This is an introductory course in theory and methods for finite-dimensional continuous optimization. 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. In this course, we begin by briefly examining applications in engineering design and system management. These applications are introduced in order to exhibit the variety of phenomena that must be addressed in our development of the theoretical apparatus designed to aid in the analysis and solution of these problems.
Once the context for the theoretical developments has been established, we begin a discussion of problem formulation isolating the key geometric and analytic features that allow for successful modeling and solution procedures. After this discussion, we begin the theoretical development in earnest. The theory is motivated by numerous examples each step of the way. Numerical methodology is presented side by side with the theoretical constructs as a means to illustrate their nature and use in application.
It should be emphasized that numerical methods are only given a cursory treatment in 515. They are discussed primarily for the purpose of clarifying and justifying the theoretical development. The full power of the theory developed in 515 is brought to bare in Math 516 to develop a rich theory of numerical methods for continuous optimization problems. Math 516 (Numerical Methods of Optimization) is designed as a sequel to 515. In this course students are introduced to state of the art methods for the implementation and design of numerical routines for the solution of finite-dimensional continuous variable optimization problems.
Attached below are links to postscript files for the course notes, problem sets, and Matlab notes.