v2010.10.26 - Convex Optimization

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