Math Problem Statement

Nous souhaitons trouver le minimum global de la fonction suivante : f(x, y) = y^2 + (y - x)^2. Par deux méthodes : analytiquement et par la méthode de descente de gradient.

Solution

It appears that you're dealing with a problem related to gradient descent and function minimization. Let's break down the questions:

  1. Question 3.1: You're asked to determine the coordinates (x∗,y∗)(x^*, y^*) that correspond to the global minimum of the given function f(x,y)=y2+(y−x)2f(x, y) = y^2 + (y - x)^2.

  2. Question 3.2: You need to recall the method of the gradient descent algorithm to minimize a given function.

  3. Question 3.3: This involves performing the first step of the gradient descent algorithm for specific step sizes (η=0.01,0.1,10\eta = 0.01, 0.1, 10) and calculating the coordinates and distance between initial and minimum points. This will help demonstrate how the step size influences the descent process.

  4. Question 3.4: Discuss the implications of using different step sizes in gradient descent and strategies for handling them effectively.

  5. Question 3.5: Implement the gradient descent algorithm using a specified programming framework to minimize the given function and test it with various step sizes.

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Math Problem Analysis

Mathematical Concepts

Optimization
Gradient Descent
Multivariable Functions

Formulas

f(x, y) = y^2 + (y - x)^2

Theorems

Gradient Descent Algorithm

Suitable Grade Level

Undergraduate

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