- #1

Nugso

Gold Member

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- 10

## Homework Statement

[tex] \text{Minimize } f(x) [/tex]

[tex] \text{Subject to } \Omega [/tex]

where [tex] f:R^2 → R \text{ is given by } f(x) = -3x_1 \text{ where } x = [2,0]^\top \text{ and } \Omega = \{x: x_1 + 2x_2^2 \leq 2\}[/tex]

[tex] \text{Does the point } x^* = [2,0]^\top \text{satisfy F.O.N.C?} [/tex]

## Homework Equations

[tex] d^T\nabla{f(x^*)} \geq 0 [/tex]

has to be satisfied for FONC.

## The Attempt at a Solution

I know FONC, but what I'm having trouble understanding is determining feasible directions. For example, in this case, the point is on the boundary.

[tex] x + ad = [2,0]^\top + a [d_1, d_2]^\top[/tex]

Now, applying the constraint, I have

[tex] 2ad_1 + 2(ad_2)^2 \leq 0[/tex]

From there, it seems like d_2 is an arbitrary choice, while d_1 has to be chosen accordingly. But, also, since the point is on the corner of the boundary, doesn't that mean that d_2 has to be in some specific direction as well?