Question

Show that maximization of the class separation criterion given by (4.23) with respect to w, using a Lagrange multiplier to enforce the constraint $w^Tw = 1$, leads to the result that $w \propto (m_2 - m_1)$. hint: Lagrange multipliers¹ provides a strategy for finding the maxima and minima of a function subject to constraints. For example, maximize $f(x, y)$ subject to $g(x,y) = c$. We introduce a new variable ($\lambda$) called a Lagrange multiplier, and define the Lagrange function $\Lambda(x, y, \lambda) = f(x, y) + \lambda \cdot (g(x, y) - c)$. The extrema can be found by solving the equation array $\frac{\partial \Lambda}{\partial x} = 0$ $\frac{\partial \Lambda}{\partial y} = 0$ $\frac{\partial \Lambda}{\partial \lambda} = 0$

          Show that maximization of the class separation criterion given by (4.23) with respect to
w, using a Lagrange multiplier to enforce the constraint $w^Tw = 1$, leads to the result that
$w \propto (m_2 - m_1)$.
hint: Lagrange multipliers¹ provides a strategy for finding the maxima and minima of a
function subject to constraints. For example, maximize $f(x, y)$ subject to $g(x,y) = c$. We
introduce a new variable ($\lambda$) called a Lagrange multiplier, and define the Lagrange function
$\Lambda(x, y, \lambda) = f(x, y) + \lambda \cdot (g(x, y) - c)$.
The extrema can be found by solving the equation array
$\frac{\partial \Lambda}{\partial x} = 0$
$\frac{\partial \Lambda}{\partial y} = 0$
$\frac{\partial \Lambda}{\partial \lambda} = 0$

Show that maximization of the class separation criterion given by (4.23) with respect to
w, using a Lagrange multiplier to enforce the constraint w^Tw = 1, leads to the result that
w ∝ (m2 - m1).
hint: Lagrange multipliers¹ provides a strategy for finding the maxima and minima of a
function subject to constraints. For example, maximize f(x, y) subject to g(x,y) = c. We
introduce a new variable (λ) called a Lagrange multiplier, and define the Lagrange function
Λ(x, y, λ) = f(x, y) + λ· (g(x, y) - c).
The extrema can be found by solving the equation array
(∂Λ)/(∂ x) = 0
(∂Λ)/(∂ y) = 0
(∂Λ)/(∂λ) = 0

Added by Irene G.

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