Introduction to Semidefinite Programming (SDP)
Robert M. Freund
March, 2004
2004 Massachusetts Institute of Technology.
1 1 Introduction
Semidefinite programming (SDP ) is the most exciting development in math- ematical programming in the 1990’s. SDP has applications in such diverse fields as traditional convex constrained optimization, control theory, and combinatorial optimization. Because SDP is solvable via interior point methods, most of these applications can usually be solved very efficiently in practice as well as in theory.
2 Review of Linear Programming
Consider the linear programming problem in standard form:
LP : minimize c · x
s.t. ai · x = bi,i=1 ,...,m
n x ∈+.
x n c · x Here is a vector of variables, and we write “ ” for the inner-product n “ j=1 cj xj ”, etc.
n n n Also, + := {x ∈ | x ≥ 0},and we call + the nonnegative orthant. n In fact, + is a closed convex cone, where K is called a closed a convex cone if K satisfies the following two conditions:
• If x, w ∈ K, then αx + βw ∈ K for all nonnegative scalars α and β.
• K is a closed set.
2 In words, LP is the following problem:
“Minimize the linear function c·x, subject to the condition that x must solve m given equations ai · x = bi,i =1,...,m, and that x must lie in the closed n convex cone K = +.”
We will write the standard linear programming dual problem as: