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

1

Exercises 2.1, 2.9, 2.12a-d and 2.12g, 2.15a-g, 2.28, 3.2, and 3.15 from the textbook.

2

Exercises 3.10, 3.16, 3.24, 3.36a, 4.1, 4.3, 4.4 from the textbook. In addition, try out CVX on the problems in 4.1 and 4.3, checking that results are consistent with your (analytical) solutions.

3

Part 1: Exercises 3.42, 3.48, 3.49a-c, 4.8a-e, and 4.17 from the textbook.

Part 2 (PDF)

4

Part 1: Exercises 3.57, 4.13, 4.16, 4.43, and 5.1 from the textbook.

Part 2 (PDF)

simple_portfolio_data.m (M)

5

Part 1: Exercises 4.15, 5.5, 5.13, and 6.9 from the textbook.

Part 2 (PDF)

6

Part 1: Exercises 5.14, 5.27, 5.38, and 7.1 from the textbook.

Part 2 (PDF)

planning_data.m (M)

7

Part 1: Exercises 7.7 and 8.8 from the textbook.

Part 2 (PDF)

sparse_lds_data.m (M)

team_data.m (M)

sep3way_data.m (M)

8

(PDF)

conf_map_data.m (M)

rel_pwr_flow_data.m (M)

energy_portfolio_data.m (M)

9

(PDF)

nonlin_meas_data.m (M)

10

Part 1: Exercises 9.30 and 9.31 in the textbook.

Part 1 notes (PDF)

Part 2 (PDF)

Midterm (PDF)

fba_data.m (M)

plot_2D_filt.m (M)

trans_ship_data.m (M)

treatment_planning_data.m (M)

Final (PDF)

blkdiag.m (M)

gen_dispatch_data.m (M)

inertia_dens_data.m (M)

rob_min_vol_ellips_data.m (M)

vfield_fit_data.m (M)