Applications of Semidefinite Programming
BY
L. Vandenberghe and S. Boyd
A wide variety of nonlinear convex optimization problems can be cast as problems involving linear matrix inequalities (LMIs), and hence efficiently solved using recently developed interior-point methods. In this paper, we will consider two classes of optimization problems with LMI constraints:
- The semidefinite programming problem, i.e., the problem of
minimizing a linear function subject to a linear matrix inequality.
Semidefinite programming is an important numerical tool for
analysis and synthesis in systems and control theory. It has also
been recognized in combinatorial optimization as a valuable
technique for obtaining bounds on the solution of NP-hard
problems.
- The problem of maximizing the determinant of a positive definite matrix
subject to linear matrix inequalities. This problem has applications in
computational geometry, experiment design, information and communication
theory, and other fields. We review some of these applications, including some
interesting applications that are less well known and arise in statistics,
optimal experiment design and VLSI.
