## Convex and Conic Optimization

### Spring 2021, Princeton University (graduate course)

### (This is the Spring 2021 version of this course. For previous versions, click here.)

#### Useful links

- Zoom (password has been emailed to registered students)
- Lectures (Tue/Thu 1:30pm-2:50pm EST). Join here.
- You can follow our live notes during lecture.
- All office hours (see schedule here). Join here.

- Piazza (used for Q&A )
- The course syllabus
- Course description
- Blackboard
- Download MATLAB
- Download CVX
- Download YALMIP

#### References

- A. Ben-Tal and A. Nemirovski, Lecture Notes on Modern Convex Optimization [link]
- S. Boyd and L. Vandenberghe, Convex Optimization [link]
- M. Laurent and F. Vallentin, Semidefinite Optimization [link]
- R. Vanderbei, Linear Programming and Extentions [link]

#### Lectures

The lecture notes below summarize most of what I cover on the whiteboard during class. Please complement them with your own notes. Some lectures take one class session to cover, some others take two.

- Lecture 1: A taste of P and NP: scheduling on Doodle + maximum cliques and the Shannon capacity of a graph.

[pdf] - Lecture 2: Mathematical background.

[pdf] - Lecture 3: Local and global minima, optimality conditions, AMGM inequality, least squares.

[pdf]

- Lecture 4: Convex sets and functions, epigraphs, quasiconvex functions, convex hullls, Caratheodory's theorem, convex optimization problems.

[ pdf ], [cvx_examples.m] - Lecture 5: Separating hyperplane theorems, the Farkas lemma, and strong duality of linear programming.

[ pdf ] - Lecture 6: Bipartite matching, minimum vetex cover, Konig's theorem, totally unimodular matrices and integral polyhedra.

[ pdf ]

- Lecture 7: Characterizations of convex functions, strict and strong convexity, optimality conditions for convex problems.

[ pdf ]

- Lecture 8: Convexity-preserving rules, convex envelopes, support vector machines.

[ pdf ]

- Lecture 9: LP, QP, QCQP, SOCP, SDP.

[pdf] - Lecture 10: Some applications of SDP in dynamical systems and eigenvalue optimization.

[pdf] - Lecture 11: Some applications of SDP in combinatorial optimization: stable sets, the Lovasz theta function, and Shannon capacity of graphs.

[pdf] - Lecture 12: Nonconvex quadratic optimization and its SDP relaxation, the S-Lemma.

[pdf] - Lecture 13: Computational complexity in numerical optimization.

[pdf] - Lecture 14: Complexity of local optimization, the Motzkin-Straus theorem, matrix copositivity.

[pdf] - Lecture 15: Sum of squares programming and relaxations for polynomial optimization.

[pdf]

[YALMIP_Demos] - Lecture 16: Robust optimization.

[pdf] - Guest lecture: Introduction to optimal control. (William Pierson Field Lecture by Sumeet Singh, Google Brain.)

[pdf] - Lecture 17: Convex relaxations for NP-hard problems with worst-case approximation guarantees.

[pdf] - Lecture 18: Approximation algorithms (ctnd.), limits of computation, concluding remarks.

[pdf]

#### Problem sets and exams

Solutions are posted on Blackboard.

- Homework 1: Image compression and SVD, matrix norms, existence of optimal solutions, descent directions, dual and induced norms, properties of positive semidefinite matrices.

[pdf], [conway.jpg] - Homework 2: Convex analysis true/false questions, optimal control, theory-applications split in a course.

[pdf] - Homework 3: Support vector machines (Hillary or Bernie?), norms defined by convex sets, totally unimodular matrices, radiation treatment planning.

[pdf], [Hillary_vs_Bernie]

- Practice midterms.

See Blackboard.

- Midterm:

[pdf]

- Homework 4: A nuclear program for peaceful reasons, distance geometry, stability of a pair of matrices, SDPs with rational data and irrational feasible solutions.

[pdf] - Homework 5: The Lovasz sandwich theorem, SDP and LP relaxations for the stable set problem, Shannon capacity.

[pdf], [Graph.mat]

- Homework 6: Equivalence of search and decision, complexity of rank-constrained SDPs, monotone and convex regression with SOS optimization, Zoom's stock and distributionally-robust optimization.

[pdf], [regression_data.mat]

- Practice final exams.

See Blackboard.

- Final exam:

[pdf], [dependency_matrix.mat]