 1,...,m Ax = b minimizing over some variables is equivalent to where f 0 (x 1 ) = inf x2 f 0 (x 1,x 2 ) minimize f 0 (x 1,x 2 ) subject to f i (x 1 ) 0, i = 1,...,m minimize f0 (x 1 ) subject to f i (x 1 ) 0, i = 1,...,m Convex optimization problems 4 13

10 Linear program (LP) minimize c T x+d subject to Gx h Ax = b convex problem with affine objective and constraint functions feasible set is a polyhedron P x c Convex optimization problems 4 17

11 Examples diet problem: choose quantities x 1,..., x n of n foods one unit of food j costs c j, contains amount a ij of nutrient i healthy diet requires nutrient i in quantity at least b i to find cheapest healthy diet, minimize c T x subject to Ax b, x 0 piecewise-linear minimization equivalent to an LP minimize max i=1,...,m (a T i x+b i) minimize t subject to a T i x+b i t, i = 1,...,m Convex optimization problems 4 18

12 Quadratic program (QP) minimize (1/2)x T Px+q T x+r subject to Gx h Ax = b P S n +, so objective is convex quadratic minimize a convex quadratic function over a polyhedron f 0 (x ) x P Convex optimization problems 4 22

13 Examples least-squares minimize Ax b 2 2 analytical solution x = A b (A is pseudo-inverse) can add linear constraints, e.g., l x u linear program with random cost minimize c T x+γx T Σx = Ec T x+γvar(c T x) subject to Gx h, Ax = b c is random vector with mean c and covariance Σ hence, c T x is random variable with mean c T x and variance x T Σx γ > 0 is risk aversion parameter; controls the trade-off between expected cost and variance (risk) Convex optimization problems 4 23

14 Quadratically constrained quadratic program (QCQP) minimize (1/2)x T P 0 x+q T 0x+r 0 subject to (1/2)x T P i x+q T i x+r i 0, i = 1,...,m Ax = b P i S n +; objective and constraints are convex quadratic if P 1,...,P m S n ++, feasible region is intersection of m ellipsoids and an affine set Convex optimization problems 4 24

15 Second-order cone programming minimize f T x subject to A i x+b i 2 c T i x+d i, i = 1,...,m Fx = g (A i R n i n, F R p n ) inequalities are called second-order cone (SOC) constraints: (A i x+b i,c T i x+d i ) second-order cone in R n i+1 for n i = 0, reduces to an LP; if c i = 0, reduces to a QCQP more general than QCQP and LP Convex optimization problems 4 25

16 Semidefinite program (SDP) with F i, G S k minimize c T x subject to x 1 F 1 +x 2 F 2 + +x n F n +G 0 Ax = b inequality constraint is called linear matrix inequality (LMI) includes problems with multiple LMI constraints: for example, x 1ˆF1 + +x nˆfn +Ĝ 0, x 1 F 1 + +x n Fn + G 0 is equivalent to single LMI x 1 [ ˆF1 0 0 F1 ] +x 2 [ ˆF2 0 0 F2 ] + +x n [ ˆFn 0 0 Fn ] [ Ĝ G ] 0 Convex optimization problems 4 36

17 LP and SOCP as SDP LP and equivalent SDP LP: minimize c T x subject to Ax b SDP: minimize c T x subject to diag(ax b) 0 (note different interpretation of generalized inequality ) SOCP and equivalent SDP SOCP: minimize f T x subject to A i x+b i 2 c T i x+d i, i = 1,...,m SDP: minimize f [ T x (c T subject to i x+d i )I A i x+b i (A i x+b i ) T c T i x+d i ] 0, i = 1,...,m Convex optimization problems 4 37

18 Some nonconvex problems slight modifications of convex problems can be very hard convex maximization, concave minimization: maximize Ax b 2 subject to x 1 nonlinear equality constraints: minimize c T x subject to x T Q ix + qi T x + c i = 0, 1 i K, where Q i 0 minimizing over integer constraints: find x such that Ax b, x i is integer Convex optimization problems

19 5. Lagrangian duality theory

20 Lagrangian standard form problem (not necessarily convex) minimize f 0 (x) subject to f i (x) 0, i = 1,...,m h i (x) = 0, i = 1,...,p variable x R n, domain D, optimal value p Lagrangian: L : R n R m R p R, with doml = D R m R p, L(x,λ,ν) = f 0 (x)+ m λ i f i (x)+ i=1 p ν i h i (x) i=1 weighted sum of objective and constraint functions λ i is Lagrange multiplier associated with f i (x) 0 ν i is Lagrange multiplier associated with h i (x) = 0 Duality 5 2

