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minimize $2x_1^2+x_2^2-2x_1x_2-4x_2$

subject to $x_1x_2\leq 4$

  1. Is the objective function convex?

  2. Does the point $(1,4)$ satisfy the FONC for a local minimizer?

  3. Do optimal solutions of this given problem change if the constraint is replaced by the new constraint $x_1x_2\geq 4$?

I have done the following:

This is a quadratic function, rewriting it gives the Hessian to be

  1. $\begin{bmatrix} 4&-2\\-2&2 \end{bmatrix}$ which has eigenvalues 5.2 and 0.76 so the function is convex.
  1. The Lagrangian can be written as

$L(x,\mu)=2x_1^2+x_2^2-2x_1x_2-4x_2 -\mu(x_1x_2 -4)$

The KKT conditions are therefore:

$\nabla x L(x,\mu)=[4x_1-2x_2-\mu x_2=0,2x_2-2x_1-4-\mu x_1=0]$

$\mu(x_1x_2-4)=0$

Filling in $(1,4)$ in the first equation of $\nabla xL(x,\mu)$ gives

$4*1-2*4-4\mu =0$ so $\mu = -1$

Now using the value found for $\mu$ we fill in $(1,4)$ in the second equation of $\nabla xL(x,\mu)$. This gives

$8-2-4-(-1)1= 3 \neq 0$

Thus the point $(1,4)$ does not satisfy the FONC for a local minimizer.

  1. I think the answer is yes, since the constraints are then changed.

Can someone please tell me if the answers I've given are correct or not? If not please explain why and what the right answer should be. Thanks a lot !

Alice
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1 Answers1

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hint

i agree with (1) and (2), but (3) is non-trivial. You are optimizing a convex function over a convex interval. If the extreme values lie on the boundary, they satisfy $x_1x_2 = 4$, so that point will satisfy both sets of constraints.

gt6989b
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