769F: Convex Optimization

Course Goals:

Convex analysis and optimization have been well studied for several decades, as a subdiscipline of applied mathematics. In recent years, the interest in convex optimization has surged, mainly for two reasons: (i) the development, since the mid-1980s, of interior-point methods, which proved very powerful for the solution of a large class of convex optimization problems; (ii) the realization that many more practical problems than previously thought can be cast as convex optimization problems. The goal of this course is to introduce students to the state of the art in convex optimization, in particular to the fundamentals of interior-point methods, enabling them to pursue research in convex optimization, and to apply recent and future research results to specific application areas.

Course Prerequisite(s):

ENEE 664, or equivalent.

Topics Prerequisite(s):

The course assumes a solid background in linear algebra and advanced calculus, preferably including elements of convex analysis and of optimization theory.

Textbook(s)

No textbook will be used. A good reference for much of the course is:
  • Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
    Another key reference will likely be
  • Arkadi Nemirovski, Interior Point Polynomial Time Methods in Convex Programming, Lecture notes, http://www2.isye.gatech.edu/~nemirovs/ .

    • Reference(s):

  • Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
  • Dimitri P. Bertsekas, with Angela Nedic and Asuman E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.
  • Jonathan M. Borwein and Adrian S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, Springer-Verlag, 2000.
  • R. Tyrrell Rockafellar, Convex Analysis, Princeton University Press, 1972.
  • Stephen J. Wright, Primal-Dual Interior-Point Methods, SIAM, 1997.

    • Tentative Topics:

    1. Generalities
      • Convex Sets
      • Convex Functions
      • Convex Optimization Problems
      • Duality
    2. Algorithms: Preliminaries
      • Unconstrained Problems
      • Equality Constrained Problems
      • Cutting Plane Methods
    3. Algorithms: Interior-Point Methods
      • Barrier Method
      • Primal-Dual Methods
      • Linear Programming
      • Semi-Definite Programming
    4. Some Applications

    Course Structure:

    Lectures

    Grading Method:

    Homework 40%;
    Project 40%;
    Instructor's general impression: 20%.

    Last Updated:

    6 July 2006, Andre Tits