EE523 Convex optimization

Total Study flow of this course

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Contents(4 folded)

1. Figure out what QP is and show its convexity

1-1. What is QP?

$$ \begin{align}\min_{x} \quad & w^T x + x^T Q x \\\text{subject to} \quad & Ax - b \leq 0 \\& Cx - e = 0\end{align} $$

which Q matrix is PSD condition.

A symmetric matrix Q is “positive semi-definite” if $v^TQv\geq 0 \hspace{3mm} \forall v\in \real^d$

$$ v^TQv\geq 0 \hspace{3mm} \forall v\in \real^d $$

위 조건은 All the eigenvalue’s of Q matrix are non-negative와 동치!

which is eigenvalue decomposition.

1-2. Show Convexity

Convexity 를 보이는 방법은 3가지

  1. def of convex function을 사용 (dom f convex set, convex inequality 어쩌구)
  2. 1st- order condition of convex (PS1 에서 보임)