v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
272 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.log-convex ⇒ convex ⇒ quasiconvex.concave ⇒ quasiconcave ⇐ log-concave ⇐ positive concave. 3.294. The line theorem (3.7.3.0.1) translates identically to quasiconvexity(quasiconcavity). [61,3.4.2]5.g convex ⇔ −g concaveg quasiconvex ⇔ −g quasiconcaveg log-convex ⇔ 1/g log-concave(translation, homogeneity) Function convexity, concavity,quasiconvexity, and quasiconcavity are invariant to offset andnonnegative scaling.6. (affine transformation of argument) Composition g(h(X)) of convex(concave) function g with any affine function h : R m×n → R p×kremains convex (concave) in X ∈ R m×n , where h(R m×n ) ∩ domg ≠ ∅ .[199,B.2.1] Likewise for the quasiconvex (quasiconcave) functions g .7. – Nonnegatively weighted sum of convex (concave) functionsremains convex (concave). (3.1.2.1.1)– Nonnegatively weighted nonzero sum of strictly convex(concave) functions remains strictly convex (concave).– Nonnegatively weighted maximum (minimum) of convex 3.30(concave) functions remains convex (concave).– Pointwise supremum (infimum) of convex (concave) functionsremains convex (concave). (Figure 74) [307,5] –Sum of quasiconvex functions not necessarily quasiconvex.– Nonnegatively weighted maximum (minimum) of quasiconvex(quasiconcave) functions remains quasiconvex (quasiconcave).– Pointwise supremum (infimum) of quasiconvex (quasiconcave)functions remains quasiconvex (quasiconcave).3.29 Log-convex means: logarithm of function f is convex on dom f .3.30 Supremum and maximum of convex functions are proven convex by intersection ofepigraphs.
Chapter 4Semidefinite programmingPrior to 1984, linear and nonlinear programming, 4.1 one a subsetof the other, had evolved for the most part along unconnectedpaths, without even a common terminology. (The use of“programming” to mean “optimization” serves as a persistentreminder of these differences.)−Forsgren, Gill, & Wright (2002) [145]Given some practical application of convex analysis, it may at first seempuzzling why a search for its solution ends abruptly with a formalizedstatement of the problem itself as a constrained optimization. Theexplanation is: typically we do not seek analytical solution because there arerelatively few. (3.5.2,C) If a problem can be expressed in convex form,rather, then there exist computer programs providing efficient numericalglobal solution. [167] [384] [385] [383] [348] [336] The goal, then, becomesconversion of a given problem (perhaps a nonconvex or combinatorialproblem statement) to an equivalent convex form or to an alternation ofconvex subproblems convergent to a solution of the original problem:4.1 nascence of polynomial-time interior-point methods of solution [363] [381].Linear programming ⊂ (convex ∩ nonlinear) programming.2001 Jon Dattorro. co&edg version 2010.10.26. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.10.26.273
- Page 221 and 222: 3.1. CONVEX FUNCTION 221f 1 (x)f 2
- Page 223 and 224: 3.1. CONVEX FUNCTION 223Rf(b)f(X
- Page 225 and 226: 3.2. PRACTICAL NORM FUNCTIONS, ABSO
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- Page 235 and 236: 3.3. INVERTED FUNCTIONS AND ROOTS 2
- Page 237 and 238: 3.4. AFFINE FUNCTION 237rather]x >
- Page 239 and 240: 3.4. AFFINE FUNCTION 239f(z)Az 2z 1
- Page 241 and 242: 3.5. EPIGRAPH, SUBLEVEL SET 241{a T
- Page 243 and 244: 3.5. EPIGRAPH, SUBLEVEL SET 243Subl
- Page 245 and 246: 3.5. EPIGRAPH, SUBLEVEL SET 245wher
- Page 247 and 248: 3.5. EPIGRAPH, SUBLEVEL SET 247part
- Page 249 and 250: 3.5. EPIGRAPH, SUBLEVEL SET 249that
- Page 251 and 252: 3.6. GRADIENT 251respect to its vec
- Page 253 and 254: 3.6. GRADIENT 253Invertibility is g
- Page 255 and 256: 3.6. GRADIENT 2553.6.1.0.2 Theorem.
- Page 257 and 258: 3.6. GRADIENT 257f(Y )[ ∇f(X)−1
- Page 259 and 260: 3.6. GRADIENT 259αβα ≥ β ≥
- Page 261 and 262: 3.6. GRADIENT 2613.6.4 second-order
- Page 263 and 264: 3.7. CONVEX MATRIX-VALUED FUNCTION
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- Page 269 and 270: 3.8. QUASICONVEX 269exponential alw
- Page 271: 3.9. SALIENT PROPERTIES 2713.8.0.0.
- Page 275 and 276: 4.1. CONIC PROBLEM 275(confer p.162
- Page 277 and 278: 4.1. CONIC PROBLEM 277PCsemidefinit
- Page 279 and 280: 4.1. CONIC PROBLEM 279is the affine
- Page 281 and 282: 4.1. CONIC PROBLEM 281faces of S 3
- Page 283 and 284: 4.1. CONIC PROBLEM 2834.1.2.3 Previ
- Page 285 and 286: 4.2. FRAMEWORK 285Semidefinite Fark
- Page 287 and 288: 4.2. FRAMEWORK 287On the other hand
- Page 289 and 290: 4.2. FRAMEWORK 2894.2.2.1 Dual prob
- Page 291 and 292: 4.2. FRAMEWORK 291For symmetric pos
- Page 293 and 294: 4.2. FRAMEWORK 293has norm ‖x ⋆
- Page 295 and 296: 4.2. FRAMEWORK 295minimize 1 TˆxX
- Page 297 and 298: 4.2. FRAMEWORK 297asminimize ‖ỹ
- Page 299 and 300: 4.3. RANK REDUCTION 2994.3 Rank red
- Page 301 and 302: 4.3. RANK REDUCTION 301A rank-reduc
- Page 303 and 304: 4.3. RANK REDUCTION 303(t ⋆ i)
- Page 305 and 306: 4.3. RANK REDUCTION 3054.3.3.0.1 Ex
- Page 307 and 308: 4.3. RANK REDUCTION 3074.3.3.0.2 Ex
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Chapter 4Semidefinite programmingPrior to 1984, linear and nonlinear programming, 4.1 one a subsetof the other, had evolved for the most part along unconnectedpaths, without even a common terminology. (The use of“programming” to mean “optimization” serves as a persistentreminder of these differences.)−Forsgren, Gill, & Wright (2002) [145]Given some practical application of convex analysis, it may at first seempuzzling why a search for its solution ends abruptly with a formalizedstatement of the problem itself as a constrained optimization. Theexplanation is: typically we do not seek analytical solution because there arerelatively few. (3.5.2,C) If a problem can be expressed in convex form,rather, then there exist computer programs providing efficient numericalglobal solution. [167] [384] [385] [383] [348] [336] The goal, then, becomesconversion of a given problem (perhaps a nonconvex or combinatorialproblem statement) to an equivalent convex form or to an alternation ofconvex subproblems convergent to a solution of the original problem:4.1 nascence of polynomial-time interior-point methods of solution [363] [381].Linear programming ⊂ (convex ∩ nonlinear) programming.2001 Jon Dattorro. co&edg version 2010.10.26. All rights reserved.citation: Dattorro, <strong>Convex</strong> <strong>Optimization</strong> & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, <strong>v2010.10.26</strong>.273