Lec1

Optimization steps

Problem analysis: real world $\iff$ math relationship

$\min f(x)$ objective function (modeling)

s.t $x \in S$ , S = feasible set.

study the math properties of the model: (model analysis)

apply an algorithm to compute an optimal solution $x^*$ (solution method)

verification and simulation

Linear Optimization

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is linear if:

1. $f(x+y)=f(x)+f(y) \quad \forall x, y \in \mathbb{R}^n$

2. $f(\lambda x)=\lambda f(x) \quad \forall x \in \mathbb{R}^n, \forall \lambda \in \mathbb{R}$ Equivalently, $f$ is linear if it can be written as:

$$c^{\prime} x=\sum_{i=1}^n c_i x_i=c_1 x_1+c_2 x_2+\cdots+c_n x_n,$$

where $c_1, c_2, \ldots, c_n$ are real numbers.

real-world relationships $\Leftrightarrow$ math relationships

$\min d^{\prime} x \quad$ LINEAR objective function

s.t. $\quad a_i^{\prime} x \geq b_i \quad i \in M_1$

$a_i^{\prime} x \leq b_i \quad i \in M_2 \quad$ LINEAR inequalities

$a_i^{\prime} x=b_i \quad i \in M_3$

Example

$$\begin{aligned} \operatorname{minimize} & 2 x_1-x_2+4 x_3 \\ \text { subject to } & x_1+x_2+x_4 \leq 2 \\ & 3 x_2-x_3=5 \\ & x_3+x_4 \geq 3 \\ & x_1 \geq 0 \\ & x_3 \leq 0 \end{aligned}$$

Strict inequalities like $x_3+x_4>3$ are not allowed!

We learn about model analysis and solution methods.

Optimal conditions, duality. - model analysis

Simplex algorithm - solution methods.