Proposed New C&O Comprehensive Exam Syllabus for
Continuous Optimization

Examiners:

    TBA

Disclaimer:

    The questions on the exam emphasize breadth of knowledge, rather than depth. An individual who has successfully taken one of the two courses CO663 or CO666, and who has done some additional reading, should do well.

References:

    The material is covered (with some overlap) in the two books:

1. Nonlinear Programming, Dimitri P. Bertsekas
2. The Mathematics of Nonlinear Programming, Peressini, Sullivan, Uhl

Outline:

    The exam covers the basic theory (e.g. attainment, uniqueness of optimal solutions, characterization of optimal solutions) and fundamental algorithms (e.g. Newton's Method, Quasi-Newton Methods, Steepest Descent).

Unconstrained Optimization:

   Basic first and second order algorithms; steepest descent, Newton method, quasi-Newton methods (no memorization of formulae of updates is required), conjugate gradient methods; sufficient decrease criteria and intervals of uncertainty; convergence theorems for Newton's method, line search and trust region methods (global convergence analysis).

Linear Programming:

    Simplex Method, Geometry, Duality and Sensitivity.

Constrained Optimization:

   Equality and inequality constraints; penalty (quadratic and ) and barrier methods, primal-dual interior-point algorithms, gradient projection methods, sequential quadratic programming methods, augmented Lagrangians; quadratic programming.

Convex Analysis:

    Separating hyperplanes, polarity, subgradients, Lagrange multipliers, duality, convex sets and functions.

Optimality Conditions:

    First and second order optimality conditions, Karush-Kuhn-Tucker theorem, local and global optimality, existence and uniqueness of optima (e.g. coercive functions, strict convexity).

 
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