Question

Let :ℝ³→ℝ be defined by (x₁,x₂,x₃) ≡ 12x₁ + 10x₂ ― (x₁)² ―2(x₂)²―3(x₃)² +2(x₁)(x₂) + 2(x₂)(x₃) +100. Show that can be written as a vector quadratic function in the standard form, and hence solve the unconstrained problem max x (x).

Answer 1

The given vector quadratic function can be written in standard form using matrices. It is then solved for the maximum by setting its gradient to zero, although no maximum is found since the function is not bounded above (A is not positive definite).

Step-by-step explanation:

The function given is a vector quadratic function which can be written in the standard form by reorganizing the given terms into a matrix representation. First, we need to express it as xTAx + bx + c, where x is the vector of variables, A is a symmetric matrix representing the quadratic terms, b is the vector of linear coefficients, and c is the constant term. For this function, we would have:

A = \begin{bmatrix} -1 & 1 & 0 \\ 1 & -2 & 1 \\ 0 & 1 & -3 \end{bmatrix}, b = \begin{bmatrix} 12 \\ 10 \\ 0 \end{bmatrix}, c = 100.

To solve the unconstrained problem of maximizing this function, we can take the gradient of the function, set it equal to zero, and solve for the vector x that maximizes the function. Alternatively, recognizing that the matrix A is not positive definite, we can infer that the function does not have a maximum due to the presence of negative eigenvalues which indicate that the function will go to negative infinity in some directions.