Gain insights into solving optimization problems in Python. Delve into essential mathematical tools, metaheuristic methods, and techniques for constrained optimization. Become proficient in efficient Python solutions.

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Optimization theory seeks the best solution, which is pivotal for machine learning, cost-cutting in manufacturing, refining logistics, and boosting finance profits.

This course provides a detailed description of different optimization problems and the techniques used to solve them. You’ll begin with the formal definition of an optimization problem and an overview of essential mathematical tools: derivatives, gradients, and Hessian. With this knowledge, you’ll implement solutions for several optimization problems. You’ll also learn approximate metaheuristic population methods like particle swarm optimization and genetic algorithms, and solve constrained optimization problems using techniques like penalty methods and constraint simplification. Finally, you’ll solve linear programs and integer programs. 

By the end of this course, you’ll become proficient in formulating different kinds of problems as optimization problems and become skilled in the Python tools that efficiently solve these problems.

Mastering Optimization with Python

## Exercise 1: Fractional knapsack

In the fractional knapsack problem, we don't have to choose the entire item, but we can get a part of it. That way, we aren't solving an integer problem anymore. The constraints are the same, but the solution is different from the original knapsack problem. In the original version, a solution is a zero or a one for each item, representing whether we take the item or not. Now a solution is a number between zero and one for each item, representing the fraction of that item we take. So the problem can be written as follows:

$$
\max_{\mathbf{x}} \mathbf{v}\mathbf{x}^\top\\
s.t.: \mathbf{w}\mathbf{x}^\top \leq w\\
0 \leq x_i \leq 1
$$

Let's see the code:

# Exercise 1: Fractional knapsack

In the fractional knapsack problem, we don't have to choose the entire item, but we can get a part of it. That way, we aren't solving an integer problem anymore. The constraints are the same, but the solution is different from the original knapsack problem. In the original version, a solution is a zero or a one for each item, representing whether we take the item or not. Now a solution is a number between zero and one for each item, representing the fraction of that item we take. So the problem can be written as follows:

$$
\max_{\mathbf{x}} \mathbf{v}\mathbf{x}^\top\\
s.t.: \mathbf{w}\mathbf{x}^\top \leq w\\
0 \leq x_i \leq 1
$$

Let's see the code:

Solve the exercises about linear and integer programming.

Introduction

Derivatives and Gradients

First Optimization Algorithms

Population Methods

Adding Constraints

Linear Constrained Optimization

Summary and Conclusion

Appendix

Solutions

Exercise 1: Fractional knapsack