Appendix C. SVM Dual Problem

To understand duality, you first need to understand the Lagrange multipliers method. The general idea is to transform a constrained optimization objective into an unconstrained one, by moving the constraints into the objective function. Let’s look at a simple example. Suppose you want to find the values of x and y that minimize the function f(x, y) = x2 + 2y, subject to an equality constraint: 3x + 2y + 1 = 0. Using the Lagrange multipliers method, we start by defining a new function called the Lagrangian (or Lagrange function): g(x, y, α) = f(x, y) – α(3x + 2y + 1). Each constraint (in this case just one) is subtracted from the original objective, multiplied by a new variable called a Lagrange multiplier.

Joseph-Louis Lagrange showed that if (x^,y^) is a solution to the constrained optimization problem, then there must exist an α^ such that (x^,y^,α^) is a stationary point of the Lagrangian (a stationary point is a point where all partial derivatives are equal to zero). In other words, we can compute the partial derivatives of g(x, y, α) with regard to x, y, and α; we can find the points where these derivatives are all equal to zero; and the solutions to the constrained optimization problem (if they exist) must be among these stationary points.

In this example the partial derivatives are: xg(x,y,α)=2x-3αyg(x,y,α)=2-2ααg(x,y,α)=-3x-2y-1

When all these partial derivatives are equal to 0, we find that 2 x ^ - 3 α ^ = 2 - 2 α ^ = -3 x ^ - 2

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