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

Here, we’ll review the solutions to the exercise problems of constrained optimization.

# Exercise 1: A simple contained problem

The first problem was:

$$
\min_{x, y} x^3 - y \\
s.t.: 3x - y = 1 \\
x^3 > -1
$$

Here, we have two constraints. The first one is a good constraint. We can use it to reduce the number of variables. Let's use it to get rid of the $x$ variable:

$$
\min_x x^3 - 3x + 1 \\
s.t.: x^3 > -1
$$

Now we have a univariate problem with a single constraint. This constraint is a bad constraint. But we know a method to solve this kind of problem. As $x^3 > -1$, then $x > -1$. If we look at the graph of the function, we see that it's bitonic for those values of $x$ and that the minimum lies somewhere between $0$ and $2$. So we can solve the problem with a ternary search.

Solve the proposed problems about constrained optimization.

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

Solutions

Exercise 1: A simple contained problem