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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

Let's solve some continuous and discrete linear optimization problems.

## Exercise 1: Fractional knapsack

The **fractional knapsack** is the continuous version of the knapsack problem. In this version, we can get a fraction of the item. The value would be the same fraction of the item's value and weight of the same fraction of the weight. For example, consider an item of value $6$ and a weight of $9$kg. If we decide to take one-third of the item, then we get value $\frac{1}{3} * 6 = 2$ from the object, and the weight added to the load will be $\frac{1}{3} * 9 = 3$kg. As in the knapsack problem, the constraint is not to load more than the maximum weight allowed and the goal is to get as much value as possible.

In this exercise, we have to solve the fractional version of the knapsack problem.



Let's solve some continuous and discrete linear optimization problems.

# Exercise 1: Fractional knapsack

The **fractional knapsack** is the continuous version of the knapsack problem. In this version, we can get a fraction of the item. The value would be the same fraction of the item's value and weight of the same fraction of the weight. For example, consider an item of value $6$ and a weight of $9$kg. If we decide to take one-third of the item, then we get value $\frac{1}{3} * 6 = 2$ from the object, and the weight added to the load will be $\frac{1}{3} * 9 = 3$kg. As in the knapsack problem, the constraint is not to load more than the maximum weight allowed and the goal is to get as much value as possible.

In this exercise, we have to solve the fractional version of the knapsack problem.



Introduction

Derivatives and Gradients

First Optimization Algorithms

Population Methods

Adding Constraints

Linear Constrained Optimization

Summary and Conclusion

Appendix

Exercises

Exercise 1: Fractional knapsack