Download Convex Optimization, Solutions Manual by Stephen Boyd,Lieven Vandenberghe PDF

By Stephen Boyd,Lieven Vandenberghe

Show description

Read Online or Download Convex Optimization, Solutions Manual PDF

Best nonfiction_5 books

The Social Origins of Christian Architecture, Vol. I: Building God's House in the Roman World: Architectural Adaptation Among Pagans, Jews, and Christians

[White's] really apt mixture of archaeological is still, textual facts, non secular philosophies, and Roman social background. .. An enthusiastic, well-written presentation measured with erudition and sound examine. --Classical international.

Nazi-Deutsch Nazi German: An English Lexicon of the Language of the Third Reich

Created and used as an device of coercion and indoctrination, the Nazi language, Nazi-Deutsch, finds how the Nazis governed Germany and German-occupied Europe, fought global struggle II, and dedicated mass homicide and genocide, making use of language to encode and euphemize those activities. Written through students focusing on socio-linguistic and old problems with the Nazi interval, this publication presents a different, broad, meticulously researched dictionary of the language of the 3rd Reich.

Extra info for Convex Optimization, Solutions Manual

Example text

3. Prove the following. 21) if and only if there exists a σ such that ∇2 f (x) + σ∇f (x)∇f (x)T 0. 22) for all y = 0 if and only if there exists a σ such ∇2 f (x) + σ∇f (x)∇f (x)T 0. 27) Hint. We can assume without loss of generality that ∇2 f (x) is diagonal. 21) if and only if either ∇f (x) = 0 and ∇2 f (x) or ∇f (x) = 0 and the matrix H(x) = ∇2 f (x) ∇f (x)T 0, ∇f (x) 0 has exactly one negative eigenvalue. 22) for all y = 0 if and only if H(x) has exactly one nonpositive eigenvalue. Hint. You can use the result of part (a).

Therefore the affine function h(x) = (c − aT x)/b lies between f and g. 13 Kullback-Leibler divergence and the information inequality. 17). Prove the information inequality: Dkl (u, v) ≥ 0 for all u, v ∈ Rn ++ . Also show that Dkl (u, v) = 0 if and only if u = v. Hint. The Kullback-Leibler divergence can be expressed as Dkl (u, v) = f (u) − f (v) − ∇f (v)T (u − v), 3 Convex functions n where f (v) = i=1 vi log vi is the negative entropy of v. Solution. The negative entropy is strictly convex and differentiable on Rn ++ , hence f (u) > f (v) + ∇f (v)T (u − v) for all u, v ∈ Rn ++ with u = v.

Let f be a convex function. Define the function g as f (αx) . g(x) = inf α>0 α (a) Show that g is homogeneous (g(tx) = tg(x) for all t ≥ 0). (b) Show that g is the largest homogeneous underestimator of f : If h is homogeneous and h(x) ≤ f (x) for all x, then we have h(x) ≤ g(x) for all x. (c) Show that g is convex. Solution. (a) If t > 0, f (αtx) f (αtx) = t inf = tg(x). α>0 α tα For t = 0, we have g(tx) = g(0) = 0. g(tx) = inf α>0 Exercises (b) If h is a homogeneous underestimator, then h(x) = h(αx) f (αx) ≤ α α for all α > 0.

Download PDF sample

Rated 4.42 of 5 – based on 28 votes