Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Calculus and Beyond Homework Help
Stationary points classification using definiteness of the Lagrangian
Reply to thread
Message
[QUOTE="fatpotato, post: 6629496"] [B]Homework Statement:[/B] Find the max/min/saddle points of ##f(x,y) = x^4 - y^4## subject to the constraint ##g(x,y) = x^2-2y^2 -1 =0## Use Lagrange multipliers method Classify the stationnary points (max/min/saddle) using the definiteness of the Hessian [B]Relevant Equations:[/B] Positive/Negative definite matrix Hello, I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse. So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla \mathscr{L} (x^∗,y^∗,λ^∗) = 0## Now, I want to classify my triplets as max/min/saddle points, using the positive/negative definiteness of the Hessian like I have been doing for unconstrained optimization, so I compute what I think is the Hessian of the Lagrangian: $$H_{\mathscr{L}}(x,y,λ)= \begin{pmatrix} 12x^2 - 2\lambda & 0 \\ 0 & -12y^2 - 4\lambda \end{pmatrix}$$ Evaluating the Hessian for my first triplet ##(0,\pm \frac{\sqrt{2}}{2},−\frac{1}{2})## gives me: $$H_{\mathscr{L}}(0,\pm \frac{\sqrt{2}}{2},−\frac{1}{2}) = \begin{pmatrix} 1 & 0 \\ 0 & - 4\end{pmatrix}$$ This matrix is diagonal, meaning that we immediately read its eigenvalues on the diagonal: ##\lambda_1 = 1 > 0## and ##\lambda_2 = -4 < 0##. A positive/negative definite matrix has only positive/negative eigenvalues, thus I conclude that this matrix is neither, due to its eigenvalues' opposite signs. When I was studying unconstrained optimization, I learned that we have in this case a saddle point, so I would like to think that the points ##(0,\pm \frac{\sqrt{2}}{2})## are both saddle points for my function f, however, the solution to this problem affirms these points are minimums, using the following argument: [ATTACH type="full" alt="Lagrange_Mult_Sol.PNG"]301125[/ATTACH] Using the fact that ##\nabla g(x,y) = (0,\pm \frac{\sqrt{2}}{2})## and that ##w^T \nabla g(x,y) = 0## if and only if ##w = (\alpha, 0), \alpha \in \mathbb{R}^{\ast}## I thought that it was enough to check for the definiteness of the Hessian, and now I am really confused... Here are my questions: [LIST=1] [*]When is it enough to check the definiteness of the Hessian to classify stationnary points? [*]Why is there this additional step in constrained optimization? [*]What am I missing? [/LIST] Thank you for your time. Edit: PF destroyed my LaTeX formatting. [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Calculus and Beyond Homework Help
Stationary points classification using definiteness of the Lagrangian
Back
Top