# 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.


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