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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

The special methods we've learned so far to solve linear problems work for continuous problems, in which the variables can take any real value. However, there are some problems in which variables should be integers.

## Integer problems

For example, when we want to know the number of workers needed to accomplish a task or the number of soda cans a business owner should buy for their store, the solution can't be a fraction. It should be an integer number.

Linear problems that are restricted to integer solutions are commonly called **integer problems**, and the field that studies the solution to those problems is **integer programming**. Formally, these problems are defined as linear problems with an additional constraint.

$$
\min_{\mathbf{x}} \mathbf{c}\mathbf{x}^\top\\
s.t.: P\mathbf{x}^\top \leq \mathbf{p}\\
Q\mathbf{x}^\top = \mathbf{q}\\
\mathbf{x} \in \mathbb{Z}^n
$$

Here, $\mathbf{x}$ is a vector with $n$ dimensions and the third constraint tells us that all of these dimensions have integer values.

The special methods we've learned so far to solve linear problems work for continuous problems, in which the variables can take any real value. However, there are some problems in which variables should be integers.

# Integer problems

For example, when we want to know the number of workers needed to accomplish a task or the number of soda cans a business owner should buy for their store, the solution can't be a fraction. It should be an integer number.

Linear problems that are restricted to integer solutions are commonly called **integer problems**, and the field that studies the solution to those problems is **integer programming**. Formally, these problems are defined as linear problems with an additional constraint.

$$
\min_{\mathbf{x}} \mathbf{c}\mathbf{x}^\top\\
s.t.: P\mathbf{x}^\top \leq \mathbf{p}\\
Q\mathbf{x}^\top = \mathbf{q}\\
\mathbf{x} \in \mathbb{Z}^n
$$

Here, $\mathbf{x}$ is a vector with $n$ dimensions and the third constraint tells us that all of these dimensions have integer values.

Learn to solve problems involving integer numbers.

Introduction

Derivatives and Gradients

First Optimization Algorithms

Population Methods

Adding Constraints

Linear Constrained Optimization

Summary and Conclusion

Appendix

Integer Programs

Integer problems