21 Lagrange dual function Lagrange dual function: g : R m R p R, g(λ,ν) = inf L(x,λ,ν) x D ( = inf x D f 0 (x)+ m λ i f i (x)+ i=1 ) p ν i h i (x) i=1 g is concave, can be for some λ, ν lower bound property: if λ 0, then g(λ,ν) p proof: if x is feasible and λ 0, then f 0 ( x) L( x,λ,ν) inf L(x,λ,ν) = g(λ,ν) x D minimizing over all feasible x gives p g(λ,ν) Duality 5 3

22 Least-norm solution of linear equations dual function minimize x T x subject to Ax = b Lagrangian is L(x,ν) = x T x+ν T (Ax b) to minimize L over x, set gradient equal to zero: x L(x,ν) = 2x+A T ν = 0 = x = (1/2)A T ν plug in in L to obtain g: a concave function of ν g(ν) = L(( 1/2)A T ν,ν) = 1 4 νt AA T ν b T ν lower bound property: p (1/4)ν T AA T ν b T ν for all ν Duality 5 4

23 The dual problem Lagrange dual problem maximize g(λ, ν) subject to λ 0 finds best lower bound on p, obtained from Lagrange dual function a convex optimization problem; optimal value denoted d λ, ν are dual feasible if λ 0, (λ,ν) domg often simplified by making implicit constraint (λ, ν) dom g explicit example: standard form LP and its dual (page 5 5) minimize c T x subject to Ax = b x 0 maximize b T ν subject to A T ν +c 0 Duality 5 9

24 weak duality: d p Weak and strong duality always holds (for convex and nonconvex problems) can be used to find nontrivial lower bounds for difficult problems for example, solving the SDP maximize 1 T ν subject to W +diag(ν) 0 gives a lower bound for the two-way partitioning problem on page 5 7 strong duality: d = p does not hold in general (usually) holds for convex problems conditions that guarantee strong duality in convex problems are called constraint qualifications Duality 5 10

25 Slater s constraint qualification strong duality holds for a convex problem if it is strictly feasible, i.e., minimize f 0 (x) subject to f i (x) 0, i = 1,...,m Ax = b x intd : f i (x) < 0, i = 1,...,m, Ax = b also guarantees that the dual optimum is attained (if p > ) can be sharpened: e.g., can replace intd with relintd (interior relative to affine hull); linear inequalities do not need to hold with strict inequality,... there exist many other types of constraint qualifications Duality 5 11

26 primal problem (assume P S n ++) Quadratic program minimize x T Px subject to Ax b dual function g(λ) = inf x ( x T Px+λ T (Ax b) ) = 1 4 λt AP 1 A T λ b T λ dual problem maximize (1/4)λ T AP 1 A T λ b T λ subject to λ 0 from Slater s condition: p = d if A x b for some x in fact, p = d always Duality 5 13

27 Geometric interpretation for simplicity, consider problem with one constraint f 1 (x) 0 interpretation of dual function: g(λ) = inf (u,t) G (t+λu), where G = {(f 1(x),f 0 (x)) x D} t t G G λu + t = g(λ) p g(λ) u p d u λu+t = g(λ) is (non-vertical) supporting hyperplane to G hyperplane intersects t-axis at t = g(λ) Duality 5 15

28 Complementary slackness assume strong duality holds, x is primal optimal, (λ,ν ) is dual optimal f 0 (x ) = g(λ,ν ) = inf x ( f 0 (x )+ f 0 (x ) f 0 (x)+ m λ if i (x)+ i=1 m λ if i (x )+ i=1 ) p νih i (x) i=1 p νih i (x ) i=1 hence, the two inequalities hold with equality x minimizes L(x,λ,ν ) λ i f i(x ) = 0 for i = 1,...,m (known as complementary slackness): λ i > 0 = f i (x ) = 0, f i (x ) < 0 = λ i = 0 Duality 5 17

29 Karush-Kuhn-Tucker (KKT) conditions the following four conditions are called KKT conditions (for a problem with differentiable f i, h i ): 1. primal constraints: f i (x) 0, i = 1,...,m, h i (x) = 0, i = 1,...,p 2. dual constraints: λ 0 3. complementary slackness: λ i f i (x) = 0, i = 1,...,m 4. gradient of Lagrangian with respect to x vanishes: f 0 (x)+ m λ i f i (x)+ i=1 p ν i h i (x) = 0 i=1 from page 5 17: if strong duality holds and x, λ, ν are optimal, then they must satisfy the KKT conditions Duality 5 18

