# Maximise this quadaratic form, subject to these constraints.

• Phillips101
In summary, the problem is to find the maximal value of a1*a2+a2*a3+...+an*a1 given the constraints a1+a2+a3+...+an=0 and a1^2+a2^2+...+an^2=1, where all variables are real numbers. The thought process so far has involved treating the expression as a combination of n variables and attempting to differentiate, as well as trying to factorize it. However, this approach has proven to be difficult. The original constraints can also be written in a matrix form, but it is not clear how to solve the problem using linear or quadratic programming. It appears to be a linear algebra question.
Phillips101
Question:

Given a1+a2+a3+...+an=0 and a1^2+a2^2+...+an^2=1, (all real numbers)

find the maximal value of a1*a2+a2*a3+...+an*a1

Thoughts so far:

I've treated the expression as a combination of n variables and differentiated - when it came to putting the constraints in it got to be a hideous mess.

It is easily factorisable as 0.5*( (a1+a2)^2 + (a2+a3)^2 +...+ (an+a1)^2 ) -1, but then maximising the inside is just as hard.

Help would be appreciated! (Also, sorry about the lack of LATEX knowhow).

You can write the original constraints as
$$\max \ \ a^T\begin{pmatrix} 0 &1 &\ldots &\ldots 0\\ 0 &0 &1 &\ldots 0\\ \vdots \\1 &0 &\ldots &0\end{pmatrix}a$$
subject to
$$\begin{pmatrix} a &\mathbf{1} \end{pmatrix}^T a = \begin{pmatrix}1 \\0\end{pmatrix}$$

and try to fit the problem to linear or quadratic programming frame

I know of linear programming, but I have no idea how to go about solving that using it unfortunately. And nor do I think I'm expected to know, this is a linear algebra question, which is the annoying thing...

And I'd never heard of or encountered quadratic programming before.

## 1) What is a quadratic form?

A quadratic form is an algebraic expression that consists of variables raised to the second power, also known as quadratic terms. It can be written in the form of ax^2 + bx + c, where a, b, and c are constants and x is a variable.

## 2) Why is it important to maximize a quadratic form?

Maximizing a quadratic form is important because it allows us to find the maximum value of the function, which can be useful in various applications such as optimization problems in engineering and economics.

## 3) What are constraints in a quadratic form?

Constraints in a quadratic form are conditions or limitations that must be satisfied in order to maximize the function. These can be in the form of equations or inequalities involving the variables in the quadratic form.

## 4) How do you solve a quadratic form with constraints?

To solve a quadratic form with constraints, we can use a method called Lagrange multipliers. This involves setting up a system of equations using the original quadratic form and the constraints, and then solving for the values of the variables that maximize the function.

## 5) What are some real-world applications of maximizing a quadratic form?

Maximizing a quadratic form has many real-world applications, such as finding the maximum profit or revenue for a business, maximizing the area of a given shape, or optimizing the design of a product to minimize costs. It is also commonly used in physics and engineering to find the maximum or minimum values of physical quantities.

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