Syllabus: Math 650

Background Topics:

• Calculus: Taylor’s theorem and differential calculus of several variables
• Basic results of analysis: compactness, connectedness, Heine-Borel, Bolzano-Weierstrass, Weierstrass, intermediate-value theorem, etc.
• Symmetric matrices: diagonalization, and recognizing positive (semi)definite matrices (with eigenvalue test, Sylvester’s theorem, and Descartes’s rule of sign test)

• Unconstrained optimization: first- and second-order tests for a local minimum, maximum, or saddle points of smooth functions of several variables, coercive functions and existence of global optimizers
• Basics of convexity: convex sets, convex functions, Caratheodory’s theorem, Jensen’s inequality, firstand second-order characterization of convex functions, global optimality of a local minimizer of a convex function, variational inequality principle for minimization of a function on a convex set
• Separation of convex sets: the basic separation theorem on separation of two convex sets, support plane theorem, familiarity with statement of proper separation theorem
• Linear equality/inequality systems: Farkas’s lemma and Motzkin’s transposition theorem, linear programming duality
• First-order optimality conditions in nonlinear programming: Fritz John’s theorem, Kuhn-Tucker conditions, Lagrange multipliers, constraint qualifications (Slater’s condition, Mangasarian-Fromovitz condition, etc.)
• Second-order (necessary and sufficient) optimality conditions in nonlinear programming
• Ability to solve concrete nonlinear programs through the use of the methods noted in the two preceding bullets
  
• Duality: concept of the primal and dual programs with respect to a Lagrangian function, general correspondence between saddle points and duality, dual of a nonlinear program, familiarity with strong duality theorem for convex programming and the ability to use it in concrete problems, duality in quadratic programming

1. Guler, Foundations of Optimization, 2010.
2. Bertsekas, Nonlinear Programming, 2nd Edition, 1999.
3. Borwein and Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, 2009.
4. Luenberger and Ye,  Linear and Nonlinear Programming, 3rd Edition, 2009.
5. Bazaraa, Sherali and Shetty, Nonlinear Programming: Theory and Algorithms, 2nd Edition, 1993.