Math/AMath 516: Numerical Optimization
Jim Burke
Spring 1999, Tuesday/Thursday 8:00-9:15 AM
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, and in the modeling physical and behavioral
phenomena. AMath/Math 516 is an introductory course in numerical methods
for continuous optimization in finite dimensions. The course is designed
as a sequel to AMath/Math/IND
E 515 (Fundamentals of Optimization). In 515 the basic theory for continuous
optimization problems is developed: optimality conditions are developed
for both constrained and unconstrained problems, regularity conditions
(or constraint qualifications) are developed, and the role of convexity
is explored. Some numerical techniques are also examined in 515. But this
is done primarily to illustrate and motivate the theoretical development.
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.