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Convex optimization and Euclidean distance geometry 1st Edition by Jon Dattorro 0976401304 9780976401308

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Author(s): Dattorro J.
ISBN(s): 9780976401308, 0976401304
Edition: draft
File Details: PDF, 8.63 MB
Year: 2011
Language: english
SKU: EB-2600960 Category: Tag:

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ISBN 10: 0976401304 

ISBN 13: 9780976401308

Author: Jon Dattorro

Optimization is the science of making a best choice in the face of conflicting requirements. Any convex optimization problem has geometric interpretation. If a given optimization problem can be transformed to a convex equivalent, then this interpretive benefit is acquired. That is a powerful attraction: the ability to visualize geometry of an optimization problem. Conversely, recent advances in geometry hold convex optimization within their proofs’ core. This book is about convex optimization, convex geometry (with particular attention to distance geometry), geometrical problems, and problems that can be transformed into geometrical problems. Euclidean distance geometry is, fundamentally, a determination of point conformation from interpoint distance information; e.g., given only distance information, determine whether there corresponds a realizable configuration of points; a list of points in some dimension that attains the given interpoint distances. 2005 International Edition I

Table of contents:

1 Overview
2 Convex geometry
2.1 Convex set
2.2 Vectorized-matrix inner product
2.3 Hulls
2.4 Halfspace, Hyperplane
2.5 Subspace representations
2.6 Extreme, Exposed
2.7 Cones
2.8 Cone boundary
2.9 Positive semidefinite (PSD) cone
2.10 Conic independence (c.i.)
2.11 When extreme means exposed
2.12 Convex polyhedra
2.13 Dual cone & generalized inequality

3 Geometry of convex functions
3.1 Convex function
3.2 Practical norm functions, absolute value
3.3 Inverted functions and roots
3.4 Affine function
3.5 Epigraph, Sublevel set
3.6 Gradient
3.7 Convex matrix-valued function
3.8 Quasiconvex
3.9 Salient properties

4 Semidefinite programming
4.1 Conic problem
4.2 Framework
4.3 Rank reduction
4.4 Rank-constrained semidefinite program
4.5 Constraining cardinality
4.6 Cardinality and rank constraint examples
4.7 Constraining rank of indefinite matrices
4.8 Convex Iteration rank-1

5 Euclidean Distance Matrix
5.1 EDM
5.2 First metric properties
5.3 ∃ fifth Euclidean metric property
5.4 EDM definition
5.5 Invariance
5.6 Injectivity of D & unique reconstruction
5.7 Embedding in affine hull
5.8 Euclidean metric versus matrix criteria
5.9 Bridge: Convex polyhedra to EDMs
5.10 EDM-entry composition
5.11 EDM indefiniteness
5.12 List reconstruction
5.13 Reconstruction examples
5.14 Fifth property of Euclidean metric

6 Cone of distance matrices
6.1 Defining EDM cone
6.2 Polyhedral bounds
6.3 √EDM cone is not convex
6.4 EDM definition in 11T
6.5 Correspondence to PSD cone SN−1+
6.6 Vectorization & projection interpretation
6.7 A geometry of completion
6.8 Dual EDM cone
6.9 Theorem of the alternative
6.10 Postscript

7 Proximity problems
7.1 First prevalent problem
7.2 Second prevalent problem
7.3 Third prevalent problem
7.4 Conclusion

A Linear algebra
A.1 Main-diagonal δ operator, λ, trace, vec
A.2 Semidefiniteness: domain of test
A.3 Proper statements
A.4 Schur complement
A.5 Eigenvalue decomposition
A.6 Singular value decomposition, SVD
A.7 Zeros

B Simple matrices
B.1 Rank-one matrix (dyad)
B.2 Doublet
B.3 Elementary matrix
B.4 Auxiliary V-matrices
B.5 Orthogonal matrix

C Some analytical optimal results
C.1 Properties of infima
C.2 Trace, singular and eigen values
C.3 Orthogonal Procrustes problem
C.4 Two-sided orthogonal Procrustes
C.5 Nonconvex quadratics

D Matrix calculus
D.1 Directional derivative, Taylor series
D.2 Tables of gradients and derivatives

E Projection
E.1 Idempotent matrices
E.2 I − P, Projection on algebraic complement
E.3 Symmetric idempotent matrices
E.4 Algebra of projection on affine subsets
E.5 Projection examples
E.6 Vectorization interpretation
E.7 Projection on matrix subspaces
E.8 Range/Rowspace interpretation
E.9 Projection on convex set
E.10 Alternating projection

F Notation and a few definitions

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Tags: Jon Dattorro, Convex, optimization, Euclidean