Statistics

ST706 - Nonlinear Programming

• Prerequisites: OR (IE, MA) 505 and MA 425
• Term & Frequency: Spring
• Student Audience: Graduate students in Statistics, Operations Research, Engineering, and Mathematics
• Credit: 3 credits
• Recent Texts: Mathematical Programming, Theory and Algorithms, 3rd edition by Bazaraa, Sherali, Shetty.
• Recent Instructors: Thomas Reiland, Shu-Cherng Fang
• Background and Goals: Advanced treatment of the analytical aspects of finite-dimensional nonlinear programming. The material discussed requires some mathematical maturity and working knowledge of differential calculus and, to a lesser extent, linear algebra. Other required topics, such as convexity and conjugacy, will be introduced in the course. The goal of the course is to prepare graduate students for research in mathematical programming and to apply mathematical programming to discipline-specific research.
• Content: Mathematical foundations: differentiability, convex sets and functions, topics in linear algebra. Optimality conditions: results will be presented in terms of tangent cones and their duals. Classical results in unconstrained optimization, Lagrange multipliers, and Kuhn-Tucker theory will be derived as special cases; heuristic derivations and geometric interpretations will be emphasized. Additional topics: constraint qualifications in the literature; the Lagrangian function and saddlepoints; saddlepoints and optimal solutions; Lagrangian duality; quadratic problems; separability. Convex analysis: modern theory of convexity - epigraphs, proper convex functions, conjugate functions. Conjugate duality, comparisons with Lagrangian duality. Geometric Programming. Optimality conditions; duality; saddlepoints, geometric Lagrangians; comparisons with “ordinary” programming. Extended nonlinear programming models. Extension of results to nonsmooth optimization problems; Lipschitz functions, optimality conditions, other topic coverage as time permits.
• Alternatives: OR 706, MA 706
• Subsequent Courses: None

SP 2013 Sections: