Tackle Maximization Problems with Ease

Suppose that we have a integer $m$ , and we need to choose $n \leq m$ and $x_{1}, x_{2}, . . ., x_{n}$ such that $\sum_{i = 1}^{n} x_{i} = m$ and $\prod_{i = 1}^{n} x_{i}$ is maximized, where $n$ and all $x_{i}$ 's are integers. In particular, is the maximum product $\prod_{i} x_{i}$ polynomial in $m$ ?

Rebecca Villa 2022-07-01

$\begin{matrix} (Problem 1) & \begin{array}{ll} maximize & | 2 x_{1} - 3 x_{2} | \\ subject to & 4 x_{1} + x_{2} \leq 4 \\ 2 x_{1} - x_{2} \leq 0.5 \\ x_{1}, x_{2} \geq 0 \end{array} \end{matrix}$
$\begin{matrix} (Problem 2) & \begin{array}{ll} minimize & | 2 x_{1} - 3 x_{2} | \\ subject to & 4 x_{1} + x_{2} \leq 4 \\ 2 x_{1} - x_{2} \leq 0.5 \\ x_{1}, x_{2} \geq 0 \end{array} \end{matrix}$
According to the question, one of them can be rewritten as a linear program (LP), and the other one cannot. The question asks the reader to determine which one can be rewritten as an LP, and why the other one cannot.

I've never converted to an LP an optimization problem whose objective function contains an absolute value. So, I think I have no way of determining the one that can be rewritten as an LP. But, I know the simplex method and I can solve an LP using it.

cambrassk3 2022-07-01

Suppose we have variables $x_{1}, x_{2}, y \in {0, 1}$ such that $y = 1$ if and only if $x_{1} = x_{2} = 1$ and we want maximize the value of $y$ . I know that this reduces to the following Integer programming problem:
$\begin{aligned} max y \\ x_{1}, x_{2}, y \in {0, 1} \\ y \leq x_{1} \\ y \leq x_{2} . \end{aligned}$
The linear relaxation of this problem is then given by:
$\begin{aligned} max y \\ 0 \leq x_{1}, x_{2}, y \leq 1 \\ y \leq x_{1} \\ y \leq x_{2} . \end{aligned}$
Now in the relaxation we will simply get $y = min {x_{1}, x_{2}}$ . I was wondering whether there is some alternative formulation for the Integer Program such that, when we take the linear relaxation of the problem, the solution $y$ is stricly smaller than $min {x_{1}, x_{2}}$ .

glitinosim3 2022-06-30

I am trying to derive the entropy in normal distribution. Let $p (x)$ to be the probability density function of uniform normal distribution
$p (x) = \frac{1}{\sqrt{2 π} σ} e^{- \frac{x^{2}}{2 σ^{2}}}$
hence, by using integration by parts, we have
$\int_{- \infty}^{\infty} x^{2} p (x) d x = x^{2} \int_{- \infty}^{\infty} p (x) d x - \int_{- \infty}^{\infty} 2 x (\int_{- \infty}^{\infty} p (x) d x) d x$
Because
$\int_{- \infty}^{\infty} p (x) d x = 100 %$
we have
$\int_{- \infty}^{\infty} x^{2} p (x) d x = x^{2} - x^{2} + C = C$
However, lots of relevant proofs online says that
$\int_{- \infty}^{\infty} x^{2} p (x) d x = σ^{2}$
Does anyone know the reason?

Ellie Benjamin 2022-06-30

If $x + y = a$ , then the maximum value of $x * y$ is when $x = y = a / 2$ . ( $x, y$ are positive numbers)

ttyme411gl 2022-06-30

If a minimization task is a convex optimization problem, is the maximization of the same objective function also always a convex optimization problem?

