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UZxiV Let us define p(u,v) Now, it can be observed that if $1 and VutExxuvvizxxv 82 are any two positive scalars, then p(81U, S2v) p(u,v). Hence , the optimization ...

Question

UZxiV Let us define p(u,v) Now, it can be observed that if $1 and VutExxuvvizxxv 82 are any two positive scalars, then p(81U, S2v) p(u,v). Hence , the optimization problem is not affected by multiplication with positive scalars; i.e , it does not change under scaling; and hence has infinitely many solutions. To fix the scaling we further constrain the problem by introducing constraints uTExxu vZxxv =1 This leads to constrained optimization of the following form:maxProblem € :subject touTzxxu =

uZxiV Let us define p(u,v) Now, it can be observed that if $1 and VutExxuvvizxxv 82 are any two positive scalars, then p(81U, S2v) p(u,v). Hence , the optimization problem is not affected by multiplication with positive scalars; i.e , it does not change under scaling; and hence has infinitely many solutions. To fix the scaling we further constrain the problem by introducing constraints uTExxu vZxxv =1 This leads to constrained optimization of the following form: max Problem € : subject to uTzxxu = 1 vTExxV =] Answer the following question: Use the method of Lagrange multipliers to show that the solutions (x, ^) of Problem € can be referred to as the generalized eigenvalues and generalized eigenvectors, respec- tively. uTExxv



Answers

Let $A$ be an $n \times n$ symmetric matrix, let $M$ and $m$ denote the maximum and minimum values of the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ , where $\mathbf{x}^{T} \mathbf{x}=1,$ and denote corresponding unit eigenvectors by $\mathbf{u}_{1}$ and $\mathbf{u}_{n} .$ The following calculations show that given any number $t$ between $M$ and $m,$ there is a unit vector $\mathbf{X}$ such that $t=\mathbf{x}^{T} A \mathbf{x} .$ Verify that $t=(1-\alpha) m+\alpha M$ for some number
$\alpha$ between 0 and $1 .$ Then let $\mathbf{x}=\sqrt{1-\alpha} \mathbf{u}_{n}+\sqrt{\alpha} \mathbf{u}_{1},$ and show that $\mathbf{x}^{T} \mathbf{x}=1$ and $\mathbf{x}^{T} A \mathbf{x}=t$

For this problem, we are asked to explain the method of lagrange multipliers for solving constrained optimization problems. So we're given some F of X. Y. Z. And we are given G of X. Y. Zed equals K. So these first step is that we want to set the gradient of F to be equal to lambda times the gradient of G. So we then need to solve. Or actually I'll rephrase this, we will set lambda F equal to lambda G. And we add on the equation G of X. Y. Zed equals K. Now this first equation will give us three or this first quote unquote equation actually gives us three equations with four unknowns. But then adding on the constraint gives us the last equation needed, so that now we have four equations and four unknowns altogether. The second step is to solve for X. Y. Zed and lambda. And then lastly flood values in to F of X Y. Z.

Okay, so we have two parts to this problem. We're trying to find a mineral value of X. Plus, why would it can change its like 16 to start off? So, FX, why is first going to be X plus y g of x y? It's gonna equal to x rely on my 16 equals zero Tell that was me with lander times the G. So we're gonna have, uh, we're gonna have one I had. Plus one shade is gonna be equal to well landed times lie I had plus will end because Sorry, no place to land it. Thanks, Jihad. Okay, so that means one is gonna be equal. Teoh Lander wy. And that means one's gonna secrets of land. Uh, thanks. So in that case, and we use the land is you know, one over why're, um and then it also means that we can put that lambda in the other side. That means one is gonna be equal to X over Lie. Why is going to go to X so we could use that to plug into our g of X? My equation. So that means rescuer has substitute that. Why's he Could X is gonna be X rayed. It's gonna be you go to 16. Oh, I forgot extra security in zero while it's a great and zero. So that means when X squared 0 to 16 x is going to be could afford not positive or negative for and then have it is wise to be able to x which is only got so that he's alive and also be good for Okay, So for the next part of heart, be forget we have a Max X Y and experts wise equal of 16. So, um f of x y it was gonna be equal to X y g g o x y It's really good to express wide by 16 which equals zero. Okay, so we're gonna have why I had plus X J hair is gonna be equal to Lander. I had us a little The J hat set of Lies and secrets a lander x agree equal till Andrea certainties of why is equal to X. So again put the ontology of X equation. And we didn't have expos explosive exes only equal Teoh 16 to accidentally your 16 acceptably good eight. And then why is gonna be equal T as Well, so let's look at the jumping geometry of these two. When you have X Y, X Y is usually a hyper Birla on both sides, and then you're gonna have that line experts, allies, every negative downward slope, and they're gonna have the 16 point right there in the middle of that. So for the first part is just gonna have y 0 to 4. X equals four and section for us will be dead in a section, will be a little bit smaller on the second part. Is gonna be a high purpose. Largely Well, why physically and executed eight instead.

Okay. So we need to find a maximum and minimum bodies for the given objective functions subjected to the given constraint we have to find a maximum more value for Z. is equal to 40 X. Plus 45. Why objective function subjected to the following constraints we have. Thanks Greater than it is. There we have four Y. With uh reports is there we have eight x. Last night. Why slaves than? It was 7200. We have eight X. Class nine white. Yes. We didn't go to 3000 600. Um So from wild graph. So you're going to get at the four practices of the region formed by the constraints suit At the four practices please. Yeah. Yeah. Yeah. So the friends practices at Is there a 400? Yeah. We are going to get to see is equal to 40 to zero. That's 45. Thanks 400. Which is equal to Mhm. 18,000. Okay So this is 18,000 minimum value of C. So let's go to the nice Invictus. So at 0 800 You're going to get Z. is equal to 40. Can you share plus 45 Times 800? So this will be equal to 36,000. Maximum value of Z. So it is this minimum let me just shut in its minimum. This is maximum value of C. So identix and practice we have at Okay 900. Mhm. Sarah. I'm going to get to see it was 40. Thanks 900. Last 45. Thank you Sarah. This is equal justice. 36,000. Mhm. Maximum value of C. So at the last practice we have at 12 50 0. Yeah. Being together. See Yes it was 40 nine is 4 50 plus. 45. Thanks Sarah. This is equal to 18,000. So this is 18,000 minimum value of C. Yeah. So in this case we can conclude that the objective function has maximum and minimum value not only at the practice is 0 800 And then expertise which is 900. We have 0 400 and 4-15-0 respectively. But this okay at any point on the line segment connecting the concerned victories. So so the maximum value of Z. So we have the maximum value. So let me right max O. Z. You're going to get see is you go to 36,000. And this. Okay. Um At the line segment joining the practices at Is there 800 and Yes. 900. Sarah. The minimum value. So I have minimum value of C. Okay at C. is equal to 18,000. And these are kids at the line segment joining the practices. So these are the practice you have 0 400 and we have 4 57. Mhm.

Everyone trigger going to solve program number 15, they have to minimize function of with Jessica's Ex Square that's like Square Pleasure Square, subject to X plus y Pleasant minus one equals zero. They had to define F ex like lambda equals X squared plus y squared plus that square minus lambda in the X Plus y. Plus that minus smart, if with respect, Lexus two. X Men is Lambda equals zero if with respect to ways to my mind slammed up because zero, if with respect to that is two's admirers, Lambda equals zero. If with respect to Lambda because X plus y plus that minus Monaco's zero real executes vehicles that and X plus y plus that minus one equals zero. So X comma comma is that it was one by three comma one by three comma one by three Thank you


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