\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)
Convex optimization
Lange, ch.5.
Convex programs. Convexity of the set of optima.
Constraint qualification: Slater condition.
Duality
BoydVandenberghe, 5.13,5.6.
Lagrange dual functions. Dual optimization problems. Fenchel duals (conjugate) functions.
Weak duality for general convex programs. Strong duality.
Sample problems

Find Fenchel conjugate function for the following functions:

\[
f(x)=\ln({e^x+e^{x}});
\]Answer: \(1/2((1p)\ln(1+p)+(1+p)\ln(1p))\ln2\).

\[
f(x)=x1+x+1;
\]
Answer: \(p2\) for \(2\leq p\leq 2\); \(+\infty\) elsewhere. 
\[
f(x)=x^2\ \mathrm{for}\ x\leq 1; +\infty \ \mathrm{otherwise}.
\]
Answer: \(p^2/4\) for \(2\leq p\leq 2\); \(p\) elsewhere.

\[
 Consider the problem
\[
\begin{array}{c}
ax+by\to\min,\ \mathrm{subject\ to} \\x^2+y^2\leq 1; \\x\leq 0.
\end{array}
\] Is it convex?
 Solve it. (Consider \(a,b\) as unknown parameters; the solutions of the primal and dual problems depend on these parameters.)
 Formulate the dual problem and solve it.
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