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thisisourconcerndude
thisisourconcerndude
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The Statement of the Problem:

Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier conditions for the following two problems:

$$(1) \text{ minimize } f(\mathbf x) \text{ subject to constraint } g(\mathbf x)=b $$

$$(2) \text{ maximize } g(\mathbf x) \text{ subject to constraint } f(\mathbf x)=c $$

Given a solution $\mathbf y$ to first and second order conditions for the minimization problem $(1)$, for some value of $b$, set $c=f(\mathbf y)$ and consider the maximization problem $(2)$; when is $\mathbf y$ a solution to this problem as well?

My Questions:

I believe I have the first-order conditions for $(1)$:

$$ \frac{\partial \mathcal{L}(\mathbf x)}{\partial x_i} = \frac{\partial f(\mathbf x)}{\partial x_i}-\lambda \frac{\partial g(\mathbf x)}{\partial x_i}$$$$ \frac{\partial \mathcal{L}(\mathbf x)}{\partial x_i} = \frac{\partial f(\mathbf x)}{\partial x_i}-\lambda \frac{\partial g(\mathbf x)}{\partial x_i}=0$$

But what are the second-order conditions? I know that, in order to minimize $f$, $\mathcal{L}$ must be convex and that the constraint set must be convex, but are those the "second-order conditions"? If so, how do I determine them? By way of bordered Hessians? That doesn't seem right. Anyway, any help here would be appreciated. Thanks!

The Statement of the Problem:

Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier conditions for the following two problems:

$$(1) \text{ minimize } f(\mathbf x) \text{ subject to constraint } g(\mathbf x)=b $$

$$(2) \text{ maximize } g(\mathbf x) \text{ subject to constraint } f(\mathbf x)=c $$

Given a solution $\mathbf y$ to first and second order conditions for the minimization problem $(1)$, for some value of $b$, set $c=f(\mathbf y)$ and consider the maximization problem $(2)$; when is $\mathbf y$ a solution to this problem as well?

My Questions:

I believe I have the first-order conditions for $(1)$:

$$ \frac{\partial \mathcal{L}(\mathbf x)}{\partial x_i} = \frac{\partial f(\mathbf x)}{\partial x_i}-\lambda \frac{\partial g(\mathbf x)}{\partial x_i}$$

But what are the second-order conditions? I know that, in order to minimize $f$, $\mathcal{L}$ must be convex and that the constraint set must be convex, but are those the "second-order conditions"? If so, how do I determine them? By way of bordered Hessians? That doesn't seem right. Anyway, any help here would be appreciated. Thanks!

The Statement of the Problem:

Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier conditions for the following two problems:

$$(1) \text{ minimize } f(\mathbf x) \text{ subject to constraint } g(\mathbf x)=b $$

$$(2) \text{ maximize } g(\mathbf x) \text{ subject to constraint } f(\mathbf x)=c $$

Given a solution $\mathbf y$ to first and second order conditions for the minimization problem $(1)$, for some value of $b$, set $c=f(\mathbf y)$ and consider the maximization problem $(2)$; when is $\mathbf y$ a solution to this problem as well?

My Questions:

I believe I have the first-order conditions for $(1)$:

$$ \frac{\partial \mathcal{L}(\mathbf x)}{\partial x_i} = \frac{\partial f(\mathbf x)}{\partial x_i}-\lambda \frac{\partial g(\mathbf x)}{\partial x_i}=0$$

But what are the second-order conditions? I know that, in order to minimize $f$, $\mathcal{L}$ must be convex and that the constraint set must be convex, but are those the "second-order conditions"? If so, how do I determine them? By way of bordered Hessians? That doesn't seem right. Anyway, any help here would be appreciated. Thanks!

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thisisourconcerndude
thisisourconcerndude
  • 121
  • 4

First and Second Order Lagrangian-Multiplier Conditions for Optimization

The Statement of the Problem:

Let $f,g$ be two functions on $\mathbb R^n$ and assume $\nabla f $and$ \nabla g$ are nonzero everywhere. Write the first and second order Langrangian-multiplier conditions for the following two problems:

$$(1) \text{ minimize } f(\mathbf x) \text{ subject to constraint } g(\mathbf x)=b $$

$$(2) \text{ maximize } g(\mathbf x) \text{ subject to constraint } f(\mathbf x)=c $$

Given a solution $\mathbf y$ to first and second order conditions for the minimization problem $(1)$, for some value of $b$, set $c=f(\mathbf y)$ and consider the maximization problem $(2)$; when is $\mathbf y$ a solution to this problem as well?

My Questions:

I believe I have the first-order conditions for $(1)$:

$$ \frac{\partial \mathcal{L}(\mathbf x)}{\partial x_i} = \frac{\partial f(\mathbf x)}{\partial x_i}-\lambda \frac{\partial g(\mathbf x)}{\partial x_i}$$

But what are the second-order conditions? I know that, in order to minimize $f$, $\mathcal{L}$ must be convex and that the constraint set must be convex, but are those the "second-order conditions"? If so, how do I determine them? By way of bordered Hessians? That doesn't seem right. Anyway, any help here would be appreciated. Thanks!