Convex optimization & euclidean distance geometry by Jon Dattorro

By Jon Dattorro

Convex research is the calculus of inequalities whereas Convex Optimization is its program. research is inherently the area of the mathematician whereas Optimization belongs to the engineer. In layman's phrases, the mathematical technological know-how of Optimization is the learn of the way to make a sensible choice while faced with conflicting requisites. The qualifier Convex skill: while an optimum answer is located, then it really is bound to be a top answer; there is not any more sensible choice. As any Convex Optimization challenge has geometric interpretation, this ebook is ready convex geometry (with specific realization to distance geometry), and nonconvex, combinatorial, and geometrical difficulties that may be cozy or reworked into convex difficulties. A digital flood of recent functions follows by way of epiphany that many difficulties, presumed nonconvex, will be so remodeled. Revised & Enlarged foreign Paperback version III

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In chapter 7, EDM proximity, we explore methods of solution to a few fundamental and prevalent Euclidean distance matrix proximity problems; the problem of finding that Euclidean distance matrix closest to a given matrix in some sense. We discuss several heuristics for the problems when compounded with rank minimization: (944) √ ◦ minimize D − H 2F minimize −V (D − H)V 2F √ ◦ D D subject to rank V D V ≤ ρ D ∈ EDMN minimize D D−H 2 F subject to rank V D V ≤ ρ D ∈ EDMN subject to rank V D V ≤ ρ √ ◦ D ∈ EDMN minimize √ ◦ D √ −V ( ◦ D − H)V 2 F subject to rank V D V ≤ ρ √ ◦ D ∈ EDMN We offer a new geometrical proof of a famous result discovered by Eckart & Young in 1936 [69] regarding Euclidean projection of a point on that nonconvex subset of the positive semidefinite cone comprising all positive semidefinite matrices having rank not exceeding a prescribed bound ρ .

Given ∆ matrix Y = [ y1 y2 · · · yN ] ∈ RN ×N , consider the mapping ∆ T (Γ) : Rn×N → Rn×N = Γ Y (5) whose domain is the set of all matrices Γ ∈ Rn×N holding a linearly independent set columnar. Linear independence of {Γyi ∈ Rn , i = 1 . . N } demands, by definition, there exists no nontrivial solution ζ ∈ RN to Γy1 ζi + · · · + ΓyN −1 ζN −1 + ΓyN ζN = 0 (6) By factoring Γ , we see that is ensured by linear independence of {yi ∈ RN }. 48 CHAPTER 2. 3 Orthant: name given to a closed convex set that is the higher-dimensional generalization of quadrant from the classical Cartesian partition of R2 .

The set of all symmetric matrices SM forms a proper subspace in RM ×M , so for it there exists a standard orthonormal basis in isometrically isomorphic RM (M +1)/2 {Eij ∈ SM } =   ei eTi ,  √1 2  i = j = 1. M  ei eTj + ej eTi , 1 ≤ i < j ≤ M  (49) where M (M +1)/2 standard basis matrices Eij are formed from the standard basis vectors ei ∈ RM . M Yii , √ Yij 2 , 1 ≤ i < j ≤ M correspond to entries of the symmetric vectorization (46). 2. 1 Definition. Hollow subspaces. 1) δ(A) ∈ RM (1036) Operating on a vector, δ naturally returns a diagonal matrix; δ 2(A) is a diagonal matrix.

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