30 KKT conditions for convex problem if x, λ, ν satisfy KKT for a convex problem, then they are optimal: from complementary slackness: f 0 ( x) = L( x, λ, ν) from 4th condition (and convexity): g( λ, ν) = L( x, λ, ν) hence, f 0 ( x) = g( λ, ν) if Slater s condition is satisfied: x is optimal if and only if there exist λ, ν that satisfy KKT conditions recall that Slater implies strong duality, and dual optimum is attained generalizes optimality condition f 0 (x) = 0 for unconstrained problem Duality 5 19

31 6. Disciplined convex programming and CVX

32 Convex optimization solvers LP solvers lots available (GLPK, Excel, Matlab s linprog,... ) cone solvers typically handle (combinations of) LP, SOCP, SDP cones several available (SDPT3, SeDuMi, CSDP,... ) general convex solvers some available (CVXOPT, MOSEK,... ) plus lots of special purpose or application specific solvers could write your own (we ll study, and write, solvers later in the quarter) Disciplined Convex Programming and CVX 2

33 Transforming problems to standard form you ve seen lots of tricks for transforming a problem into an equivalent one that has a standard form (e.g., LP, SDP) these tricks greatly extend the applicability of standard solvers writing code to carry out this transformation is often painful modeling systems can partly automate this step Disciplined Convex Programming and CVX 3

34 Disciplined convex programming describe objective and constraints using expressions formed from a set of basic atoms (convex, concave functions) a restricted set of operations or rules (that preserve convexity) modeling system keeps track of affine, convex, concave expressions rules ensure that expressions recognized as convex (concave) are convex (concave) but, some convex (concave) expressions are not recognized as convex (concave) problems described using DCP are convex by construction Disciplined Convex Programming and CVX 6

35 CVX uses DCP runs in Matlab, between the cvx_begin and cvx_end commands relies on SDPT3 or SeDuMi (LP/SOCP/SDP) solvers refer to user guide, online help for more info the CVX example library has more than a hundred examples Disciplined Convex Programming and CVX 7

36 Example: Constrained norm minimization A = randn(5, 3); b = randn(5, 1); cvx_begin variable x(3); minimize(norm(a*x - b, 1)) subject to -0.5 <= x; x <= 0.3; cvx_end between cvx_begin and cvx_end, x is a CVX variable statement subject to does nothing, but can be added for readability inequalities are intepreted elementwise Disciplined Convex Programming and CVX 8

37 What CVX does after cvx_end, CVX transforms problem into an LP calls solver SDPT3 overwrites (object) x with (numeric) optimal value assigns problem optimal value to cvx_optval assigns problem status (which here is Solved) to cvx_status (had problem been infeasible, cvx_status would be Infeasible and x would be NaN) Disciplined Convex Programming and CVX 9

38 Some functions function meaning attributes norm(x, p) x p cvx square(x) x 2 cvx square_pos(x) (x + ) 2 cvx, nondecr pos(x) x + cvx, nondecr sum_largest(x,k) x [1] + + x [k] cvx, nondecr sqrt(x) x (x 0) ccv, nondecr inv_pos(x) 1/x (x > 0) cvx, nonincr max(x) max{x 1,..., x n } cvx, nondecr quad_over_lin(x,y) x 2 /y (y > 0) cvx, nonincr in y lambda_max(x) λ max (X) (X = X T ) cvx { x 2, x 1 huber(x) cvx 2 x 1, x > 1 Disciplined Convex Programming and CVX 11

39 References Course Convex Optimization I by Prof. Stephen Boyd at Stanford University, CA. Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge, U.K.: Cambridge University Press, 2004.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

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

Lecture Note 5: Semidefinite Programming for Stability Analysis

Lecture Note 5: Semidefinite Programming for Stability Analysis ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State

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

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics 1. Introduction Convex Optimization Boyd & Vandenberghe mathematical optimization least-squares and linear programming convex optimization example course goals and topics nonlinear optimization brief history

More information

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics 1. Introduction Convex Optimization Boyd & Vandenberghe mathematical optimization least-squares and linear programming convex optimization example course goals and topics nonlinear optimization brief history

More information

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics 1. Introduction mathematical optimization least-squares and linear programming convex optimization example course goals and topics nonlinear optimization brief history of convex optimization 1 1 Mathematical

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

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

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

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

Convex Optimization and Modeling

Convex Optimization and Modeling Convex Optimization and Modeling Convex Optimization Fourth lecture, 05.05.2010 Jun.-Prof. Matthias Hein Reminder from last time Convex functions: first-order condition: f(y) f(x) + f x,y x, second-order

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

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

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 Problems. Prof. Daniel P. Palomar

