CS675: Convex and Combinatorial Optimization (Fall 2019)

Basic information


Schedule by Week

Course Description

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.


Requirements and Grading

Homework assignments will count for 75% of the grade. There will be 4-6 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 5 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, Fifth edition by Korte and Vygen, available online through USC libraries. Additional references include Combinatorial Optimization by Schrijver, Linear and Nonlinear Programming by Luenberger and Ye, Fourth edition, 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.