Semidefinite Programming

TLDR

Semidefinite programming is a convex optimization framework that generalizes linear and quadratic programming, unifying many engineering and combinatorial problems, and is efficiently solvable by interior‑point methods with polynomial worst‑case complexity. This paper surveys the theory and applications of semidefinite programs and introduces primal‑dual interior‑point methods for their solution. The authors present primal‑dual interior‑point algorithms adapted from linear programming to solve semidefinite programs efficiently.

Abstract

In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.