Convex Optimization Problems. Prof. Daniel P. Palomar Conve Optimization Problems Prof. Daniel P. Palomar The Hong Kong University of Science and Technology (HKUST) MAFS6010R- Portfolio Optimization with R MSc in Financial Mathematics Fall 2018-19, HKUST,

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. Convex optimization problems (part 1: general)

4. Convex optimization problems (part 1: general) EE/AA 578, Univ of Washington, Fall 2016 4. Convex optimization problems (part 1: general) optimization problem in standard form convex optimization problems quasiconvex optimization 4 1 Optimization problem

More information

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

HW1 solutions. 1. α Ef(x) β, where Ef(x) is the expected value of f(x), i.e., Ef(x) = n. i=1 p if(a i ). (The function f : R R is given.

HW1 solutions. 1. α Ef(x) β, where Ef(x) is the expected value of f(x), i.e., Ef(x) = n. i=1 p if(a i ). (The function f : R R is given. HW1 solutions Exercise 1 (Some sets of probability distributions.) Let x be a real-valued random variable with Prob(x = a i ) = p i, i = 1,..., n, where a 1 < a 2 < < a n. Of course p R n lies in the standard

More information

Convex Optimization in Communications and Signal Processing

Convex Optimization in Communications and Signal Processing Convex Optimization in Communications and Signal Processing Prof. Dr.-Ing. Wolfgang Gerstacker 1 University of Erlangen-Nürnberg Institute for Digital Communications National Technical University of Ukraine,

More information

Introduction 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 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 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

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

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

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics 1. Introduction Convex Optimization Boyd & Vandenberghe mathematical optimization least-squares and linear programming convex optimization example course goals and topics nonlinear optimization brief history

More information

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics

1. Introduction. mathematical optimization. least-squares and linear programming. convex optimization. example. course goals and topics 1. Introduction Convex Optimization Boyd & Vandenberghe mathematical optimization least-squares and linear programming convex optimization example course goals and topics nonlinear optimization brief history

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 Overview (cnt d)

Convex Optimization Overview (cnt d) Conve Optimization Overview (cnt d) Chuong B. Do November 29, 2009 During last week s section, we began our study of conve optimization, the study of mathematical optimization problems of the form, minimize

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

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

The Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem:

The 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 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

Lecture Notes on Support Vector Machine

Lecture Notes on Support Vector Machine Lecture Notes on Support Vector Machine Feng Li fli@sdu.edu.cn Shandong University, China 1 Hyperplane and Margin In a n-dimensional space, a hyper plane is defined by ω T x + b = 0 (1) where ω R n is

More information

Primal/Dual Decomposition Methods

Primal/Dual Decomposition Methods Primal/Dual Decomposition Methods Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong Outline of Lecture Subgradients

More information

Support Vector Machines: Maximum Margin Classifiers

Support 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 Fu-Jie Huang 1 Outline What is behind

More information

Optimization for Machine Learning

Optimization 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 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

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

Machine Learning. Lecture 6: Support Vector Machine. Feng Li.

Machine 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 information

Karush-Kuhn-Tucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36-725

Karush-Kuhn-Tucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36-725 Karush-Kuhn-Tucker Conditions Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 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 information

COM S 578X: Optimization for Machine Learning

COM S 578X: Optimization for Machine Learning COM S 578X: Optimization for Machine Learning Lecture Note 4: Duality Jia (Kevin) Liu Assistant Professor Department of Computer Science Iowa State University, Ames, Iowa, USA Fall 2018 JKL (CS@ISU) COM

More information

Duality. Geoff Gordon & Ryan Tibshirani Optimization /

Duality. Geoff Gordon & Ryan Tibshirani Optimization / Duality Geoff Gordon & Ryan Tibshirani Optimization 10-725 / 36-725 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 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 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

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

Subgradients. subgradients and quasigradients. subgradient calculus. optimality conditions via subgradients. directional derivatives

Subgradients. 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 information

Lectures 9 and 10: Constrained optimization problems and their optimality conditions

Lectures 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 information

ELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications

ELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications ELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications Professor M. Chiang Electrical Engineering Department, Princeton University February

More information

Course Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979)

Course Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979) Course Outline FRTN Multivariable Control, Lecture Automatic Control LTH, 6 L-L Specifications, models and loop-shaping by hand L6-L8 Limitations on achievable performance L9-L Controller optimization:

More information

Subgradients. 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 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 information

Generalization to inequality constrained problem. Maximize

Generalization 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 information

Convex Optimization for Signal Processing and Communications: From Fundamentals to Applications

