MS&E 311 Optimization Winter 2013-2014

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MS&E 311 emphasizes classical theories, efficient algorithms and recent progresses in Linear and Nonlinear Optimization---one of the central mathematical decision models in Management Science and Operations Research. MS&E 311 is a continuation course of MS&E211. Although there are some minor overlaps with MS&E 211, MS&E 311 covers more advanced application models, in-depth optimization theories, and state of the art algorithms which were not able to be covered by MS&E211. 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 topics are Information Aggregation, Economic Equilibrium, Pricing Model, Core of Game, Financial Decision and Risk Management, Sparse and Low Rank Opimization and its 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