Learn to solve optimization problems in Python using essential math tools, metaheuristic methods, and constrained optimization techniques. Master efficient solutions for real-world applications.

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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

Now, it's time to deal with bad constraints. Unfortunately, these are the most common kinds of constraints. They're difficult to get rid of, and it's also difficult to generate solutions that fulfill them.

For example, let's see the following problem:

$$
\min_{x, y} 2x + 3y\\
s.t.: x < y
$$

We don't know any exact or approximate method to solve this problem in this form. To get rid of the constraint, we'd need to make some analysis of the valid region of solutions in the plane. It's not too hard to do so, but that's because this is a simple problem.

If we use a genetic algorithm to solve this problem, we can generate valid initial solutions with a simple method. We first generate a random $y$ and then generate a random $x$, making sure it's less than $y$. But then we'd need to make sure that mutated solutions and crossover solutions maintain the constraint. With PSO, this can be even harder.

We can see how difficult it is to solve a simple problem involving inequality constraints. What if we further complicate things?

$$
\min_{a, b, c, d} 2a + 0.5b + 10c + d \\
s.t.: ab < c + d \\
a + c < \frac{d}{b}
$$

Now we're almost disarmed. We need to change our mindset and approach the problem in a different way. Let's see how.

Learn how to deal with constraints using penalty methods.

Introduction

Derivatives and Gradients

First Optimization Algorithms

Population Methods

Adding Constraints

Solving Sudoku and the 8-Queens Puzzle as Constraint Satisfaction

Linear Constrained Optimization

Summary and Conclusion

Appendix

Maze Solver Using the Ant Colony Optimization Algorithm

Penalty Methods