Convex Optimization for Signal Processing and Communications: From Fundamentals to Applications Convex Optimization for Signal Processing and Communications: From Fundamentals to Applications Chong-Yung Chi Institute of Communications Engineering & Department of Electrical Engineering National Tsing

More information

Optimality, Duality, Complementarity for Constrained Optimization

Optimality, Duality, Complementarity for Constrained Optimization Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of Wisconsin-Madison May 2014 Wright (UW-Madison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear

More information

Chapter 2: Linear Programming Basics. (Bertsimas & Tsitsiklis, Chapter 1)

Chapter 2: Linear Programming Basics. (Bertsimas & Tsitsiklis, Chapter 1) Chapter 2: Linear Programming Basics (Bertsimas & Tsitsiklis, Chapter 1) 33 Example of a Linear Program Remarks. minimize 2x 1 x 2 + 4x 3 subject to x 1 + x 2 + x 4 2 3x 2 x 3 = 5 x 3 + x 4 3 x 1 0 x 3

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

Agenda. 1 Cone programming. 2 Convex cones. 3 Generalized inequalities. 4 Linear programming (LP) 5 Second-order cone programming (SOCP)

Agenda. 1 Cone programming. 2 Convex cones. 3 Generalized inequalities. 4 Linear programming (LP) 5 Second-order cone programming (SOCP) Agenda 1 Cone programming 2 Convex cones 3 Generalized inequalities 4 Linear programming (LP) 5 Second-order cone programming (SOCP) 6 Semidefinite programming (SDP) 7 Examples Optimization problem in

More information

Convex Optimization Overview (cnt d)

Convex Optimization Overview (cnt d) Convex Optimization Overview (cnt d) Chuong B. Do October 6, 007 1 Recap During last week s section, we began our study of convex optimization, the study of mathematical optimization problems of the form,

More information

Lecture 7: Weak Duality

Lecture 7: Weak Duality EE 227A: Conve Optimization and Applications February 7, 2012 Lecture 7: Weak Duality Lecturer: Laurent El Ghaoui 7.1 Lagrange Dual problem 7.1.1 Primal problem In this section, we consider a possibly

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

4TE3/6TE3. Algorithms for. Continuous Optimization

4TE3/6TE3. 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 information

Support Vector Machines

Support 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 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

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

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

Semidefinite Programming Basics and Applications

Semidefinite Programming Basics and Applications Semidefinite Programming Basics and Applications Ray Pörn, principal lecturer Åbo Akademi University Novia University of Applied Sciences Content What is semidefinite programming (SDP)? How to represent

More information

minimize x x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x 2 u 2, 5x 1 +76x 2 1,

minimize x x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x 2 u 2, 5x 1 +76x 2 1, 4 Duality 4.1 Numerical perturbation analysis example. Consider the quadratic program with variables x 1, x 2, and parameters u 1, u 2. minimize x 2 1 +2x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x

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

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

Nonlinear Programming and the Kuhn-Tucker Conditions

Nonlinear Programming and the Kuhn-Tucker Conditions Nonlinear Programming and the Kuhn-Tucker Conditions The Kuhn-Tucker (KT) conditions are first-order conditions for constrained optimization problems, a generalization of the first-order conditions we

More information

Robust and Optimal Control, Spring 2015

Robust and Optimal Control, Spring 2015 Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) D. Linear Matrix Inequality D.1 Convex Optimization D.2 Linear Matrix Inequality(LMI) D.3 Control Design and LMI Formulation

More information

Lecture 8. Strong Duality Results. September 22, 2008

Lecture 8. Strong Duality Results. September 22, 2008 Strong Duality Results September 22, 2008 Outline Lecture 8 Slater Condition and its Variations Convex Objective with Linear Inequality Constraints Quadratic Objective over Quadratic Constraints Representation

More information

Convex Optimization. Lecture 12 - Equality Constrained Optimization. Instructor: Yuanzhang Xiao. Fall University of Hawaii at Manoa

Convex Optimization. Lecture 12 - Equality Constrained Optimization. Instructor: Yuanzhang Xiao. Fall University of Hawaii at Manoa Convex Optimization Lecture 12 - Equality Constrained Optimization Instructor: Yuanzhang Xiao University of Hawaii at Manoa Fall 2017 1 / 19 Today s Lecture 1 Basic Concepts 2 for Equality Constrained

More information

Shiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 4. Subgradient

Shiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 4. Subgradient Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 4 Subgradient Shiqian Ma, MAT-258A: Numerical Optimization 2 4.1. Subgradients definition subgradient calculus duality and optimality conditions Shiqian

More information

Optimality Conditions for Constrained Optimization

Optimality 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 information