quiz drop: used for quiz 4. hw drop: used for hw8.

(44+44+43+33+46+54+38+30+35.5) / (4010) * 20 + (91+71) / 100 15 + 0/100 * 40 + (10+7+10+0+10) / (10*5) * 10 = 50.19

Reminder

Overview

Def. A nonlinear “program” (optimization problem) P:

$$ \argmin_{\vec x \in S \subset \R^n} f(\vec x), s.t.\ \forall 1 \le i \le m, g_i(\vec x) \le 0 $$
- Def. (Feasible region F of P) The set of $\vec x$ (feasible/candidate solution) satisfying all constraints.
- Def. (Value of P) $MP=\inf f(x)$ over feasible region F.
- When a problem has no feasible solutions, we call it infeasible or inconsisent, which will yield a value of $+\infty$.
We are going to:
- look at special cases which we do know how to solve,
- reduce complicated cases to simpler ones, and
- classify problems into cases which can or cannot be handled.

Chain rule of $f \circ g$ when $f: \R^n \to \R, g: \R \to \R^n$ differentiable:
- Need $\frac{\partial f}{\partial x_i}$ to be continuous.
- $\frac{d}{dt} f(g(t)) = \sum_{i=1}^n \frac{\partial f}{\partial x_i} (g(t)) \frac{d}{dt} g_i(t)$
Gradient

$$ \nabla f(\vec x) = [ \frac{\partial f}{\partial x_1} (\vec x) \dots \frac{\partial f}{\partial x_n} (\vec x) ]^T $$
Hessian matrix

${\bf H}_f(\boldsymbol{x}) = \begin{bmatrix} \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1 \partial x_2} & \ldots & \dfrac{\partial^2 f}{\partial x_1 \partial x_n} \\ \dfrac{\partial^2 f}{\partial x_2 \partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \ldots & \dfrac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial^2 f}{\partial x_n \partial x_1} & \dfrac{\partial^2 f}{\partial x_n \partial x_2} & \ldots & \dfrac{\partial^2 f}{\partial x_n^2} \end{bmatrix}$