Convex and Conic Optimization

Spring 2021, Princeton University (graduate course)

This is the Spring 2021 version of this course. See the current version, or the previous versions.

Useful links

• Zoom (password has been emailed to registered students)
  • Lectures (Tue/Thu 1:30pm-2:50pm EST).
  • You can follow our live notes during lecture.
  • Office hours schedule and remote office hours link.
• 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]