v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
774 APPENDIX F. NOTATION AND A FEW DEFINITIONS: as in f : R n → R m meaning f is a mapping,or sequence of successive integers specified by bounds as in i:j = i ... j(if j < i then sequence is descending)f : M → Rmeaning f is a mapping from ambient space M to ambient R , notnecessarily denoting either domain or range| as in f(x) | x∈ C means with the condition(s) or such that orevaluated for, or as in {f(x) | x∈ C} means evaluated for each andevery x belonging to set Cg| xpexpression g evaluated at x pA, B as in, for example, A, B ∈ S N means A ∈ S N and B ∈ S N(A, B) open interval between A and B in R ,or variable pair perhaps of disparate dimension[A, B ]closed interval or line segment between A and B in R( ) hierarchal, parenthetical, optional{ } curly braces denote a set or list, e.g., {Xa | a≽0} the set comprisingXa evaluated for each and every a≽0 where membership of a tosome space is implicit, a union〈 〉 angle brackets denote vector inner-product (33) (38)[ ] matrix or vector, or quote insertion, or citation[d ij ] matrix whose ij th entry is d ij[x i ] vector whose i th entry is x ix pparticular value of xx 0 particular instance of x , or initial value of a sequence x ix 1 first entry of vector x , or first element of a set or list {x i }x εextreme point
775x +vector x whose negative entries are replaced with 0 ; x + = 1 (x + |x|)2(535) or clipped vector x or nonnegative part of xx −ˇxx ⋆x ∗x − 1(x − |x|) or nonpositive part of x = x 2 ++ x −known dataoptimal value of variable x . optimal ⇒ feasiblecomplex conjugate or dual variable or extreme direction of dual conef ∗ convex conjugate function f ∗ (s)= sup{〈s , x〉 − f(x) | x∈domf }P C x or PxP k xδ(A)δ 2 (A)δ(A) 2λ i (X)λ(X) iλ(A)σ(A)Σ∑π(γ)ΞΠprojection of point x on set C , P is operator or idempotent matrixprojection of point x on set C k or on range of implicit vector(a.k.a diag(A) ,A.1) vector made from main diagonal of A if A isa matrix; otherwise, diagonal matrix made from vector A≡ δ(δ(A)). For vector or diagonal matrix Λ , δ 2 (Λ) = Λ= δ(A)δ(A) where A is a vectori th entry of vector λ is function of Xi th entry of vector-valued function of Xvector of eigenvalues of matrix A , (1461) typically arranged innonincreasing ordervector of singular values of matrix A (always arranged in nonincreasingorder), or support function in direction Adiagonal matrix of singular values, not necessarily squaresumnonlinear permutation operator (or presorting function) arrangesvector γ into nonincreasing order (7.1.3)permutation matrixdoublet or permutation operator or matrix
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- Page 769 and 770: Appendix FNotation and a few defini
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- Page 787 and 788: Bibliography[1] Edwin A. Abbott. Fl
- Page 789 and 790: BIBLIOGRAPHY 789[23] Dror Baron, Mi
- Page 791 and 792: BIBLIOGRAPHY 791[49] Leonard M. Blu
- Page 793 and 794: BIBLIOGRAPHY 793[74] Yves Chabrilla
- Page 795 and 796: BIBLIOGRAPHY 795[102] Etienne de Kl
- Page 797 and 798: BIBLIOGRAPHY 797[129] Carl Eckart a
- Page 799 and 800: BIBLIOGRAPHY 799[154] James Gleik.
- Page 801 and 802: BIBLIOGRAPHY 801[182] Johan Håstad
- Page 803 and 804: BIBLIOGRAPHY 803[212] Viren Jain an
- Page 805 and 806: BIBLIOGRAPHY 805[237] Monique Laure
- Page 807 and 808: BIBLIOGRAPHY 807[265] Sunderarajan
- Page 809 and 810: BIBLIOGRAPHY 809Notes in Computer S
- Page 811 and 812: BIBLIOGRAPHY 811[319] Anthony Man-C
- Page 813 and 814: BIBLIOGRAPHY 813[346] Pham Dinh Tao
- Page 815 and 816: BIBLIOGRAPHY 815[375] Bernard Widro
- Page 817 and 818: Index∅, see empty set0-norm, 229,
- Page 819 and 820: INDEX 819bees, 30, 413Bellman, 276b
- Page 821 and 822: INDEX 821circular, 130construction,
- Page 823 and 824: INDEX 823measure, 295convex, 21, 36
774 APPENDIX F. NOTATION AND A FEW DEFINITIONS: as in f : R n → R m meaning f is a mapping,or sequence of successive integers specified by bounds as in i:j = i ... j(if j < i then sequence is descending)f : M → Rmeaning f is a mapping from ambient space M to ambient R , notnecessarily denoting either domain or range| as in f(x) | x∈ C means with the condition(s) or such that orevaluated for, or as in {f(x) | x∈ C} means evaluated for each andevery x belonging to set Cg| xpexpression g evaluated at x pA, B as in, for example, A, B ∈ S N means A ∈ S N and B ∈ S N(A, B) open interval between A and B in R ,or variable pair perhaps of disparate dimension[A, B ]closed interval or line segment between A and B in R( ) hierarchal, parenthetical, optional{ } curly braces denote a set or list, e.g., {Xa | a≽0} the set comprisingXa evaluated for each and every a≽0 where membership of a tosome space is implicit, a union〈 〉 angle brackets denote vector inner-product (33) (38)[ ] matrix or vector, or quote insertion, or citation[d ij ] matrix whose ij th entry is d ij[x i ] vector whose i th entry is x ix pparticular value of xx 0 particular instance of x , or initial value of a sequence x ix 1 first entry of vector x , or first element of a set or list {x i }x εextreme point