CS675: Convex and Combinatorial Optimization (Fall 2014)
- Lecture time: Tuesdays and Thursdays 11 am - 12:20 pm
- Lecture place: VKC 151
- Instructor: Shaddin Dughmi
- Email: email@example.com
- Office: SAL 234
- Office Hours: Thursday 1:30pm - 2:30pm
- TA: Ruixin Qiang
- Email: firstname.lastname@example.org
- Office Hours: Monday 3pm-4pm in LVL 3B
- Course Homepage: www-bcf.usc.edu/~shaddin/cs675fa14
- Dec 2: HW3 solutions are out.
- Nov 17: HW4 is out. It is due on Tuesday Dec 2 at the beginning of class. Shaddin reserves the right to add one or two more problems over the next few days, based on what transpires in class.
- Oct 28: HW2 solutions are out.
- Oct 23: HW3 is out. It is due on Tuesday Nov 11 at the beginning of class.
- Oct 7: Project instructions are up.
- Sep 22: HW1 solutions are out.
- Sep 21: HW2 is out. It is due on Thursday Oct 9th at the beginning of class.
- Sep 5: HW1 is out. It is due on Thursday Sep 18th at the beginning of class.
- Aug 29: We have a new room (again): VKC 151
- Aug 29: Shaddin's office hours have been set. See above.
- Aug 29: You should receive email from me in the next few days. If you don't, please email Ruixin with your contact info.
- Aug 29: Course website is up!
Schedule by Week
- Week 1: Introduction to Optimization. Linear Programming and Duality.
- Week 2: Wrapping up LP duality. Convex Sets.
- Week 3: Convex Functions. Geometric Duality
- Week 4: Convex optimization Problems
- Reading: BV Chapter 4
- Homework 1 Due
- Homework 2 released
- Week 5: Duality of Convex Optimization Problems. KKT conditions.
- Weeks 6-7: Combinatorial problems as linear and convex programs
- Weeks 8-9: Algorithms: Simplex method, ellipsoid method and its consequences.
- Slides: Simplex, Ellipsoid, Consequences of Ellipsoid.
- Reading: Korte Vygen Chapter 3 (Simplex Algorithm) and Chapter 4 (Ellipsoid Algorithm).
- Project team selection and proposal due by end of week 8
- Homework 3 released in week 8
- Additional Reading: Luenberger and Ye Chapter 3 and Vince Conitzer's lecture notes for an algebraic treatment of the simplex method
- Highly recommended: Ryan O'Donnell's lecture notes on the Ellipsoid method: here and here.
- Ben Tal and Nemirovski's lecture notes on the ellipsoid method and polynomial sovability of convex programming (Chapter 8). These lecture notes describe the most useful and general polynomial-solvability guarantee for convex programming which I could find, and Shaddin is puzzled as to why this isn't the standard "solvability statement" presented in optimization textbooks.
- For equivalence of separation and optimization, I recommend the impeccably-written breakthrough paper by Grotschel, Lovasz and Schrijver. You may have to use the USC library proxy to get access, or find it elsewhere online.
- Week 10-11: Matroid theory. Optimization over matroids and matroid intersections.
- Week 12-13: Submodular functions and optimization.
- Homework 3 due early week 12
- Homework 4 by end of week
- Lecture notes by Jan Vondrak (lectures 16-19)
- A survey article by your instructor on continuous extensions of submodular functions and their algorithmic applications.
- The original paper by Laszlo Lovasz on submodular functions and convexity. You may have to use the USC library proxy to get access, or find it elsewhere online.
- The Calinescu et al paper on maximizing a monotone submodular functions subject to a matroid constraint.
- The recent paper by Buchbinder et al on unconstrained maximization of non-monotone submodular functions.
- Slides from a recent tutorial by Jan Vondrak, with an overview of many of the "state of the art" results.
- For those interested in applications of submodularity to machine learning, see the materials and references on this page maintained by Andreas Krause and Carlos Guestrin.
- Week 14: Semidefinite Programming and Constraint Satisfaction Problems
- Week 15: Additional topics
Over the past half century or so, computer science and mathematical optimization have witnessed the development and maturity of two different paradigms for algorithm design.
The first approach, most familiar to computer scientists, is combinatorial in nature. The tools of discrete mathematics are used to understand the structure of the problem, and algorithms effectively exploit this structure to search over a large yet finite set of possible solutions. The second approach, standard in much of the operations research and mathematical optimization communities, primarily employs the tools of continuous mathematics, high dimensional geometry, and convex analysis. Problems are posed as a search over a set of points in high-dimensional Euclidean space, which can be performed efficiently when the search space and objective function are ``convex.''
Whereas many optimization problems are best modeled either as a discrete or convex optimization problem, researchers have increasingly discovered that many problems are best tackled by a combination of combinatorial and continuous techniques. The ability to seamlessly transition between the two views has become an important skill to every researcher working in algorithm design and analysis. This course intends to instill this skill by presenting a unified treatment of both approaches, focusing on algorithm design tasks that employ techniques from both. The intended audience for this course are PhD students, Masters students, and advanced undergraduates interested in research questions in algorithm design, mathematical optimization, or related disciplines.
- Prerequisite Courses: CSCI 570 or CSCI 670 or permission of instructor.
- Recommended Preparation: Mathematical maturity and a solid grounding in linear algebra.
Requirements and Grading
Homework assignments will count for 75% of the grade. There will be 4 assignments, which will be proof-based, and are intended to be very challenging. Collaboration and discussion among students is allowed, even encouraged, though students must write up their solutions independently.
The remaining 25% of the grade will be allocated to a final project. Students will have to choose a related research topic, read several papers in that area, and write a survey of the area.
Late Homework Policy: Students will be allowed 3 late days for homework, to be used in integer amounts and distributed as the student sees fit. No additional late days are allowed.
We will refer to two main texts: Convex Optimization by Boyd and Vandenberghe, available free online, and Combinatorial Optimization by Korte and Vygen, available online through USC libraries. Additional references include Combinatorial Optimization by Schrijver, Linear and Nonlinear Programming by Luenberger and Ye, available online through usc libraries, as well as research papers and lecture notes from related courses elsewhere which will be linked on the course website.