Improved Maximum
Approximation
Algorithms
for Using
Cut and Satisfiability Programming
Problems
Semidefinite
MIC13EL
X.
GOEMANS
Massachusetts Institute of Technology, Cambridge, Massachusetts
AND DAVID
IBM
P. WILLIAMSON
T. J. Watson Research Center, Yorkt
UC Berkeley Department of Electrical Engineering and Computer Sciences Department of Statistics EE 227A / STAT 260 : NONLINEAR AND CONVEX OPTIMIZATION Problem Set 2 Fall 2004 Issued: Thursday, September 16, 2004 Due: Tuedsay, September 28, 2004
Problem 2.
498
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 2, FEBRUARY 2001
Factor Graphs and the Sum-Product Algorithm
Frank R. Kschischang, Senior Member, IEEE, Brendan J. Frey, Member, IEEE, and Hans-Andrea Loeliger, Member, IEEE
AbstractAlgorithms that
i
Lecture Notes on Optimization
Pravin Varaiya
ii
Contents
1 INTRODUCTION
1
2 OPTIMIZATION OVER AN OPEN SET
7
3 Optimization with equality constraints
15
4 Linear Programming
27
5 Nonlinear Programming
49
6 Discrete-time optimal control
75
7 Continuous-
SIAM J. OPTIM. Vol. 9, No. 1, pp. 3352
c 1998 Society for Industrial and Applied Mathematics
ROBUST SOLUTIONS TO UNCERTAIN SEMIDEFINITE PROGRAMS
LAURENT EL GHAOUI , FRANCOIS OUSTRY , AND HERVE LEBRET Abstract. In this paper we consider semidenite program
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 24 December 2
Lecturer: Martin Wainwright Scribe: Jon P. Entwistle
Lecture Overview
Barrier and Interior Point Methods
24.1
Barrier and Interior Point Methods
min f (x) subject to gj
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 23 November 29
Lecturer: Martin Wainwright Scribe: Bobak A. Nazer, Benjamin I. P. Rubinstein
Announcements
Homework #6 problems to be graded: 6.1, 6.3, 6.4 and 6.6 Project write-up (
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 22 November 23
Lecturer: Martin Wainwright Scribe: M Moskewicz & N R Satish
22.1
Outline
SDP Relaxations Log-determinant problems
22.2
22.2.1
SDP Relaxations
Formulation
Let Q Sn Con
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 21 November 18
Lecturer: Martin Wainwright Scribe: Wei He, Paolo Minero, Albert To
21.1
Semidenite programs
Any semidenite program (SDP) can be expressed in either of the following fo
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 20 November 16
Lecturer: Martin Wainwright Outline: Lagrangian Duality for Conic Program (CP) Semidenite Program (SDP) Scribe: Xiaoyan Liu and Li Yang
20.1
Lagrangian Duality for Coni
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 19 November 09
Lecturer: Martin Wainwright Scribe: Yuen-Hui Chee, Jianhui Zhang
Second-order cone program.
19.1
Second-order cone programs (SOCP)
K = cfw_(x, t) | x t, x Rn , t R.
Rec
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 18 November 4
Lecturer: Martin Wainwright Scribe: Pannag R Sanketi
Homework # 5 due on Tuesday, Nov 9. See the website for instructions on connecting and using the discussion forum fo
EE227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 18 November 4
Lecturer: Martin Wainwright Scribe: Mark Hoemmen
18.1
Feasibility problems
For the functions g1 , . . . , gr : Rn R, consider the inequality constraints gi (x) 0, i = 1,
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 17 Nov 2
Lecturer: Martin Wainwright Scribe: Zile Wei
17.1
Geometric view of duality
min f (x),
We are focusing on optimization problems with only inequality constraints. Consider:
xR
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 16 October 26
Lecturer: Martin Wainwright Scribe: Thaisiri Watewai
Outline
1. Lagrangian Duality 2. Examples and Reformulations 3. Geometric Interpretation of Weak Duality
16.1
16.1.1
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 15 October 21
Lecturer: Martin Wainwright Scribe: Jane A. Student
Outline:
1. conjugate duality in applications; examples 2. Fenchels inequality 3. Lagrangian duality
15.1
Conjugate d
STAT260 / EE 227A: Convex Optimization
Fall 2004
Lecture 14 October 19
Lecturer: Martin Wainwright Scribe: Guilherme V. Rocha
Outline
Conjugate functions Denition and examples Properties Application to large deviations
14.1
14.1.1
Conjugate Functions
Den
EE 227A / CS 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 13 October 14
Lecturer: Martin Wainwright Scribe: Walter Richter , Ko Boakye
HW 4 distributed last Tuesday, due Thursday October, 21 2004. Problem 4.5 is optional (but highly recommen
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 12 October 19
Lecturer: Martin Wainwright Scribe: Yan Huang
12.1
Consequences of Separation
In this lecture, we begin by exploring some consequences of the separation principle develo
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 12 October 12
Lecturer: Martin Wainwright Scribe: Sivakumar Rathinam
12.1
Consequences of Separations
In this lecture, we begin by exploring some consequences of the separation princi
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 11 October 7
Lecturer: Martin Wainwright Scribe: Farhana Sheikh
Homework #4 will be handed out next Tuesday (Oct 12) Homework #3 problems to be graded are posted on the class homepag
LP Decoding
Jon Feldman
Industrial Engineering and Operations Research Columbia University, New York, NY, 10027 jonfeld@ieor.columbia.edu
David R. Karger
Laboratory for Computer Science MIT, Cambridge, MA, 02139 karger@theory.lcs.mit.edu
Martin J. Wainwri
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 10 October 5
Lecturer: Martin Wainwright Scribe: Yeonsik Kang
10.1
Karush-Kuhn-Tucker(KKT) Conditions
For a regular local minimum x of a ICP problem, there exists a unique vector Rm a
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 9 September 28
Lecturer: Martin Wainwright Scribe: Jingyi Shao
9.1
9.1.1
First-order Necessary Conditions
Recap from last lecture
Recall from last lecture, we were interested in solvi
EE 227A: Nonlinear and Convex Optimization
Fall 2004
Lecture 8 September 23
Lecturer: Martin Wainwright Scribe: Lenny Grokop and Minghua Chen
Lagrange multipliers Necessary conditions
8.1
Lagrangian theory and methods
hi (x) = 0, i = 1, . . . , m gj (x)
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 7 September 21
Lecturer: Martin Wainwright Scribe: David Bindel
7.1
Projection onto a convex set
Let X Rn be convex, closed, and non-empty. Recall that for z Rn , we dened PX (z), cal
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 6 September 16
Lecturer: Martin Wainwright Scribe: Kenneth Hsu and Sandipan Mishra
6.1
Outline
1. Constrained Optimization 2. Projections onto a Convex Set
6.2
Constrained Optimizatio
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 4 9 Sep 2004
Lecturer: Martin Wainwright Outline: Recap. Problem(s) with successive reduction. The Armijo rule for the generalized gradient method. Convergence of the gradient method.
EE 227A / STAT 260: Nonlinear and Convex Optimization
Fall 2004
Lecture 3 September 7
Lecturer: Martin Wainwright Scribe: Salman Avestimehr
3.1
Recap
Let f : S R be continuously dierentiable on some open set S. The rst order condition, f (x ) = 0, is a ne