Discover Maximization Techniques in Geometry Triangles

How can I find ${\mathrm{\lambda <!--\; \lambda \; -->}}_H$ and ${\mathrm{\lambda <!--\; \lambda \; -->}}_T$ such that
$\underset{0\le <!--\; \le \; -->{\mathrm{\lambda <!--\; \lambda \; -->}}_H,{\mathrm{\lambda <!--\; \lambda \; -->}}_T\le <!--\; \le \; -->1}{max}\{\frac{4.6575342\times <!--\; \times \; -->{10}^{-<!--\; -\; -->4}}{2.1722965\times <!--\; \times \; -->{10}^{-<!--\; -\; -->4}+{\mathrm{\lambda <!--\; \lambda \; -->}}_H},\frac{1.0958904\times <!--\; \times \; -->{10}^{-<!--\; -\; -->2}}{3.4311896\times <!--\; \times \; -->{10}^{-<!--\; -\; -->4}+{\mathrm{\lambda <!--\; \lambda \; -->}}_T}\}<1?$
Is this problem equivalent to finding $x$ and $y$ such that
$\underset{0\le <!--\; \le \; -->x,y\le <!--\; \le \; -->1}{min}\{.4664048+(2147.0588450)x,0.0313096+(91.2500009)y\}\ge <!--\; \ge \; -->1?$

A huge conical tank to be made from a circular piece of sheet metal of radius 10m by cutting out a sector with vertex angle theta and welding the straight edges of the straight edges of the remaining piece together. Find theta so that the resulting cone has the largest possible volume.

Specifically, the question is asked in the context of wanting derivatives, multiple max/min equations, and hopefully more calc rather than trig or geo.

I have gotten as far as using 10m as the hypotenuse for a triangle formed by the height of the cone, radius of the base of the cone, and slant. I'm not sure where to go from there, because I can't determine how to find height and/or radius, without which I'm not sure I can continue.

Show that $f(x_1^{\ast <!--\; \ast \; -->},...x_n^{\ast <!--\; \ast \; -->})=max\{f(x_1,...,x_n):(x_1,...,x_n)\in <!--\; \in \; -->\mathrm{\Omega <!--\; \Omega \; -->}\}$ if and only if $-<!--\; -\; -->f(x_1^{\ast <!--\; \ast \; -->},...x_n^{\ast <!--\; \ast \; -->})=min\{-<!--\; -\; -->f(x_1,...,x_n):(x_1,...,x_n)\in <!--\; \in \; -->\mathrm{\Omega <!--\; \Omega \; -->}\}$
I am not exactly sure how to approach this problem -- it is very general, so I can't assume anything about the shape of $f$. It seems obvious that flipping the max problem with a negative turns it into a min problem. Thoughts?

I want to solve the following optimization problem
$$\begin{gathered} \text{maximize }f(X)=-<!--\; -\; -->\log <!--\; \; -->\mathrm{d}\mathrm{e}\mathrm{t}(X+Y)-<!--\; -\; -->a^T(X+Y{)}^{-<!--\; -\; -->1}a \\ \text{subject to }X⪰<!--\; ⪰\; -->W, \end{gathered}$$
where the design variable $X$ is symmetric positive semidefinite, $W,Y$ are fixed symmetric positive semidefinite matrices, and $a$ is a given vector. The inequality $X⪰<!--\; ⪰\; -->W$ means that $X-<!--\; -\; -->W$ is symmetric positive semidefinite.

I was wondering if there's hope to find an analytical solution to either the constrained or unconstrained problem. And if there is none, could I use convex optimization techniques to solve this numerically?

Let be $t\in <!--\; \in \; -->\mathbb{R}$, $n=1,2,\cdots <!--\; \cdots \; -->$, $p\in <!--\; \in \; -->[0,1]$ and $a\in <!--\; \in \; -->(p,1]$.

Show that
$\underset{t}{sup}(ta-<!--\; -\; -->\log <!--\; \; -->(p{e}^t+(1-<!--\; -\; -->p))=a\log <!--\; \; -->\left(\frac{a}{p}\right)+(1-<!--\; -\; -->a)\log <!--\; \; -->\left(\frac{1-<!--\; -\; -->a}{1-<!--\; -\; -->p}\right)$
So, I've try to derivate, but I did'nt get sucess, since my result is different. I've get $\log <!--\; \; -->\left(\frac{a(1-<!--\; -\; -->p)}{p(1-<!--\; -\; -->a)}\right)$. Any ideas?

Suppose we have some functions $f_1(x),f_2(x),\dots <!--\; \dots \; -->,f_n(x)$ with $x\in <!--\; \in \; -->{\mathbb{Z}}^n$.

We can denote the subset $X_1$ of ${\mathbb{Z}}^n$ that maximizes $f_1(x)$ as:
$X_1=\underset{x\in <!--\; \in \; -->{\mathbb{Z}}^n}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}{f}_1(x)$

Now, suppose there is a kind of "priority" in which I also want to maximize $f_2$, as long as I keep maximizing $f_1$. This could be represented as:
$X_2=\underset{x\in <!--\; \in \; -->X_1}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}{f}_2(x)$

The same for $f_3$:
$X_3=\underset{x\in <!--\; \in \; -->X_2}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}{f}_3(x)$

So on and so forth...

