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

As derivatives are functions, they have their own derivatives. The derivative of a derivative is called a **second-order derivative**. In the following code, we calculate the second-order derivative of $f = x^2$.

from sympy import symbols, diff

x = symbols('x')

f = x**2

df = diff(f, x) # first-order derivative

ddf =  diff(df, x) # second-order derivative

print("First-order:", df)
print("Second-order:", ddf)

# Second-order derivatives in multiple dimensions

In cases with more than one variable, we can calculate the second-order derivative with respect to the same first variable or with respect to another variable. For example, we denote as $\frac{\partial^2 f}{\partial x^2}$ the operation of taking the derivative of $f$ with respect to $x$ and then taking again the derivative with respect to $x$. But we could take the second derivative with respect to another variable. We denote as $\frac{\partial^2 f}{\partial x\partial y}$ when we first take the derivative with respect to $x$ and then with respect to $y$. See an example with multiple variables below. We'll use the function $f(x, y) = x^2 + x^y$.


Learn what second-order derivatives are and how to calculate the Hessian of a function.

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

Second-Order Derivatives and the Hessian

Second-order derivatives in multiple dimensions