v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
List of Tables2 Convex geometryTable 2.9.2.3.1, rank versus dimension of S 3 + faces 106Table 2.10.0.0.1, maximum number of c.i. directions 121Cone Table 1 167Cone Table S 167Cone Table A 169Cone Table 1* 1724 Semidefinite programmingfaces of S 3 + correspond to faces of S 3 + 2315 Euclidean Distance Matrixaffine dimension in terms of rank Précis 5.7.2 338B Simple matricesAuxiliary V -matrix Table B.4.4 530D Matrix calculusTable D.2.1, algebraic gradients and derivatives 571Table D.2.2, trace Kronecker gradients 572Table D.2.3, trace gradients and derivatives 573Table D.2.4, log determinant gradients and derivatives 575Table D.2.5, determinant gradients and derivatives 576Table D.2.6, logarithmic derivatives 577Table D.2.7, exponential gradients and derivatives 5772001 Jon Dattorro. CO&EDG version 2007.09.13. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.18
Chapter 1OverviewConvex OptimizationEuclidean Distance GeometryPeople are so afraid of convex analysis.−Claude Lemaréchal, 2003In layman’s terms, the mathematical science of Optimization is the studyof how to make a good choice when confronted with conflicting requirements.The qualifier convex means: when an optimal solution is found, then it isguaranteed to be a best solution; there is no better choice.Any convex optimization problem has geometric interpretation. If a givenoptimization problem can be transformed to a convex equivalent, then thisinterpretive benefit is acquired. That is a powerful attraction: the ability tovisualize geometry of an optimization problem. Conversely, recent advancesin geometry and in graph theory hold convex optimization within their proofs’core. [299] [238]This book is about convex optimization, convex geometry (withparticular attention to distance geometry), and nonconvex, combinatorial,and geometrical problems that can be relaxed or transformed into convexproblems. A virtual flood of new applications follow by epiphany that manyproblems, presumed nonconvex, can be so transformed. [8] [9] [44] [63] [99][101] [203] [220] [227] [269] [270] [296] [299] [27,4.3, p.316-322]2001 Jon Dattorro. CO&EDG version 2007.09.13. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.19
- Page 1 and 2: DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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- Page 5 and 6: for Jennie Columba♦Antonio♦♦&
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- Page 13 and 14: List of Figures1 Overview 191 Orion
- Page 15 and 16: LIST OF FIGURES 1559 Quadratic func
- Page 17: LIST OF FIGURES 17E Projection 5791
- Page 21 and 22: ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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- Page 25 and 26: ases (biorthogonal expansion). We e
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- Page 29 and 30: its membership to the EDM cone. The
- Page 31 and 32: 31appendicesProvided so as to be mo
- Page 33 and 34: Chapter 2Convex geometryConvexity h
- Page 35 and 36: 2.1. CONVEX SET 35Figure 9: A slab
- Page 37 and 38: 2.1. CONVEX SET 372.1.6 empty set v
- Page 39 and 40: 2.1. CONVEX SET 392.1.7.1 Line inte
- Page 41 and 42: 2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
- Page 43 and 44: 2.1. CONVEX SET 43This theorem in c
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- Page 53 and 54: 2.3. HULLS 53Figure 12: Convex hull
- Page 55 and 56: 2.3. HULLS 55Aaffine hull (drawn tr
- Page 57 and 58: 2.3. HULLS 57The union of relative
- Page 59 and 60: 2.4. HALFSPACE, HYPERPLANE 59of dim
- Page 61 and 62: 2.4. HALFSPACE, HYPERPLANE 61H +ay
- Page 63 and 64: 2.4. HALFSPACE, HYPERPLANE 63Inters
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List of Tables2 <strong>Convex</strong> geometryTable 2.9.2.3.1, rank versus dimension of S 3 + faces 106Table 2.10.0.0.1, maximum number of c.i. directions 121Cone Table 1 167Cone Table S 167Cone Table A 169Cone Table 1* 1724 Semidefinite programmingfaces of S 3 + correspond to faces of S 3 + 2315 Euclidean Distance Matrixaffine dimension in terms of rank Précis 5.7.2 338B Simple matricesAuxiliary V -matrix Table B.4.4 530D Matrix calculusTable D.2.1, algebraic gradients and derivatives 571Table D.2.2, trace Kronecker gradients 572Table D.2.3, trace gradients and derivatives 573Table D.2.4, log determinant gradients and derivatives 575Table D.2.5, determinant gradients and derivatives 576Table D.2.6, logarithmic derivatives 577Table D.2.7, exponential gradients and derivatives 5772001 Jon Dattorro. CO&EDG version 2007.09.13. All rights reserved.Citation: Jon Dattorro, <strong>Convex</strong> <strong>Optimization</strong> & Euclidean Distance Geometry,Meboo Publishing USA, 2005.18
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