Solve the quadratic optimization problem:

To solve a quadratic optimization problem, we need to follow these steps:

1. Write down the objective function and constraints in standard form.
2. Formulate the Lagrangian function.
3. Take the partial derivatives of the Lagrangian with respect to all variables and set them equal to zero to find the optimal solution.
4. Solve for the Lagrange multipliers.
5. Plug in the optimal solution and Lagrange multipliers to find the optimal value of the objective function.

Let's consider an example quadratic optimization problem:

minimize 2x^2 + 3y^2 - 4xy
subject to x + y = 1

1. Writing in standard form:
minimize 2x^2 + 3y^2 - 4xy
subject to x + y = 1

2. The Lagrangian function is:
L(x,y,λ) = 2x^2 + 3y^2 - 4xy + λ(1 - x - y)

3. Taking the partial derivatives and setting them equal to zero:
∂L/∂x = 4x - 4y - λ = 0
∂L/∂y = 6y - 4x - λ = 0
∂L/∂λ = 1 - x - y = 0

Solving these equations, we get:
x = 2/5, y = 3/5, λ = -4/5

4. The Lagrange multipliers are λ = -4/5.

5. Plugging in the optimal solution and Lagrange multipliers, we get the optimal value of the objective function:
f(x,y) = 2(2/5)^2 + 3(3/5)^2 - 4(2/5)(3/5) = 1.6

Therefore, the solution to the quadratic optimization problem is x = 2/5, y = 3/5, and the minimum value of the objective function is 1.6.