My guess is yes since minimization of the negative of the objective function is maximization, but wondering if there are cases that outdo, and disprove, this sign 'trick'

sebadillab0 2022-06-29

Each concert a singer makes profit of 0.75, but after every concert a singert can fall into bad mood with probability = 0.5. To get a singer out of these mood producer can send her flowers. If flowers cost $x$ money, then singer can get out of bad mood with probability $\sqrt{x}$ . Which $x$ should producer choose in order to maximize his expected profit. (All the profit from concert goes to producer. One can make assumprion, that singer's career lasts $n$ days.)

Amber Quinn 2022-06-28

Let $f (x, y)$ be a non-separable, non-negative real-valued function, that is jointly concave in $x$ and $y$ . We want to maximize $f (x, y)$ over $x$ and $y$ .

Is the sequential maximization
$max_{x} max_{y} f (x, y)$
always equal to the simultaneous maximization
$max_{x, y} f (x, y)$
Or is there a simple counter-example for this?

I know that if
is separable, then the sequential maximization and simultaneous maximization are equal. Are there any conditions on a non-separable function such that this result still holds?

Davon Irwin 2022-06-28

Let $A$ be a matrix and $Q$ be an orthogonal matrix such that $A Q^{T}$ is symmetric, positive semidefinite. Show that
$| | A + Q | |_{F} \geq | | A + P | |_{F}$
for any orthogonal matrix $P$ . Here, $| | \cdot | |_{F}$ is the Frobenius norm.

Leah Pope 2022-06-28

Actually i don't know how to distinguish maximization and minimization of nonlinear function with Newton Raphson Method.

What i know is, for finding the optimization points, we have to calculate this iteration:
$x_{i + 1} = x_{i} - [H f (x_{i})]^{- 1} \nabla f (x_{i})$
Then, what is actually the difference between maximization and minimization using this method?

Bailee Short 2022-06-27

I have a problem with finding:

$max_{a \in R^{p}} \frac{a^{'} C a}{a^{'} B a},$

where $C$ and $B$ are $p \times p$ symmetric matrices.

After differentiation I get the following result:

$\frac{2 a^{'} C (a^{'} B a) - 2 a^{'} B (a^{'} C a)}{(a^{'} B a) (a^{'} B^{'} a)} = 0$

I can’t solve the above equation for $a$ . How to find it?

Armeninilu 2022-06-26

Edit: Removed solved in title, because I realize I need someone to check my work.
Ok, so the problem is a lot more straight forward than I originally approached it (which was a false statement -- so it was excluded).
Let R,S, x $\in$ N with x $\leq$ R*S and 0 $0 <$ R $\leq$ S. Next, define B as a multiplicative factor of x - c with c $\geq 0$ 0 and B $\leq$ S such that $\frac{x - c}{B} = A \leq R$ and A $\in$ N. What value of B maximizes A?

Jeffery Clements 2022-06-26

Without using Lagrange multipliers, find the maximum volume of a rectangular box inscribed in tetrahedron bounded by the coordinate planes and the plane 2x/5 + y + z = 1

I am self-teaching vector calculus but I got stuck on this question. Thanks for any help! :)

hawatajwizp 2022-06-26

I have to maximize $U (x, y) = M i n (a x + y, b y + x)$ s.a $p_{1} x + p_{2} y = m$ . I try the traditional solution for a leontieff $(a x_{1} + y = b y_{1} + x)$ function but I'm not sure.. beacause exist regions where one plan is under the other and only one of them is a minimun...

Davion Harding 2022-06-26

$max min [α x_{1}, β x_{2}, γ x_{3}] s.t. λ_{1} x_{1} + λ_{2} x_{2} + λ_{3} x_{3} = c, α, β, γ, λ_{i}, c are constants$
Well, that function is not differentiable , so what methods can be applied to solve for for the optimal values of $x_{1}$ , $x_{2}$ and $x_{3}$ ? Is knowledge of the $λ^{'} s$ and $c$ necessary, to at least some degree, or does a general approach/solution exist?

