Discover Maximization Techniques in Geometry Triangles

Is it possible to solve the value of lambda that maximizes the following equation analytically?:
$\frac{1-<!--\; -\; -->e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}}{30\mathrm{\lambda <!--\; \lambda \; -->}}-<!--\; -\; -->e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}$
So then the derivative is equal to:
$\frac{e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}}{\mathrm{\lambda <!--\; \lambda \; -->}}-<!--\; -\; -->\frac{30}{900{\mathrm{\lambda <!--\; \lambda \; -->}}^2}+\frac{30e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}}{900{\mathrm{\lambda <!--\; \lambda \; -->}}^2}+30e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}=0$
Is it possible to get an analytical solution for lambda or does it need to be solved numerically?

If ($x,y,z$) be the lengths of perpendiculars from any interior point P of a triangle $ABC$ on sides $BC$,$CA$ and $AB$ respectively then find the minimum value of :
$x^2+y^2+z^2$
The sides of triangle being $a,b,c$.

I thought of using Lagrange's method of multipliers but am not able to find another function in terms of $x,y,z$ and $a,b,c$Any help will be appreciated. Thanks.

If we have that the contours of a response surface are elliptical and the response is given by the following function:
$\exp <!--\; \; -->(-<!--\; -\; -->(w^2+\frac{1}{4}{l}^2-<!--\; -\; -->\frac{1}{4}\cdot <!--\; \cdot \; -->w\cdot <!--\; \cdot \; -->l))$
then if we maximize this function w.r.t l holding w fixed at 1/2.
And if we call the maximizer l-star, then holding l-star fixed, maximize over w. How to show that the overall max isn't achieved?
My approach: I got the partial of the above function w.r.t. l and then tried to evaluate it at 1/2, but got stuck. It most likely will involve some analysis of Hessians.

I need help solving this following equation, a Lagrangian problem that I encountered during my studies in principal component analysis (PCA).

One should maximize the variance with respect to the first principal component

Following is the case: We know that $W={\mathbf{v}}_1^T{\mathbf{S}\mathbf{v}}_1$ (eq 1) and that ${\mathbf{v}}_{\mathbf{1}}$ is an orthonormal basis.

Therefore $||{\mathbf{v}}_1|{|}^2={\mathbf{v}}_1^T{\mathbf{v}}_1=1$ (eq 2). The maximization of $W$ under this constraint can be done by introducing the Lagrangian multiplier $\mathrm{\lambda <!--\; \lambda \; -->}$ and maximizing the Lagrangian:
$\mathcal{L}=W+\mathrm{\lambda <!--\; \lambda \; -->}(1-<!--\; -\; -->||{\mathbf{v}}_1|{|}^2)={\mathbf{v}}_1^T(\mathbf{S}-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}\mathbf{I}){\mathbf{v}}_1+\mathrm{\lambda <!--\; \lambda \; -->}$ (eq 3)
So expanding $W$ using eq. 1, and the laws of distributivity we end up with eq 3.
Next, taking the derivative with respect to vector and $\mathrm{\lambda <!--\; \lambda \; -->}$ we obtain:
$\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\lambda <!--\; \lambda \; -->}}\mathcal{L}=1-<!--\; -\; -->{\mathbf{v}}_1^T{\mathbf{v}}_1=0$
${\mathrm{\nabla <!--\; \nabla \; -->}}_{\mathbf{V}}{}_1\mathcal{L}=(\mathbf{S}-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}\mathbf{I}){\mathbf{v}}_1=0$
This can later be shown to be an eigenvalue problem. Can anyone explain to me why the derivatives look like that? I can't seem to figure it out.

I got stuck in a some step
$\ge <!--\; \ge \; -->[\underset{z}{\sum <!--\; \sum \; -->}P(z|X,{\mathrm{\theta <!--\; \theta \; -->}}_n)\cdot <!--\; \cdot \; -->\ln <!--\; \; -->(\frac{P(X|z,\mathrm{\theta <!--\; \theta \; -->})P(z|\mathrm{\theta <!--\; \theta \; -->})}{P(z|X,{\mathrm{\theta <!--\; \theta \; -->}}_n)})]-<!--\; -\; -->\ln <!--\; \; -->(P(X|{\mathrm{\theta <!--\; \theta \; -->}}_n))$
which in the next line results in
$=\underset{z}{\sum <!--\; \sum \; -->}P(z|X,{\mathrm{\theta <!--\; \theta \; -->}}_n)\cdot <!--\; \cdot \; -->\ln <!--\; \; -->(\frac{P(X|z,\mathrm{\theta <!--\; \theta \; -->})P(z|\mathrm{\theta <!--\; \theta \; -->})}{P(z|X,{\mathrm{\theta <!--\; \theta \; -->}}_n)P(X|{\mathrm{\theta <!--\; \theta \; -->}}_n)})$
What happened in between those lines?

