v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
460 CHAPTER 7. PROXIMITY PROBLEMSConfinement of G to the geometric center subspace provides numericalstability and no loss of generality (confer (989)); implicit constraint G1 = 0is otherwise unnecessary.To include constraints on the list X ∈ R n×N , we would first rewrite (1162)minimize − tr(V (D(G) − 2Y )V )G∈S N c , Y ∈ S N h , X∈ Rn×N [ ]〈Φij , G〉 y ijsubject to≽ 0 ,y ijh 2 ij[ ] I XX T ≽ 0GX ∈ Cj > i = 1... N −1(1163)and then add the constraints, realized here in abstract membership to someconvex set C . This problem realization includes a convex relaxation of thenonconvex constraint G = X T X and, if desired, more constraints on G couldbe added. This technique is discussed in5.4.2.2.4.7.2.2 Minimization of affine dimension in Problem 2When desired affine dimension ρ is diminished, the rank function becomesreinserted into problem (1157) that is then rendered difficult to solve becausethe feasible set {D , Y } loses convexity in S N h × R N×N . Indeed, the rankfunction is quasiconcave (3.3) on the positive semidefinite cone; (2.9.2.6.2)id est, its sublevel sets are not convex.7.2.2.1 Rank minimization heuristicA remedy developed in [90] [191] [91] [89] introduces convex envelope (cenv)of the quasiconcave rank function: (Figure 110)7.2.2.1.1 Definition. Convex envelope. [146]The convex envelope of a function f : C →R is defined as the largest convexfunction g such that g ≤ f on convex domain C ⊆ R n . 7.14 △7.14 Provided f ≢+∞ and there exists an affine function h ≤f on R n , then the convexenvelope is equal to the convex conjugate (the Legendre-Fenchel transform) of the convexconjugate of f ; id est, the conjugate-conjugate function f ∗∗ . [147,E.1]
7.2. SECOND PREVALENT PROBLEM: 461rankXcenv rankXFigure 110: Abstraction of convex envelope of rank function. Rank isa quasiconcave function on the positive semidefinite cone, but its convexenvelope is the smallest convex function enveloping it.[90] [89] Convex envelope of rank function: for σ a singular value,cenv(rankA) on {A∈ R m×n | ‖A‖ 2 ≤κ} = 1 ∑σ(A) i (1164)κcenv(rankA) on {A∈ S n + | ‖A‖ 2 ≤κ} = 1 tr(A) (1165)κA properly scaled trace thus represents the best convex lower bound on rankfor positive semidefinite matrices. The idea, then, is to substitute the convexenvelope for rank of some variable A∈ S M + (A.6.5)rankA ← cenv(rankA) ∝trA = ∑ iσ(A) i = ∑ iiλ(A) i = ‖λ(A)‖ 1which is equivalent to the sum of all eigenvalues or singular values.(1166)[89] Convex envelope of the cardinality function is proportional to the1-norm:cenv(cardx) on {x∈ R n | ‖x‖ ∞ ≤κ} = 1 κ ‖x‖ 1 (1167)
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7.2. SECOND PREVALENT PROBLEM: 461rankXcenv rankXFigure 110: Abstraction of convex envelope of rank function. Rank isa quasiconcave function on the positive semidefinite cone, but its convexenvelope is the smallest convex function enveloping it.[90] [89] <strong>Convex</strong> envelope of rank function: for σ a singular value,cenv(rankA) on {A∈ R m×n | ‖A‖ 2 ≤κ} = 1 ∑σ(A) i (1164)κcenv(rankA) on {A∈ S n + | ‖A‖ 2 ≤κ} = 1 tr(A) (1165)κA properly scaled trace thus represents the best convex lower bound on rankfor positive semidefinite matrices. The idea, then, is to substitute the convexenvelope for rank of some variable A∈ S M + (A.6.5)rankA ← cenv(rankA) ∝trA = ∑ iσ(A) i = ∑ iiλ(A) i = ‖λ(A)‖ 1which is equivalent to the sum of all eigenvalues or singular values.(1166)[89] <strong>Convex</strong> envelope of the cardinality function is proportional to the1-norm:cenv(cardx) on {x∈ R n | ‖x‖ ∞ ≤κ} = 1 κ ‖x‖ 1 (1167)