Sattelhofsk 2022-06-26

I am given the following problem:
$A = [\begin{matrix} 4 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 2 \end{matrix}]$
Find
$max_{x} \frac{| (A x, x) |}{(x, x)}$
where $(., .)$ is a dot product of vectors and the maximization is performed over all $x = {[\begin{matrix} x_{1} & x_{2} & x_{3} \end{matrix}]}^{T} \in R^{3}$ , such that $\sum_{i = 1}^{3} x_{i} = 0$
I have found the eigenvectors for $A$ and they happen to match the sum criterion:
$E (λ_{1}) = span ({[\begin{matrix} - 2 & 1 & 1 \end{matrix}]}^{T})$
for $λ_{1} = 3$ and
$E (λ_{2}) = span ({[\begin{matrix} 0 & - 1 & 1 \end{matrix}]}^{T})$
for $λ_{2} = - 1$ .
(For $λ_{3} = 6$ there are no eigenvectors).
Can the above eigenvectors and eigenvalues be used for solving this maximization problem?

enrotlavaec 2022-06-26

In a book I'm reading, a claim is made that
$f (a) = r \cos (a) - m \sin (a)$
has the maximum
$\sqrt{r^{2} + m^{2}}$
(where a, r and m are real numbers).
But I'm not sure how to prove it or even if it's true. Setting $\frac{d}{d a} f (a) = 0$ gives $- r \sin (a) - m \cos (a) = 0$ or $\tan (a) = - m / r$ , so $a = \arctan (- m / r)$ . Plugging that in into wolfram does not give the claimed result...

excluderho 2022-06-26

$x^{2} y \to$ max,
such that $x^{2} + 4 x y \leq 1, x \geq 0$ and $y \geq 0$ .

I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone please give me an example of how this should be done? And should I include the constraints $x \geq 0$ and $y \geq 0$ into the Lagrangian function?

opepayflarpws 2022-06-25

I am maximizing
$f (x, y) = - x$
given the constraint
$g (x, y) = x^{2} - y^{2} = 0$
To satisfy the non degenerate constraint qualification I have:
$D g (x, y) = [2 x - 2 y]$
and the set of $(x, y)$ that satisfy it is having $x = y$ .

However on setting up the Lagrange multiplier:
$L (x, y, λ) = - x + λ (x^{2} - y^{2})$
and getting the first order conditions:
$L_{x} = - 1 + 2 λ x = 0$ and
$L_{y} = - 2 λ y = 0$
$L_{λ} = x^{2} - y^{2} = 0$
I have a contradiction since for $x = y$

The first equation will give:
$- 2 λ x = - 1$
The second however shows:
$- 2 λ x = 0$
Is there anywhere I have gotten wrong here?

Gaaljh 2022-06-25

I have a multivariable function that I have defined the cost function and its gradient with respect to the variable vector. Let's call the cost function $f (\vec{x})$ , and variable vector $\vec{x}$ . I have been using nonlinear conjugate gradient descenet method for minimization. Algorithm is as follows:
$k = 0 x = 0 g_{0} = \nabla_{\vec{x}} f (x_{0}) Δ x_{0} = - g_{0}$
$x_{k + 1} \leftarrow x_{k} + t Δ x_{k} g_{k + 1} \leftarrow \nabla f (x_{k + 1}) Δ x_{k + 1} \leftarrow - g_{k + 1} + γ Δ x_{k}$
I know that I need to update formula so the solution "ascends" instead of descension. How can I update the algorithm? Also I wonder how the wolfe condition changes for maximization problems. Thank you and have a nice day.

Even when you have an engineering maximization problem calculator with the values, you may still have certain questions that will not help you find a solution that will work. As a way to assist you with your geometry problems, you should start with the exploration of the answers that have been provided below. Most maximization problems are related to high school geometry that you will already remember. If things do not seem clear or you are feeling confused, look for the maximum values and coordination axes as you study your task's objectives or the word problem by offering several explanations.

Discover Maximization Techniques in Geometry Triangles