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

Ellie Benjamin
2022-06-30
Answered

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

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asked 2022-06-10

I have an optimization problem of the form

Max ${x}_{1}+{x}_{2}+{x}_{3}+\cdots +{x}_{n}$

subject to ${x}_{0}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{n}^{2}+{x}_{12}^{2}+{x}_{13}^{2}+{x}_{14}^{2}+\cdots +{x}_{1n}^{2}+{x}_{23}^{2}+\cdots +{x}_{2n}^{2}+\cdots +{x}_{nn-1}^{2}+{x}_{123}^{2}+\cdots +{x}_{\mathrm{12..}n}^{2}=1$

${x}_{0}+{x}_{1}+{x}_{2}+\cdots +{x}_{n}+{x}_{12}+{x}_{13}$+${x}_{14}+\cdots +{x}_{1n}+{x}_{23}+\cdots +{x}_{2n}+\cdots +{x}_{nn-1}+{x}_{123}+\cdots +{x}_{\mathrm{12..}n}=1$

where ${x}_{0},{x}_{1},{x}_{2},\dots ,{x}_{123\dots n}$ are unrestricted.

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

Max ${x}_{1}+{x}_{2}+{x}_{3}+\cdots +{x}_{n}$

subject to ${x}_{0}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{n}^{2}+{x}_{12}^{2}+{x}_{13}^{2}+{x}_{14}^{2}+\cdots +{x}_{1n}^{2}+{x}_{23}^{2}+\cdots +{x}_{2n}^{2}+\cdots +{x}_{nn-1}^{2}+{x}_{123}^{2}+\cdots +{x}_{\mathrm{12..}n}^{2}=1$

${x}_{0}+{x}_{1}+{x}_{2}+\cdots +{x}_{n}+{x}_{12}+{x}_{13}$+${x}_{14}+\cdots +{x}_{1n}+{x}_{23}+\cdots +{x}_{2n}+\cdots +{x}_{nn-1}+{x}_{123}+\cdots +{x}_{\mathrm{12..}n}=1$

where ${x}_{0},{x}_{1},{x}_{2},\dots ,{x}_{123\dots n}$ are unrestricted.

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

asked 2022-07-08

I'm facing this problem:

$\underset{x\in {\mathbb{R}}_{+}^{3}}{min}max\{\frac{\sum _{i=1}^{3}{x}_{i}^{2}-2{x}_{1}{x}_{3}}{{\left(\sum _{i=1}^{3}{x}_{i}\right)}^{2}},\frac{\sum _{i=1}^{3}{x}_{i}^{2}+2({x}_{1}{x}_{3}-{x}_{1}{x}_{2}+{x}_{2}{x}_{3})}{{\left(\sum _{i=1}^{3}{x}_{i}\right)}^{2}}\}$

I don't know how to deal with inner max and choose one of two!

I'm trying to use $max(A,B)\ge \frac{1}{2}(A+B)$! Do you have any idea?

$\underset{x\in {\mathbb{R}}_{+}^{3}}{min}max\{\frac{\sum _{i=1}^{3}{x}_{i}^{2}-2{x}_{1}{x}_{3}}{{\left(\sum _{i=1}^{3}{x}_{i}\right)}^{2}},\frac{\sum _{i=1}^{3}{x}_{i}^{2}+2({x}_{1}{x}_{3}-{x}_{1}{x}_{2}+{x}_{2}{x}_{3})}{{\left(\sum _{i=1}^{3}{x}_{i}\right)}^{2}}\}$

I don't know how to deal with inner max and choose one of two!

I'm trying to use $max(A,B)\ge \frac{1}{2}(A+B)$! Do you have any idea?

asked 2022-06-21

My initial investment is $\mathrm{\$}100$, and I earn $1\mathrm{\%}$ interest per day. I can opt for any number of compoundings per day (if twice per day, then the interest rate per compounding period is $0.5\mathrm{\%},$ and so on), but I have to pay $\mathrm{\$}0.01$ each time my interest is compounded. After 365 days I will close the account.

What would this equation look like, and how should I include this to maximize my total deposit? How to generalize and figure out a good or optimal maximization?

What would this equation look like, and how should I include this to maximize my total deposit? How to generalize and figure out a good or optimal maximization?

asked 2022-06-24

let $T\ge $1 be some finite integer, solve the following maximization problem.

Maximize $\sum _{t=1}^{T}$($\frac{1}{2}$)$\sqrt{{x}_{t}}$ subject to $\sum _{t=1}^{T}$, ${x}_{t}\le 1$, ${x}_{t}\ge 0$, t=1,...,T

I have never had to maximize summations before and I do not know how to do so. Can someone show me a step by step break down of the solution?

Maximize $\sum _{t=1}^{T}$($\frac{1}{2}$)$\sqrt{{x}_{t}}$ subject to $\sum _{t=1}^{T}$, ${x}_{t}\le 1$, ${x}_{t}\ge 0$, t=1,...,T

I have never had to maximize summations before and I do not know how to do so. Can someone show me a step by step break down of the solution?

asked 2022-07-03

I am trying to find the maximum of a hermitian positive definite quadratic form $xQ{x}^{H}$ (where $Q={Q}^{H}$ and all eigenvalues of $Q$ are non-negative) over the complex unit cube $|{x}_{i}|\le 1$, $i=1,\dots ,n$ where $x=({x}_{1},{x}_{2},\dots ,{x}_{n})\in {\mathbb{C}}^{n}$.

This is the problem of minimizing a concave function over a convex domain. I have read that this problem is NP-hard but there exist some bounds on the optimum. What global optimization technique would your recommend to tackle this problem numerically? Since I am new to the field of optimization, I would appreciate every answer, thanks!

This is the problem of minimizing a concave function over a convex domain. I have read that this problem is NP-hard but there exist some bounds on the optimum. What global optimization technique would your recommend to tackle this problem numerically? Since I am new to the field of optimization, I would appreciate every answer, thanks!

asked 2022-05-09

Show that $f({x}_{1}^{\ast},...{x}_{n}^{\ast})=max\{f({x}_{1},...,{x}_{n}):({x}_{1},...,{x}_{n})\in \mathrm{\Omega}\}$ if and only if $-f({x}_{1}^{\ast},...{x}_{n}^{\ast})=min\{-f({x}_{1},...,{x}_{n}):({x}_{1},...,{x}_{n})\in \mathrm{\Omega}\}$

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 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?

asked 2022-05-07

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.

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.