Course Schedule

High level plan: Week 1-6 = "Theory + toy applications", Week 7-8 = "Serious applications", Week 9-10 = "Algorithms"

Low level plan (tentative, subject to change):

Week 1: Introduction (9/27, 10/02)

Course logistics, usage of parser software (CVX in MATLAB, CVXPY in Python, Convex.jl in Julia), review of vectors, matrices and functions of them. [Lecture 1] [Lecture 2]

Week 2: Convex sets (10/04, 10/09) [Chap. 2]

Affine and convex sets, some important examples, operations that preserve set convexity, generalized inequalities, separating and supporting hyperplanes, dual cones. [Lecture 3] [Lecture 4]

Week 3: Convex functions (10/11, 10/16, 10/18) [Chap. 3]

Properties, examples, operations that preserve function convexity, Fenchel conjugate, friends of convex: quasi-convex, log-convex, operator convex. [Lecture 5] [Lecture 6] [Lecture 7]

Additional material supporting the lecture notes:

Geometric interpretation of the convex conjugate: Link 1 (both Legendre and Legendre-Fenchel transform) and Link 2 (Legendre transform)

Pseudo-convex functions

Week 4: Convex optimization problems (10/23, 10/25, 10/30) [Chap. 4]

Standard form and transformations, LP, QP, QCQP, GP, CP, SOCP, SDP. [Lecture 8] [Lecture 9] [Lecture 10]

In-class Midterm Exam (11/01) [Midterm Solutions] [Lecture 10.5]

Week 5: Duality (11/06, 11/08) [Chap. 5]

Lagrange dual function, Lagrange dual problem, weak and strong duality, Slater's condition, complimentary slackness, KKT conditions, Lagrange duality with generalized inequality constraints. [Lecture 11] [Lecture 12]

Proof that negative log det is convex over positive definite cone

Week 6: Subgradient calculus (11/13)

Subdifferential and subgradient, weak and strong subgradient calculus; operations: scaling, sum and integral, expectation, affine transformation of domain, pointwise max, pointwise sup; duality and optimality conditions. [Lecture 13]

Week 7: Approximation and statistical estimation problems [Chap. 6 and 7]

Approximation problems: norm approximation, least norm problems, regularized approximation, robust approximation, function approximation and interpolation. (11/15)  [Lecture 14]

Statistical estimation problems: parametric: ML and MAP estimation, nonparametric: ML, MaxEnt, MinKL estimation. (11/20)  [Lecture 15]

Week 8: Geometric problems (11/27,11/29) [Chap. 8]

Projection on a set, distance between sets, Euclidean distance and angle problems, extremal volume ellipsoids [Lecture 16] [Lecture 17]

Week 9: Miscellaneous problems (12/04)

Classification problems, image in-painting, miscellaneous cvx examples [Lecture 18]

Week 10: Algorithms for non-smooth convex optimization (contd.) (12/06)

Basic subgradient algorithm, convergence, Polyak’s step length, alternating projections, projected subgradient 

Take-home Final Exam (assigned on 12/07, due on 12/14)

Final Solution