Convex Optimization and Modeling

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Additional Material

Read the book, it is very well written: Convex Optimization – Boyd and Vandenberghe
http://videolectures.net/mlss07_vandenberghe_copt/ - this is a nice video of one of the authors of the book we are using
Material of the first introductory tutorial: Optimization: Rayleigh-Ritz and Linear Algebra
Here is my alternative proof for Ex. 3.57 in Boyd, i.e. a proof that $f(X) = X^{-1}$ is matrix convex on $S_{++}^n$ .
Proof related to Boyd, Exercise 3.26 (a): $\sum_{i=1}^k \lambda_i(X) = \sup \{tr(V^T X V)| V \in R^{n \times k}, V^T V = I\}$ where $X \in S^n$ and $\lambda_1(X) \geq \lambda_2(X) \geq \ldots \geq \lambda_n(X)$ This shows that the given supremum is really equal to the sum of the k largest eigenvalues.
Additional proof for composition with affine mapping: $g(x)$ is convex/concave, if $f(x)$ is convex/concave:
$g(x) = f(Ax+b) = f(L(x)) \to f'(L(x))\cdot L'(X) \to f''(L(X))\cdot(L'(x))^2 + \underbrace{L''(x)\cdot f'(L(x))}_0$

Try to use Matlab for your plots (such as the one in Ex. 5.26 in Boyd). For this exercise, the code could be:

[X,Y] = meshgrid(-2:.1:2,-2:.1:2);
Z = X.^2+Y.^2;
C1=(X-1).^2+(Y-1).^2-1;
C2=(X-1).^2 +(Y+1).^2-1;
p=0.*X;

hold on;
  contourf(X,Y,Z);
  mesh(X,Y,C1);
  mesh(X,Y,C2);
  meshc(X,Y,p);
  axis([-2 2 -2 2 -2 2]);
hold off;

Then, rotate the plot manually.

Matlab Code For Smoothness Regularization - as discussed in Ex08

Wiki Plateau www.socher.org

Convex Optimization and Modeling

Additional Material