4. Semidefinite Programming [chapter]

2001 Lectures on Modern Convex Optimization  
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
more » ... 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. Keywords. Semide nite programming, convex optimization, interior-point methods, eigenvalue optimization, combinatorial optimization, system and control theory AMS subject classi cations. 65K05, 49M45, 93B51, 90C25, 90C27, 90C90, 15A18 To appear in SIAM Review. Associated software for semide nite programming is available via anonymous ftp into isl.stanford.edu in pub/boyd/semidef_prog.
doi:10.1137/1.9780898718829.ch4 fatcat:54pwgbfqezd4rnmqewhsccilmq