---
date: '2024-02-01'
description: economic optimization framework with decision variables, constraints, objective functions, feasible sets, and optimal value determination.
id: Optimization
modified: 2026-06-05 15:08:34 GMT-04:00
tags:
  - eng3px3
title: Economic Optimization
created: '2024-02-01'
published: '2024-02-01'
pageLayout: default
slug: thoughts/university/twenty-three-twenty-four/eng-3px3/Optimization
permalink: https://aarnphm.xyz/thoughts/university/twenty-three-twenty-four/eng-3px3/Optimization.md
generator:
  quartz: v4.6.0
  hostedProvider: Cloudflare
  baseUrl: aarnphm.xyz
full: https://aarnphm.xyz/llms-full.txt
---
See also [[thoughts/university/twenty-three-twenty-four/eng-3px3/3PX3 07 Optimization Problem Formulation.pdf|slides]]

# model-based

- conclusions from the model of the system

Components:

- decision variables
- constraints
- objectives
- functions: mathematical function that determines the objective as a function of decision variable

$$
\begin{align*}
\min_{x} \phi = f(x) & &\leftarrow &\space \text{Objective function} \\\
\text{s.t} & &\leftarrow &\space \text{Constraints} \\\
h(x) = 0 & &\leftarrow &\space \text{Equality constraints} \\\
g(x) \leq 0 & &\leftarrow &\space \text{Inequality constraints} \\\
x_{lb} \leq x \leq x_{ub} & &\leftarrow &\space \text{Bounds}
\end{align*}
$$

## decision variables

### discrete.

> limited to a fixed or countable set of values

$$
x_{\mathcal{D}} \mid a \in \mathcal{I} = \lbrace 1, 2, 3, 4, 5 \rbrace
$$

### continuous.

> can take any value within a range

$$
x_{\mathcal{C}} \subset \mathcal{R}
$$

## constraints

- physical limitations: cannot purchase negative raw materials

- model assumptions: assumptions about the system

> \[!tip\] _domain of a definition_
>
> a decision upper and lower bounds ($x^{\mathcal{U}}$ and $x^{\mathcal{L}}$)

> \[!note\] Properties
>
> - **Active/binding**: $\exists \space x^{*} \mid g(x^{*}) = 0$
> - **Inactive**: $\exists \space x^{*} \mid g(x^{*}) < 0$

### graphing models

> \[!note\] feasible set of an optimization model
>
> The collection of decision variables that satisfy all constraints
>
> $$
> \mathcal{S} \triangleq \lbrace x : g(x) \leq 0, h(x) = 0, x^L \leq x \leq x^U \rbrace
> $$

## outcomes

> \[!tip\] optimal value
>
> the optimal value $\phi^{*}$ is the value of the objective at the optimum(s)
>
> $$
> \phi^{*} \triangleq \phi(x^{*})
> $$

> Constraints satisfy, but it is not binding

Linear optimization problems

$$
\begin{aligned}
\underset{x_1,x_2}{\min} \space \phi &= 50x_1 + 37.5x_2 \\
&\text{s.t} \\\
0.3x_1 + 0.4x_2 &\geq 2000 \\\
0.4x_1 + 0.15x_2 &\geq 1500 \\\
0.2x_1 + 0.35x_2 &\leq 1000, \\\
x_1 &\leq 9000 \\\
x_2 &\leq 6000 \\\
x_i &\geq 0
\end{aligned}
$$

<!-- end date Feb 1 -->

See also [[thoughts/university/twenty-three-twenty-four/eng-3px3/Linear Optimization]]