Is there some, more concise, notation to represent this "maximization priority"?

The problem is like
$ma{x}_{\mathbf{x}}u(x_1,x_2,...,x_L)=-<!--\; -\; -->\underset{i=1}{\overset{L}{\sum <!--\; \sum \; -->}}\mid <!--\; \mid \; -->x_i-<!--\; -\; -->a_i\mid <!--\; \mid \; -->$,
$s.t.\underset{i}{\overset{L}{\sum <!--\; \sum \; -->}}{x}_i\le <!--\; \le \; -->C$
for each $i$, $a_i>0$ is a scalar;
$C$ is a constant that is strictly greater than 0;
$\mathbf{x}=(x_1,x_2,...,x_L)\in <!--\; \in \; -->{\mathbf{R}}_{+}^L$. Characterize the optimal $\mathbf{x}$ as a function of $C$ or $a_i$.
Hint: to solve the problem we should discuss the cases when $C\le <!--\; \le \; -->\underset{i}{\sum <!--\; \sum \; -->}{a}_i$ and $C\ge <!--\; \ge \; -->\underset{i}{\sum <!--\; \sum \; -->}{a}_i$.
Thank you!

Let $X\in <!--\; \in \; -->\{0,1{\}}^d$ be a Boolean vector and $Y,Z\in <!--\; \in \; -->\{0,1\}$ are Boolean variables. Assume that there is a joint distribution $\mathcal{D}$ over $Y,Z$ and we'd like to find a joint distribution ${\mathcal{D}}^{\prime }$ over $X,Y,Z$ such that:

1. The marginal of ${\mathcal{D}}^{\prime }$ on $Y$,$Z$ equals $\mathcal{D}$.
2. $X$ are independent of $Z$ under ${\mathcal{D}}^{\prime }$, i.e., $I(X;Z)=0$.
3. $I(X;Y)$ is maximized,

where $I(\cdot <!--\; \cdot \; -->;\cdot <!--\; \cdot \; -->)$ denotes the mutual information. For now I don't even know what is a nontrivial upper bound of $I(X;Y)$ given that $I(X;Z)=0$? Furthermore, is it possible we can know the optimal distribution ${\mathcal{D}}^{\prime }$ that achieves the upper bound?

My conjecture is that the upper bound of $I(X;Y)$ should have something to do with the correlation (coupling?) between $Y$ and $Z$, so ideally it should contain something related to that.

Given a right circular cone with the line of symmetry along $x=0$, and the base along $y=0$, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the cone? Maximising the volume of the paraboloid relative to the volume of the right circular cone. In 2-D, the parabola has 2 points of tangency to the triangle, one of each side of the line of symmetry. I have tried using the disk method to find the volume of the cone, and the parabola, both with arbitrary equations such as $y=b-<!--\; -\; -->ax$, and $y=c-<!--\; -\; -->d{x}^2$, but I end up with a massive equation for several variables, instead of a simple percentage answer. Any help is appreciated! Thanks in advance.

Which are the most famous problems having an objective of maximizing a nonlinear convex function (or minimizing a concave function)? As far as I know such an objective with respect to linear constraints is np-hard.

An airline will fill 100 seats of its aircraft at a fare of 200 dollars. For every 5 dollar increase in the fare, the plane loses two passengers. For every decrease of $5, the company gains two passengers. What price maximizes revenue?
I'm a bit lost as to what my equations should be. The total revenue, if all seats are filled can be 20000. So for every 5 dollar increase the total revenue becomes: 20000−400x (400x = 2 people time the price of each ticket times the number of 5 dollar increases). I don't know how to model the other situation of a 5 dollar decrease.

Let $X=(X_1,...,X_n)$ be a vector of $n$ random variables. Consider the following maximization problem:
$\underset{a,b}{max}\mathrm{C}\mathrm{o}\mathrm{v}(a\cdot <!--\; \cdot \; -->X,b\cdot <!--\; \cdot \; -->X)$ under the constraint that $\Vert <!--\; \Vert \; -->a{\Vert <!--\; \Vert \; -->}_2=\Vert <!--\; \Vert \; -->b{\Vert <!--\; \Vert \; -->}_2=1$.
($a\cdot <!--\; \cdot \; -->X$ is the dot product between $a$ and $X$). Would it be true that there is a solution to this maximization problem such that $a=b$?
Thanks.

Let $x\in <!--\; \in \; -->{\mathbb{R}}^n$ and let $P$ be a $n\times <!--\; \times \; -->n$ positive definite symmetrix matrix. It is known that the maximum of
$\begin{array}{ll}\text{maximize}& x^{T}Px\\ \text{subject to}& x^{T}x\le <!--\; \le \; -->1\end{array}$
is ${\mathrm{\lambda <!--\; \lambda \; -->}}_{\text{max}}(P)$, the largest eigenvalue of $P$. Now consider the following problem
$\begin{array}{ll}\text{maximize}& x^{T}Px\\ \text{subject to}& (x-<!--\; -\; -->a{)}^T(x-<!--\; -\; -->a)\le <!--\; \le \; -->1\end{array}$
where $a\in <!--\; \in \; -->{\mathbb{R}}^n$ is known. What is the analytical solution?

