Handbook of Semidefinite Programming

Abstract

PrefaceSemide nite programming or SDP has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity w as spurred by the discovery of important applications in combinatorial optimization and control theory, the development of e cient i n terior-point algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory.This book includes nineteen chapters on the theory, algorithms, and applications of semide nite programming. Written by the leading experts on the subject, it o ers an advanced and broad overview of the current state of the eld. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to nish the book in a timely fashion, we h a ve had to abandon hopes for separate chapters on some important topics such as a discussion of SDP algorithms in the framework of general convex programming using the theory of self-concordant barriers, or an overview of numerical implementation aspects of interior-point methods for SDP. We apologize for any important gaps in the contents, and hope that the historical notes at the end of the book provide a useful guide to the literature on the topics that are not adequately covered in this handbook.We w ould like to thank all the authors for their outstanding contributions, their editorial help, and for their patience during the many revisions of the handbook. In addition, we thank Mike T odd for his valuable editorial advice on many occasions. We w ould also like to thank Erna Unrau for her help in combining some of the bibliographies into one. 11.5 -function, n K = 8 , n A = 10, = 1 0 ,5 , time limit one hour. 334 11.6 PLDI, n 500 where the variable is X 2 S n , the space of real symmetric n n matrices. The vector b 2 IR m , and the matrices A i 2 S n and C 2 S n are given problem parameters. The inequality X 0 means X is positive semide nite. More generally, throughout this book, the notation A B will mean B,A is positive semide nite. The notation C X stands for the inner product of the symmetric matrices C and X:Some authors write the inner product as C X = T r CXwhere Tr CXis the trace of the matrix CX. In other words, C X is a linear function of the elements X ij . In 1.1.1 we minimize a linear function of the matrix variable X, subject to m linear equality constraints A i X = b i , and the positive semide nitess constraint X 0. We refer to problem 1.1.1 as a semide nite program or SDP. W e can think of an SDP as a generalization of the standard form linear programming problem minimize c T x subject to a T i x = b i for all i = 1 ; : : : ; m x 0 1.1.2 1 2 HANDBOOK OF SEMIDEFINITE PROGRAMMING in which the elementwise nonnegativity constraint x 0 is replaced by a generalized inequality with re...

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

“…In this section, we encode the synthesis of the aforementioned exponential and time-dependent stochastic barrier certificates into semidefinite programming [38] optimizations, and thus a solution thereof yields an upper bound on the failure probability over the infinite-time horizon. Specifically, an SDP problem is formulated, for each of the two barrier certificates, to encode the constraints for "being an exponential/time-dependent stochastic barrier certificate", while in the meantime optimizing the tightness of the failure probability bound.…”

“…We furthermore present an alternative method to address the reduced finite-time horizon verification problem based on the discovery of a time-dependent stochastic barrier certificate. We show that both the exponential and the time-dependent stochastic barrier certificate can be synthesized by respectively solving a pertinent semidefinite programming (SDP) [38] optimization problem. Experimental results on some interesting examples taken from the literature demonstrated the effectiveness of the reduction and that our method often produces tighter bounds on the failure probability.…”

On bounds of the Pythagoras number of the sum of square magnitudes of Laurent polynomials

“…Results on the facial structure of linear programs and semidefinite programs [17,16,18] also give an upper bound on the Pythagoras number of real polynomials but it is not as sharp as U (n, d). Several nice properties of faces of the cone of positive semidefinite matrices can be found in [20,18,17,2,10,1,16]. This unsharp upper bound is derived by considering the following primal and dual semidefinite programs, respectively,…”

“…where C ∈ Sê. Pataki [16,17,18] proved that for any feasible point X (with rank r) of the primal semidefinite program (6), the following rank inequality holds…”

Time-reversal of rank-one quantum strategy functions

“…Various developments, applications, and variants of the quantum strategy framework can be found in [1,3,5,8,10], for instance, and in a number of other sources.In the discussion of the quantum strategy framework that follows, as well as in the subsequent sections of this paper, we assume that the reader is familiar with quantum information theory and semidefinite programming. References on this material include [11,13,15,16] as well as [14], which we follow closely with respect to notation and terminology. In particular,…”