Convex Function on Many variables on an interval

In summary, the task is to show that the function f(x) = x1x2 is convex on an interval [a, ma]T, where a and m are non-negative and m is greater than or equal to 1. The definition of convexity is given and the attempt at a solution involves applying this definition to multiple variables. [a, ma]T is interpreted as a point in R2 with coordinates x1 = a and x2 = ma (or mx1).
  • #1
curiousguy23
10
0

Homework Statement


Show that f(x) = x1x2 is a convex function on [a,ma]T where a [itex]\geq[/itex] 0
and m [itex]\geq[/itex] 1.


Homework Equations


By definition f is convex iff

[itex]\forall x,y\in \Re \quad \wedge \quad \forall \lambda :\quad 0\le \lambda \le 1\quad \Rightarrow \quad f\left( \lambda x+(1-\lambda )y \right) \le \lambda f\left( x \right) +(1-\lambda )f\left( y \right)[/itex]


The Attempt at a Solution



I am not really sure how to go about this. Firstly I was thinking on how to apply the above relation to multiple variables. I would assume that it applies in general and in this case x and y would be a vector of variables, but I still don't know how the proof follows. Also I am not sure is how the interval [a,ma]T plays into the equation.
 
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  • #2
curiousguy23 said:

Homework Statement


Show that f(x) = x1x2 is a convex function on [a,ma]T where a [itex]\geq[/itex] 0
and m [itex]\geq[/itex] 1.


Homework Equations


By definition f is convex iff

[itex]\forall x,y\in \Re \quad \wedge \quad \forall \lambda :\quad 0\le \lambda \le 1\quad \Rightarrow \quad f\left( \lambda x+(1-\lambda )y \right) \le \lambda f\left( x \right) +(1-\lambda )f\left( y \right)[/itex]


The Attempt at a Solution



I am not really sure how to go about this. Firstly I was thinking on how to apply the above relation to multiple variables. I would assume that it applies in general and in this case x and y would be a vector of variables, but I still don't know how the proof follows. Also I am not sure is how the interval [a,ma]T plays into the equation.

What is meant by [a,ma]T? I doubt that [a,ma] is a 1-dimensional interval, because what in the world would possibly be meant by the transpose of an interval? My guess would be that [a,ma] means a point in R2 of the form x1 = a, x2 = ma. However, that is just a guess.

RGV
 
  • #3
Ray Vickson said:
What is meant by [a,ma]T? I doubt that [a,ma] is a 1-dimensional interval, because what in the world would possibly be meant by the transpose of an interval? My guess would be that [a,ma] means a point in R2 of the form x1 = a, x2 = ma. However, that is just a guess.

RGV

Yes you are right in this, sorry I did not stipulate this, x1= a and x2=ma=mx1
 

1. What is a convex function on many variables on an interval?

A convex function on many variables on an interval is a function where the line segment connecting any two points on the graph of the function lies above or on the graph, within the specified interval. In other words, the function is "curving upwards" and does not have any "dips".

2. How is a convex function on many variables on an interval different from a convex function on a single variable?

A convex function on many variables on an interval is similar to a convex function on a single variable in that they both have a "curving upwards" shape. However, a convex function on many variables on an interval is defined on a range of values for each variable, while a convex function on a single variable is defined on a single range of values. Additionally, a convex function on many variables on an interval can have multiple dimensions, while a convex function on a single variable only has one dimension.

3. What is the significance of a convex function on many variables on an interval in mathematics?

Convex functions on many variables on an interval have various applications in mathematics, particularly in optimization problems. They are used to find the minimum or maximum of a function within a given range of values. They are also used in economics, physics, and other fields to model real-world situations.

4. How can the convexity of a function on many variables on an interval be determined?

The convexity of a function on many variables on an interval can be determined by looking at its second derivative, also known as the Hessian matrix. If the Hessian matrix is positive semi-definite (all eigenvalues are non-negative), then the function is convex. Additionally, the function can be tested for convexity by checking if all its one-dimensional restrictions on each variable are convex.

5. Can a function be convex on some variables and concave on others?

Yes, a function can be convex on some variables and concave on others. This type of function is called a "non-convex" function and does not meet the criteria for a convex function on many variables on an interval. Non-convex functions can have multiple local minima or maxima, making them more challenging to optimize.

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