Please help me how to deal with maximization of functional like this:
$F\{a(s)\}=\underset{0}{\overset{t}{\int <!--\; \int \; -->}}(g(a(s))-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}v(s{)}^2)ds,a(s)\in <!--\; \in \; -->[0,\mathrm{\infty <!--\; \infty \; -->})$
where $g(x)=x{e}^{-<!--\; -\; -->x}$, $\mathrm{\alpha <!--\; \alpha \; -->}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, and ${\displaystyle v(t)=\underset{0}{\overset{t}{\int <!--\; \int \; -->}}a(s)ds}$
And I don't understand how to deal with $v(t{)}^2$

Can someone point out the development from Eq(1) to Eq(2) of the following maximization over $k$ ?

Eq(1): $-<!--\; -\; -->\frac{1}{2}(x-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_k{)}^T{\mathrm{\Sigma <!--\; \Sigma \; -->}}^{-<!--\; -\; -->1}(x-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_k)$

Eq(2): $C-<!--\; -\; -->\frac{1}{2}{\mathrm{\mu <!--\; \mu \; -->}}_k^T{\mathrm{\Sigma <!--\; \Sigma \; -->}}^{-<!--\; -\; -->1}{\mathrm{\mu <!--\; \mu \; -->}}_k+x^T{\mathrm{\Sigma <!--\; \Sigma \; -->}}^{-<!--\; -\; -->1}{\mathrm{\mu <!--\; \mu \; -->}}_k$ where $C$ denotes a constant

I`m trying to solve the optimization problem on $\mathbf{x}$ and $\mathbf{y}$
$\begin{array}{c}\text{max}(\mathbf{x}\otimes <!--\; \otimes \; -->\mathbf{y}{)}^{T}A(\mathbf{x}\otimes <!--\; \otimes \; -->\mathbf{y})\\ s.t\Vert <!--\; \Vert \; -->\mathbf{x}{\Vert <!--\; \Vert \; -->}_2=1\\ \Vert <!--\; \Vert \; -->\mathbf{y}{\Vert <!--\; \Vert \; -->}_2=1,\end{array}$
where $\mathbf{x}\in <!--\; \in \; -->{\mathbb{R}}^m$ and $\mathbf{y}\in <!--\; \in \; -->{\mathbb{R}}^n.$ The matrix $A$ is a rank-one symmetric matrix given by $A=\mathbf{v}{\mathbf{v}}^T,$ where $\mathbf{v}\in <!--\; \in \; -->{\mathbb{R}}^{mn}.$

I have an optimization problem of the form
$\underset{\mathrm{\gamma <!--\; \gamma \; -->}}{max}⟨<!--\; ⟨\; -->f\circ <!--\; \circ \; -->\mathrm{\gamma <!--\; \gamma \; -->},w_f⟩<!--\; ⟩\; -->⟨<!--\; ⟨\; -->g\circ <!--\; \circ \; -->\mathrm{\gamma <!--\; \gamma \; -->},w_g⟩<!--\; ⟩\; -->$
where $f\circ <!--\; \circ \; -->\mathrm{\gamma <!--\; \gamma \; -->}$ is the composition of $f$ and $\mathrm{\gamma <!--\; \gamma \; -->}$ and the inner product is defined as
$⟨<!--\; ⟨\; -->f\circ <!--\; \circ \; -->\mathrm{\gamma <!--\; \gamma \; -->},w_f⟩<!--\; ⟩\; -->=\int <!--\; \int \; -->(f\circ <!--\; \circ \; -->\mathrm{\gamma <!--\; \gamma \; -->})(t)w_f(t)dt$
In other words I am trying to find a $\mathrm{\gamma <!--\; \gamma \; -->}$ that maximizes the product of the two inner products above. I know one could pursue a gradient approach, which would be slow. Is there a more direct optimization method?

I am trying to implement a fitness function for a genetic algorithm and have problem with understanding what function maximization means. In the paper which I use as the basis of the algorithm, the fitness function is defined as (1-RC) + (1-RO), where RC and RO are floating point values between 0 and 1, which I calculate using another formulas. However in the next paragraph of the article it is written - "This function (fitness function) has to be maximized"

I am struggling to understand what that means. Could you please explain it in easy-to-understand way for a Math noob? Thank you!

