1. Introduction to Optimization Problems
Optimization problems involve finding the best solution, often minimizing or maximizing a function, subject to constraints. These problems can have one or more variables.
2. Gradient Descent for Single Variable
Gradient descent is an iterative optimization algorithm used to find the minimum of a function. In the case of a single variable, we update the variable by taking a step in the direction of the negative gradient (downhill direction).
The update rule for gradient descent is given by:
x₁ = x₀ - α * ∇f(x₀)
where α is the learning rate, and ∇f(x₀) is the gradient of the function at x₀.
3. Gradient Descent for Multiple Variables
When there are multiple variables, gradient descent updates each variable in the direction of its partial derivative, considering the entire gradient vector.
The update rule for multiple variables is:
x₁ = x₀ - α * ∇f(x₀)
where α is the learning rate, and ∇f(x₀) is the gradient vector of the function.
4. Lagrange Multipliers (Constrained Optimization)
Lagrange multipliers are used in constrained optimization problems, where we optimize a function subject to constraints. The method introduces a new variable (the Lagrange multiplier) to account for the constraint.
The Lagrangian is given by:
L(x, λ) = f(x) - λ * g(x)
where f(x) is the objective function, g(x) is the constraint, and λ is the Lagrange multiplier.