A mathematical optimization problem is one in which a given function is either minimized or maximized relative to some set or range of choices available in a given situation. Optimization problems arise in a multitude of ways as a means of solving problems in engineering design, portfolio design, system management, parameter estimation, statistics, and in the modeling physical and behavioral phenomena. AMath/Math 516 is an introductory course in numerical methods for continuous optimization in finite dimensions.
AMath/Math 516 begins with a detailed discussion of the basic convergence theory for numerical methods of optimization. Both global and local theories are developed. Special attention is given to descent methods based on exact first-order information and approximate second-order information. The basic line-search and trust region methodologies are examined. Generalizations to nondifferentiable functions in composite format are also studied. The study of constrained methods begins with a review of interior point path following methodology for linear programs, quadratic programs, and the monotone linear complementarity problem. This is followed by a review of gradient projection methods, penalty methods, and multiplier methods for constrained optimization problems.
The course is graded on students performance on a mixture of problem sets and programming assignments. All programming is to be done in Matlab.