Semideenite Programming

In semide nite programming one minimizes a linear function subject to the constraint that an a ne combination of symmetric matrices is positive semide nite. Such a constraint is nonlinear and nonsmooth, but convex, so semide nite programs are convex optimization problems. Semide nite programming uni es several standard problems (e.g., linear and quadratic programming) and nds many applications in engineering and combinatorial optimization. Although semide nite 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 semide nite 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 semide nite programs, and an introduction to primal-dual interior-point methods for their solution.