MS&E 311 Optimization Winter 2007-2008

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The field of optimization is concerned with the study of maximization and minimization of mathematical functions. Very often the arguments of (i.e., variables in) these functions are subject to side conditions or constraints. By virtue of its great utility in such diverse areas as applied science, engineering, economics, finance, medicine, and statistics, optimization holds an important place in the practical world and the scientific world. Indeed, as far back as the Eighteenth Century, the famous Swiss mathematician and physicist Leonhard Euler (1707-1783) proclaimed that ... nothing at all takes place in the Universe in which some rule of maximum or minimum does not appear. The subject is so pervasive that we even find some optimization terms in our everyday language.

Optimization is so large a subject that it cannot adequately be treated in the short amount time available in one quarter of an academic year. In this course, we shall restrict our attention mainly to some aspects of nonlinear programming and discuss linear programming as a special case. Among the many topics that will not be covered in this course are integer programming, network programming, and stochastic programming.

Optimization often goes by the name Mathematical Programming (MP). The latter name tends to be used in conjunction with finite-dimensional optimization problems, which in fact are what we shall be studying here. The word "Programming" should not be confused with computer programming which in fact it antedates. As originally used, the term refers to the timing and magnitude of actions to be carried out so as to achieve a goal in the best possible way. Mathematical Programming is one of the central quantitative decision models in Management Science and Operations Research. Highlights of this year's topics are Information Aggregation, Economic Equilibrium, Pricing Model, Core of Game, Financial Decision and Risk Management, Graph Realization, and their Computations, which you would learn during the process of the course.

Course Outline

• Part I: Math Reviews and Math. Prog. Models
  • Math. Prog. Introduction
  • Mathematical Preliminaries
  • Math. Prog. Models and Applications: financial portfolio management, support vector machines, Fisher's pricing model, Arrow-Debreu's equilibrium, combinatorial auctions, computational core of games, robust network design, etc.

• Part II: Math. Prog. Theories
  • Elements of convex analysis
  • Geometries of Math. Prog.
  • First- and second-order optimality criteria
  • Primal, dual, and duality theory
  • Sensitivity analysis
  • Duality applications

• Part III: Math. Prog. Algorithms
  • Unconstrained optimization:
    • Basic descent methods
    • Newton's method
  • Linearly constrained optimization:
    • Simplex methods
    • Interior-point algorithms and barrier methods
    • Gradient-based first-order methods
    • Sequential quadratic programming methods
  • Nonlinearly constrained optimization:
    • Penalty, cutting plane, trust-region, etc. methods (time permitting)

Course Requirements

What background is needed for MS&E 311? This is an advanced master or doctoral-level Core course in the MS&E Department.  No prior optimization background is required (although it should do no harm to have some).  In this sense, it is an introductory course, but it is not intended to be an elementary course. Students who have taken courses such as MS&E 211 (or MS&E 310) will see some repetition of material. This is unavoidable, but MS&E 311 is intended to be more theoretical and advanced than MS&E 211.

Students in this course will be expected to possess a firm background in the following mathematical subjects: multivariate differential calculus; basic concepts of analysis; linear algebra and some matrix theory. Familiarity with computers and computer programming might also be useful. Above all, it is essential to have a tolerance for mathematical discourse plus an ability to follow - and sometimes devise one's own - mathematical proofs.  These play a much larger role in the course than computer work.