Some have proposed that for a natural centrality measure, the most central a node can get is the center node in the star network. I've heard this called "star maximization." That is, for a measure $M(\cdot <!--\; \cdot \; -->)$, and a star network $g^{\star <!--\; \star \; -->}$ with center $c^{\star <!--\; \star \; -->}$,
$\{(c^{\star <!--\; \star \; -->},g^{\star <!--\; \star \; -->})\}\in <!--\; \in \; -->{\arg <!--\; \; -->max}_{(i,g)\in <!--\; \in \; -->N\times <!--\; \times \; -->\mathcal{G}(N)}M(i,g)$
where $N$ is the set of nodes and $\mathcal{G}$ considers all unweighted network structures.

I'd like to learn about some centrality measures that don't satisfy this property, but "star maximization" isn't a heavily used term, so I am having trouble in searching for many such measures. What are some such measures of centrality?

"A common theme in linear compression and feature extraction is to map a high dimensional vector $x$ to a lower dimensional vector $y=Wx$ such that the information in the vector $x$ is maximally preserved in $y$. Opten PCA is applied for this purpose. However, the optimal setting for $W$ is in generall not given by the widely used PCA. Actually, PCA is sub-optimal special case of mutual information maximisation."

Can anyone elaborate why PCA is a sub-optimal special case of mutual information maximisation ?

I have to maximize the following function -
Max A $C_1^{-<!--\; -\; -->m}/-<!--\; -\; -->m$ + (1-A) $C_2^{-<!--\; -\; -->m}/-<!--\; -\; -->m$
Subject to,
$C_1$ ≤ 5(1-x) + x
$C_2$ ≤ 3(1-x) + 7x
1≤x≤10
I wrote it as: L(x) = f(x) - ${\lambda }_1$($C_1$ - 5(1-x) + x) - ${\lambda }_2$($C_2$ - 3(1-x) + 7x) - ${\lambda }_3$(x-1) - ${\lambda }_4$(x-10)
Can I write last constraint as partioned into ${\lambda }_3$ and ${\lambda }_4$. Is there some other way to introduce such box constraints into the same problem?

Statement:

A manufacturer has been selling 1000 television sets a week at $\mathrm{\$<!--\; \$\; -->}480$ each. A market survey indicates that for each $\mathrm{\$<!--\; \$\; -->}11$ rebate offered to a buyer, the number of sets sold will increase by 110 per week.

Questions :

a) Find the function representing the revenue R(x), where x is the number of $\mathrm{\$<!--\; \$\; -->}11$ rebates offered.

For this, I got $(110x+1000)(480-<!--\; -\; -->11x)$. Which is marked correct

b) How large rebate should the company offer to a buyer, in order to maximize its revenue?

For this I got 17.27. Which was incorrect. I then tried 840999, the sum total revenue at optimized levels

c) If it costs the manufacturer $\mathrm{\$<!--\; \$\; -->}160$ for each television set sold and there is a fixed cost of $\mathrm{\$<!--\; \$\; -->}80000$, how should the manufacturer set the size of the rebate to maximize its profit?

For this, I received an answer of 10, which was incorrect.

In the one-variable case we can take a $\log$ transform of the function $f(x)\to <!--\; \to \; -->\log <!--\; \; -->(f(x))$ and know that the same value maximizes both because log is increasing. The log turns products into sums which facilitates taking derivatives. Can the same argument be applied for multivariable cases? The question is: for a function $f(x,y)$ if I wish to find the $x^{\star <!--\; \star \; -->},y^{\star <!--\; \star \; -->}$ that maximize $f(x,y)$ are these the same $x^{\star <!--\; \star \; -->\star <!--\; \star \; -->},y^{\star <!--\; \star \; -->\star <!--\; \star \; -->}$ that maximize $\log <!--\; \; -->(f(x,y))$? My intuition is yes, nothing should be different in this multivariable case, but I desire a quick sanity check before setting off with computations. I would appreciate a rigorous proof or a counterexample.

I have not yet had the privilege of studying multivariable calculus, but I have made an educated guess about how to find the minimum or maximum of a function with two variables, for example, $x$ and $y$.

Since, in three dimensions, a minimum or maximum would be represented by a tangent plane with no slope in any direction, could I treat $y$ as a constant and differentiate $z$ with respect to $x$, then treat $x$ as a constant and differentiate with respect to $y$, and find the places where both of these two are equal to zero?

Sorry if this is just a stupid assumption... it may be one of those things that just seems correct but is actually wrong.

Maximize the value of the function
$z=\frac{ab+c}{a+b+c},$
where $a,b,c$ are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, and if possible a general technique. Thank you.