I have an optimization problem of the form
Max $x_1+x_2+x_3+\cdots <!--\; \cdots \; -->+x_n$
subject to $x_0^2+x_1^2+x_2^2+\cdots <!--\; \cdots \; -->+x_n^2+x_{12}^2+x_{13}^2+x_{14}^2+\cdots <!--\; \cdots \; -->+x_{1n}^2+x_{23}^2+\cdots <!--\; \cdots \; -->+x_{2n}^2+\cdots <!--\; \cdots \; -->+x_{nn-<!--\; -\; -->1}^2+x_{123}^2+\cdots <!--\; \cdots \; -->+x_{\mathrm{12..}n}^2=1$
$x_0+x_1+x_2+\cdots <!--\; \cdots \; -->+x_n+x_{12}+x_{13}$+$x_{14}+\cdots <!--\; \cdots \; -->+x_{1n}+x_{23}+\cdots <!--\; \cdots \; -->+x_{2n}+\cdots <!--\; \cdots \; -->+x_{nn-<!--\; -\; -->1}+x_{123}+\cdots <!--\; \cdots \; -->+x_{\mathrm{12..}n}=1$
where $x_0,x_1,x_2,\dots <!--\; \dots \; -->,x_{123\dots <!--\; \dots \; -->n}$ are unrestricted.

What is the upper bound for the cost function ..? Thank you.

My question is that in general, is there a case where saddle point be the global max of a function? I am solving a game theory question which the optimal solution is the saddle point. Can I conclude that the optimal solution is at the boundaries?

For $1\le <!--\; \le \; -->i\le <!--\; \le \; -->215$ let $a_i={\displaystyle \frac{1}{2^i}}$ and $a_{216}={\displaystyle \frac{1}{2^{215}}}$. Let $x_1,x_2,\dots <!--\; \dots \; -->,x_{216}$ be positive real numbers such that ${\displaystyle \underset{i=1}{\overset{216}{\sum <!--\; \sum \; -->}}{x}_i=1}$ and
$\underset{1\le <!--\; \le \; -->i<j\le <!--\; \le \; -->216}{\sum <!--\; \sum \; -->}{x}_i{x}_j={\displaystyle \frac{107}{215}}+\underset{i=1}{\overset{216}{\sum <!--\; \sum \; -->}}{\displaystyle \frac{a_i{x}_i^2}{2(1-<!--\; -\; -->a_i)}}.$
Find the maximum possible value of $x_2.$
I simplified the condition to ${\displaystyle \underset{i=1}{\overset{216}{\sum <!--\; \sum \; -->}}{\displaystyle \frac{x_i^2}{1-<!--\; -\; -->a_i}}={\displaystyle \frac{1}{215}},}$ but I'm not sure what to do next.

For fixed $g$, I want to find maximum $b$ with
$-<!--\; -\; -->2b(3t^2(s+1)+6t(s+1)+3s+2)-<!--\; -\; -->2g(6ts+3t+6s+2)-<!--\; -\; -->3t{s}^2+6ts+3t-<!--\; -\; -->3s^2+3s+1>0$
for some nonnegative reals $t$,$s$. Here $g$,$b$ are also $\ge <!--\; \ge \; -->0$. Can it be possible to get a function $f$ such that we get the upper bound on $b$ as $f(g)$ i.e, $b<f(g)$? If one can find $t$,$s$ for which $b$ is maximum, one will get $f(g)$.

Let $f(x_1,\dots <!--\; \dots \; -->,x_n)=x_1{x}_2\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->x_n$. Let $A:=\{x\in <!--\; \in \; -->{\mathbb{R}}^n:x_1+x_2+\cdots <!--\; \cdots \; -->+x_n=n,x_i\ge <!--\; \ge \; -->0\mathrm{\forall <!--\; \forall \; -->}i\}$. I want to find the global max of $f$ under the constrain given by $A$. MY approach:
Put $F(x_1,\dots <!--\; \dots \; -->,x_n)=\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{x}_i-<!--\; -\; -->n=0$ is the constrain. We use lagrange multipliers:
$\mathrm{\nabla <!--\; \nabla \; -->}f=\mathrm{\lambda <!--\; \lambda \; -->}\mathrm{\nabla <!--\; \nabla \; -->}F⟹<!--\; ⟹\; -->(x_2\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->x_n,\dots <!--\; \dots \; -->,x_1\cdots <!--\; \cdots \; -->x_{n-<!--\; -\; -->1})=(\mathrm{\lambda <!--\; \lambda \; -->},\mathrm{\lambda <!--\; \lambda \; -->},\dots <!--\; \dots \; -->,\mathrm{\lambda <!--\; \lambda \; -->}).$
Hence we have
$x_2\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->x_n=x_1{x}_3\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->x_n=\cdots <!--\; \cdots \; -->=x_1\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->\cdot <!--\; \cdot \; -->x_{n-<!--\; -\; -->1}$
this implies that
$x_1=x_2=\cdots <!--\; \cdots \; -->=x_n$
We also know $\sum <!--\; \sum \; -->x_i=n⟹<!--\; ⟹\; -->nX=n⟹<!--\; ⟹\; -->X=1$ if we let $X=x_i$
So, max occurs at $(1,1,\dots <!--\; \dots \; -->,1)$ which is $f(1,1,\dots <!--\; \dots \; -->,1)=1$ Is this correct? I feel im missing something. thanks for any feedback.

