Does the Point [2,0] Satisfy the First Order Necessary Conditions?

[Nugso]

Homework Statement

$$\text{Minimize } f(x)$$
$$\text{Subject to } \Omega$$

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

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

Homework Equations

$$d^T\nabla{f(x^*)} \geq 0$$

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.

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

Now, applying the constraint, I have

$$2ad_1 + 2(ad_2)^2 \leq 0$$

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?

 

[Ray Vickson]

Homework Statement

$$\text{Minimize } f(x)$$
$$\text{Subject to } \Omega$$

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

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

Homework Equations

$$d^T\nabla{f(x^*)} \geq 0$$

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.

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

Now, applying the constraint, I have

$$2ad_1 + 2(ad_2)^2 \leq 0$$

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?

What does the term "FONC" mean? I get the "FO" = "first-order" part, but what is NC?

 

[Nugso]

What does the term "FONC" mean? I get the "FO" = "first-order" part, but what is NC?

Hi Ray. It's First Order Necessary Condition.

 

[Ray Vickson]

Homework Statement

$$\text{Minimize } f(x)$$
$$\text{Subject to } \Omega$$

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

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

Homework Equations

$$d^T\nabla{f(x^*)} \geq 0$$

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.

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

Now, applying the constraint, I have

$$2ad_1 + 2(ad_2)^2 \leq 0$$

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?

I suggest you draw some feasible directions $\vec{d} = [d_1,d_2]$. Remember that a point of the form $\vec{x} = [2,0] + t \vec{d}$ only need to satisfy the constraint for small, positive $t$. In particular, if $d_1=0$ then also you need $d_2 = 0$. Also, of course, you cannot have $d_1 > 0$; why not?

 

[Nugso]

I suggest you draw some feasible directions $\vec{d} = [d_1,d_2]$. Remember that a point of the form $\vec{x} = [2,0] + t \vec{d}$ only need to satisfy the constraint for small, positive $t$. In particular, if $d_1=0$ then also you need $d_2 = 0$. Also, of course, you cannot have $d_1 > 0$; why not?

When I imagine (draw), a coordinate system, with x and y, with our point on the corner of the boundary, I'm inclined to say both $d_1$ and $d_2$ needs to be nonnegative, because otherwise $d$ wouldn't be a feasible direction.

However, as I've said, if I apply $d$ to the constraint inequality, I only get $d_1 \geq0$

For example, if I had the same constraints, but this time our point was only on the boundary (not necessarily corner), I could see why it'd be only $d_1 \geq0$. But I'm having trouble understanding this.

 

[Ray Vickson]

When I imagine (draw), a coordinate system, with x and y, with our point on the corner of the boundary, I'm inclined to say both $d_1$ and $d_2$ needs to be nonnegative, because otherwise $d$ wouldn't be a feasible direction.

However, as I've said, if I apply $d$ to the constraint inequality, I only get $d_1 \geq0$

For example, if I had the same constraints, but this time our point was only on the boundary (not necessarily corner), I could see why it'd be only $d_1 \geq0$. But I'm having trouble understanding this.

By definition, $\vec{d} = [d_1,d_2]$ is a feasible direction at $\vec{x}_0$ if $\vec{x} = \vec{x}_0 + t \vec{d}$ is feasible for all small enough $t > 0$ (note the sign!). So, for $\vec{x}_0 = [2,0]$, is $\vec{d} = [1,0]$ feasible? Is $\vec{d} = [1,1]$ feasible? Is $\vec{d} = [1, -0.01]$ feasible? Is $\vec{d} = [-1,10]$ feasible? What about $\vec{d} = [-1, -1/2]?$

 

[Nugso]

By definition, $\vec{d} = [d_1,d_2]$ is a feasible direction at $\vec{x}_0$ if $\vec{x} = \vec{x}_0 + t \vec{d}$ is feasible for all small enough $t > 0$ (note the sign!). So, for $\vec{x}_0 = [2,0]$, is $\vec{d} = [1,0]$ feasible? Is $\vec{d} = [1,1]$ feasible? Is $\vec{d} = [1, -0.01]$ feasible? Is $\vec{d} = [-1,10]$ feasible? What about $\vec{d} = [-1, -1/2]?$

Okay, so, if I were to use the definition, I'd say only $d = [-1, 10]$ and $d = [-1, -1/2 ]$ are feasible. However, when I'm trying to picture it, I get the following:

with the blue curve and line [2,-2] the set red lines inside of the set, I think the feasible directions have to be inside the set because we're right at the corner. So by that logic

$d = [1,0]$ and $d = [1,1]$are also feasible. I think either I'm drawing something incorrect, or I just simply can't interpret it.

 

[Ray Vickson]

Okay, so, if I were to use the definition, I'd say only $d = [-1, 10]$ and $d = [-1, -1/2 ]$ are feasible. However, when I'm trying to picture it, I get the following:

with the blue curve and line [2,-2] the set red lines inside of the set, I think the feasible directions have to be inside the set because we're right at the corner. So by that logic

$d = [1,0]$ and $d = [1,1]$are also feasible. I think either I'm drawing something incorrect, or I just simply can't interpret it.

If you go any positive distance at all along $\vec{d} = [1,0]$ you go into the infeasible region (the unshaded region in your drawing)! Same for $\vec{d} = [1,1]$.

 

[Nugso]

If you go any positive distance at all along $\vec{d} = [1,0]$ you go into the infeasible region (the unshaded region in your drawing)! Same for $\vec{d} = [1,1]$.

Ohh! I think I got it now! Thanks!