Lecture Notes

Notes for Lecture 20 are not available on MIT OpenCourseWare.

LEC # TOPICS LECTURE NOTES
1

Introduction

Mathematical optimization; least-squares and linear programming; convex optimization; course goals and topics; nonlinear optimization.

( PDF)
2

Convex sets

Convex sets and cones; some common and important examples; operations that preserve convexity.

( PDF)
3

Convex functions

Convex functions; common examples; operations that preserve convexity; quasiconvex and log-convex functions.

( PDF)
4

Convex optimization problems

Convex optimization problems; linear and quadratic programs; second-order cone and semidefinite programs; quasiconvex optimization problems; vector and multicriterion optimization.

( PDF)
5

Duality

Lagrange dual function and problem; examples and applications.

( PDF)
6

Approximation and fitting

Norm approximation; regularization; robust optimization.

( PDF)
7

Statistical estimation

Maximum likelihood and MAP estimation; detector design; experiment design.

( PDF)
8

Geometric problems

Projection; extremal volume ellipsoids; centering; classification; placement and location problems.

( PDF)
9

Filter design and equalization

FIR filters; general and symmetric lowpass filter design; Chebyshev equalization; magnitude design via spectral factorization.

( PDF)
10

Miscellaneous applications

Multi-period processor speed scheduling; minimum time optimal control; grasp force optimization; optimal broadcast transmitter power allocation; phased-array antenna beamforming; optimal receiver location.

( PDF)
11

l1 methods for convex-cardinality problems

Convex-cardinality problems and examples; l1 heuristic; interpretation as relaxation.

( PDF)
12

l1 methods for convex-cardinality problems (cont.)

Total variation reconstruction; iterated re-weighted l1; rank minimization and dual spectral norm heuristic.

( PDF - 1.4MB)
13

Stochastic programming

Stochastic programming; “certainty equivalent” problem; violation/shortfall constraints and penalties; Monte Carlo sampling methods; validation.

( PDF)
14

Chance constrained optimization

Chance constraints and percentile optimization; chance constraints for log-concave distributions; convex approximation of chance constraints.

( PDF)
15

Numerical linear algebra background

Basic linear algebra operations; factor-solve methods; sparse matrix methods.

( PDF)
16

Unconstrained minimization

Gradient and steepest descent methods; Newton method; self-concordance complexity analysis.

( PDF)
17

Equality constrained minimization

Elimination method; Newton method; infeasible Newton method.

( PDF)
18

Interior-point methods

Barrier method; sequential unconstrained minimization; self-concordance complexity analysis.

( PDF)
19

Disciplined convex programming and CVX

Convex optimization solvers; modeling systems; disciplined convex programming; CVX.

( PDF)
20 Conclusions  

Course Info

Learning Resource Types

grading Exams
notes Lecture Notes
assignment Programming Assignments