It's been a while since I've had to do derivations so I'm just a little rusty. I need to understand the steps for deriving (Py – 2)(44 – 2Py + Px) to reach Py = 12 + 0.25Px. Py here is price of product y and Px is the price of product x.

I undertsant that I need to take product rule of the first equation and then set it = 0 and solve for Py, but it's finding that derived equation that's troubling me.

How I tried to do it was to first simplify the equation into 44Py - 2Py^2 + PxPy - 88 + 4Py + 2Px. Then taking the derivative and simplifying a little I end up with 46 - 3Py + Px which I would then set = 0 and solve for Py, giving me Py = (46 + Px)/3. Which is obviously not correct.

If you could help me through the steps to reach Py = 12 + 0.25Px I would be very greatful.

I am trying to find a function prescribed in polar coordinates $r=f(\mathrm{\theta <!--\; \theta \; -->})$ that maximizes the following quantity
$\frac{{\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}{r}^3\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}d\mathrm{\theta <!--\; \theta \; -->}}{{\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}{r}^{4}d\mathrm{\theta <!--\; \theta \; -->}}$
subject to the constraint $r\le <!--\; \le \; -->R\mathrm{\forall <!--\; \forall \; -->}\mathrm{\theta <!--\; \theta \; -->}\in <!--\; \in \; -->[0,2\mathrm{\pi <!--\; \pi \; -->})$.

I don't even know where to begin to be honest. I went through a couple of books on calculus of variations. I haven't seen anything that comes close to the problem I have at hand.

I'm a first year Finance grad student and we're learning utility maximization problem now. We have been assuming that a solution to these problems exists. Are there any general theorem about the existence of maximization solution on ${\mathbb{R}}^n$? I have encountered once that "A utility maximization problem with a continuous utility function on a compact set has solution". Thanks in advance!

Suppose we have a function of the form: $(x_1-<!--\; -\; -->x_2)+(x_3-<!--\; -\; -->x_4)+(x_5-<!--\; -\; -->x_6)$ and we have maximized this summation using linprog (using some constraints which are not important for this matter). This provides us with a value for the different x variables.

The problem I now want to solve is the maximization of the minimum $(x_i-<!--\; -\; -->x_j)$ and with the constraint that the solution $(x_1,x_2,...)$ filled in in the original summation should have a higher or the same value (constraint).

This would distribute the difference between the variables $x_1,x_2;x_3,x_4,...$
How can this problem be solved in Matlab (maximization of the minima)?

Is it true that for all symmetric $n\times <!--\; \times \; -->n$ matrix, $(a_{ij})$, such that for $x\in <!--\; \in \; -->\{\pm <!--\; \pm \; -->1{\}}^n$
$\underset{ij}{\sum <!--\; \sum \; -->}{a}_{ij}{x}_i{x}_j\le <!--\; \le \; -->1$
there exists a universal constant such that
$\underset{ij}{\sum <!--\; \sum \; -->}{a}_{ij}{x}_i{y}_j\le <!--\; \le \; -->C$
for all $x,y\in <!--\; \in \; -->\{\pm <!--\; \pm \; -->1{\}}^n$ . I tried to use the polarization identity
$⟨<!--\; ⟨\; -->Ax,y⟩<!--\; ⟩\; -->=⟨<!--\; ⟨\; -->Au,u⟩<!--\; ⟩\; -->-<!--\; -\; -->⟨<!--\; ⟨\; -->Av,v⟩<!--\; ⟩\; -->$
where $u=(x+y)/2$ and $v=(x-<!--\; -\; -->y)/2$. However, as $x$ and $y$ vary over $\pm <!--\; \pm \; -->1$ vectors, $u$ and $v$ can be vectors in $\{\pm <!--\; \pm \; -->1,0\}$.