v2010.10.26 - Convex Optimization

232 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSwhich are dual linear programs. Finding k largest entries of an n-lengthvector x is expressible as a supremum of n!/(k!(n − k)!) linear functionsof x . (Figure 74) The summation is therefore a convex function (andmonotonic in this instance,3.6.1.0.1).3.2.2.1 k-largest normLet Πx be a permutation of entries x i such that their absolute valuebecomes arranged in nonincreasing order: |Πx| 1 ≥ |Πx| 2 ≥ · · · ≥ |Πx| n .Sum of the k largest entries of |x|∈ R n is a norm, by properties of vectornorm (3.2), and is the optimal objective value of a linear program:‖x‖nkk∑ ∑|Πx| i = k π(|x|) ii=1{= sup a T i xi∈Ii=1∣ a }ij ∈ {−1, 0, 1}carda i = k= minimize k t + 1 T zz∈R n , t∈Rsubject to −t1 − z ≼ x ≼ t1 + zz ≽ 0= maximize (y 1 − y 2 ) T xy 1 , y 2 ∈R nsubject to 0 ≼ y 1 ≼ 10 ≼ y 2 ≼ 1where the norm subscript derives from a binomial coefficient(y 1 + y 2 ) T 1 = k( nk), and(527)‖x‖n n= ‖x‖ 1‖x‖n1= ‖x‖ ∞(528)‖x‖nk= ‖π(|x|) 1:k ‖ 1Sum of k largest absolute entries of an n-length vector x is expressible asa supremum of 2 k n!/(k!(n − k)!) linear functions of x ; (Figure 74) hence,this norm is convex in x . [61, exer.6.3e]minimize ‖x‖nx∈R n ksubject to x ∈ C≡minimizez∈R n , t∈R , x∈R nsubject tok t + 1 T z−t1 − z ≼ x ≼ t1 + zz ≽ 0x ∈ C(529)