Optimization

By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

# 1. Optimization¶

• an important tool in 1) engineering problem solving and 2) decision science
• people optimize
• nature optimizes

3 key components

1. objective function
2. decision variable or unknown
3. constraints

Procedures

1. The process of identifying objective, variables, and constraints for a given problem is known as "modeling"
2. Once the model has been formulated, optimization algorithm can be used to find its solutions.

In mathematical expression

\begin{align*} \min_{x} \quad &f(x) \\ \text{subject to} \quad &g_i(x) \leq 0, \qquad i=1,\cdots,m \end{align*}

$\;\;\;$where

• $x=\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix} \in \mathbb{R}^n$ is the decision variable
• $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is an objective function
• Feasible region: $\mathcal{C} = \{x: g_i(x) \leq 0. \quad i=1, \cdots,m\}$

Remarks) equivalent

\begin{align*} \min_{x} f(x) \quad&\leftrightarrow \quad \max_{x} -f(x)\\ \quad g_i(x) \leq 0\quad&\leftrightarrow \quad -g_i(x) \geq 0\\ h(x) = 0 \quad&\leftrightarrow \quad \begin{cases} h(x) \leq 0 \quad \text{and} \\ h(x) \geq 0 \end{cases} \end{align*}

• The good news: for many classes of optimization problems, people have already done all the "hardwork" of developing numerical algorithms
• A wide range of tools that can take optimization problems in "natural" forms and compute a solution

# 2. Solving Optimization Problems¶

• Starting with the unconstrained, one dimensional case

• To find minimum point $x^*$, we can look at the derivave of the function $f'(x)$
• Any location where $f'(x)$ = 0 will be a "flat" point in the function
• For convex problems, this is guaranteed to be a minimum
• Generalization for multivariate function $f:\mathbb{R}^n \rightarrow \ \mathbb{R}$

• The gradient of $f$ must be zero
$$\nabla _x f(x) = 0$$
• Gradient is a n-dimensional vector containing partial derivatives with respect to each dimension

$$x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \quad \quad \quad \quad \nabla _x f(x) = \begin{bmatrix} \partial f(x) \over \partial x_1 \\ \vdots\\ \partial f(x) \over \partial x_n \end{bmatrix}$$

• For continuously differentiable $f$ and unconstrained optimization, optimal point must have $\nabla _x f(x^*)=0$

## 2.1. How To Find $\nabla _x f(x) = 0$: Analytic Approach¶

• Direct solution

• In some cases, it is possible to analytically compute $x^*$ such that $\nabla _x f(x^*)=0$

\begin{align*} f(x) &= 2x_1^2+ x_2^2 + x_1 x_2 -6 x_1 -5 x_2\\\\ \Longrightarrow \nabla _x f(x) &= \begin{bmatrix} 4x_1+x_2-6\\ 2x_2 + x_1 -5 \end{bmatrix} = \begin{bmatrix}0\\0 \end{bmatrix}\\\\ \therefore x^* &= \begin{bmatrix} 4 & 1\\ 1 & 2 \end{bmatrix} ^{-1} \begin{bmatrix} 6 \\ 5\\ \end{bmatrix} = \begin{bmatrix} 1 \\ 2\\ \end{bmatrix} \end{align*}

• Note: Matrix derivatives

## 2.2. How To Find $\nabla _x f(x) = 0$: Iterative Approach¶

• Iterative methods

• More commonly the condition that the gradient equal zero will not have an analytical solution, require iterative methods

• The gradient points in the direction of "steepest ascent" for function $f$

• It motivates the gradient descent algorithm, which repeatedly takes steps in the direction of the negative gradient

$$x \leftarrow x - \alpha \nabla _x f(x) \quad \quad \text{for some step size } \alpha > 0$$

$$\text{Repeat : } x \leftarrow x - \alpha \nabla _x f(x) \quad \quad \text{for some step size } \alpha > 0$$

• Gradient Descent in Higher Dimension
$$\text{Repeat : } x \leftarrow x - \alpha \nabla _x f(x)$$

Example

\begin{align*} \min& \quad (x_1-3)^{2} + (x_2-3)^{2}\\\\ =\min& \quad \frac{1}{2} \begin{bmatrix} x_1 & x_2 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} - \begin{bmatrix} 6 & 6 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + 18 \end{align*}

\begin{align*} f &= \frac{1}{2}X^THX + g^TX \\ \nabla f &= HX+g \end{align*}
• update rule
$$X_{i+1} = X_{i} - \alpha_i \nabla f(X_i)$$
In [1]:
import numpy as np

H = np.matrix([[2, 0],[0, 2]])
g = -np.matrix([[6],[6]])

x =  np.zeros((2,1))
alpha = 0.2

for i in range(25):
df = H*x + g
x = x - alpha*df

print(x)

[[2.99999147]
[2.99999147]]


## 3.1. Choosing Step Size¶

• Learning rate