# 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 $(1)$:

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

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!

• Why that doesn't seem right by way of bordered Hessians? – Metta World Peace Apr 20 '15 at 15:46
• What you write as a "first-order condition", is not -it is just the first derivative of the Lagrangean. What is the condition on it? Also, convexity is a convenient property to guarantee that the second-order conditions hold, and so that the first-order conditions are sufficient and necessary for a minimum. As with univariate optimization, where we look into second derivatives, here we have to look into Hessians/border Hessians. These are the 2nd-order conditions, not a "way to verify them". – Alecos Papadopoulos Apr 20 '15 at 17:02
• Thanks. So, yeah, I forgot to put the "$=0$" in there for the "first-order conditions." And, right, so by checking the definite-ness of the bordered Hessians, I can determine whether or not $\mathcal{L}$ is concave/convex... which is the second-order condition. However, I don't really have a way of determining the definiteness of the Hessian because I don't actually have any values to work with. Furthermore, don't I have to consider "regularity"? Sorry, I'm really confused about all of this... – thisisourconcerndude Apr 20 '15 at 17:38
• Since you are given no other information, you might as well assume convexity, or just "for $y$ to be a solution, second order conditions must be satisfied". After all, this is obviously an exercise in duality. – Alecos Papadopoulos Apr 20 '15 at 18:35