v2010.10.26 - Convex Optimization

3.2. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 231What about minimizing other functions? Given some nonsingularmatrix A , geometrically describe three circumstances under which there arelikely to exist vertex solutions tominimizex∈R n ‖Ax‖ 1subject to x ∈ Poptimized over some bounded polyhedron P . 3.113.2.1 k smallest entries(524)Sum of the k smallest entries of x∈ R n is the optimal objective value from:for 1≤k ≤nn∑n∑π(x) i = minimize x T yπ(x) i = maximize k t + 1 T zy∈R ni=n−k+1 z∈R n , t∈Rsubject to 0 ≼ y ≼ 1≡subject to x ≽ t1 + z1 T y = kz ≼ 0(525)i=n−k+1which are dual linear programs, where π(x) 1 = max{x i , i=1... n} whereπ is a nonlinear permutation-operator sorting its vector argument intononincreasing order. Finding k smallest entries of an n-length vector x isexpressible as an infimum of n!/(k!(n − k)!) linear functions of x . The sum∑ π(x)i is therefore a concave function of x ; in fact, monotonic (3.6.1.0.1)in this instance.3.2.2 k largest entriesSum of the k largest entries of x∈ R n is the optimal objective value from:[61, exer.5.19]k∑k∑π(x) i = maximize x T yπ(x) i = minimize k t + 1 T zy∈R ni=1z∈Rsubject to 0 ≼ y ≼ 1≡n , t∈Rsubject to x ≼ t1 + z1 T y = kz ≽ 0(526)i=13.11 Hint: Suppose, for example, P belongs to an orthant and A were orthogonal. Beginwith A=I and apply level sets of the objective, as in Figure 67 and Figure 70. [ Or rewrite ] xthe problem as a linear program like (514) and (516) but in a composite variable ← y .t