By Stephen P Boyd

During this booklet the authors lessen a wide selection of difficulties bobbing up in procedure and keep watch over conception to a handful of convex and quasiconvex optimization difficulties that contain linear matrix inequalities. those optimization difficulties will be solved utilizing lately constructed numerical algorithms that not just are polynomial-time but in addition paintings rather well in perform; the relief consequently will be thought of an answer to the unique difficulties. This e-book opens up an incredible new examine sector within which convex optimization is mixed with procedure and keep an eye on thought, leading to the answer of a big variety of formerly unsolved difficulties

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**Additional info for Linear matrix inequalities in system and control theory**

**Sample text**

Fm ∈ Rn×n be symmetric matrices. Then there is a matrix A ∈ Rm×p with p ≤ m, a vector b ∈ Rm , and symmetric matrices F˜0 , . . , F˜p ∈ Rq×q with q ≤ n such that: p ∆ F (x) ≥ 0 if and only if x = Az + b for some z ∈ Rp and F˜ (z) = F˜0 + i=1 zi F˜i ≥ 0. In addition, if the LMI F (x) ≥ 0 is feasible, then the LMI F˜ (z) ≥ 0 is strictly feasible. This electronic version is for personal use and may not be duplicated or distributed. 42) where Fi ∈ Rn×n , i = 0, . . , m. Let X denote the feasible set {x | F (x) ≥ 0}.

Evidently this is an LMIP. We can easily recover some known results about this problem. 1). Then the set of all positive-definite completions, if nonempty, is bounded. 1). 2). , it has the same sparsity structure as the original matrix. Depending on the pattern of specified entries, this condition can lead to an “analytic solution” of the problem. But of course an arbitrary positive-definite completion problem is readily solved as an LMIP. Let us mention a few simple generalizations of this matrix completion problem that have not been considered in the literature but are readily solved as LMIPs, and might be useful in applications.

Using conjugate-gradient and other related methods to solve these least-squares systems gives two advantages. First, by exploiting problem structure in the conjugate-gradient iterations, the computational effort required to solve the leastsquares problems is much smaller than by standard “direct” methods such as QR or Cholesky factorization. Second, it is possible to terminate the conjugate-gradient iterations before convergence, and still obtain an approximation of the Newton direction suitable for interior-point methods.