Affine Function on an Open Convex Set

In summary, an affine function on an open convex set is a linear function that maps points from the set to a line while preserving convexity. It differs from a linear function by including a constant term, and can only be defined on an open convex set. Common examples include linear and polynomial functions, and they are used in optimization, regression, and geometry.
  • #1
jrand
1
0

Homework Statement



I am trying to understand the reasoning behind the following statement from Boyd's Convex Optimization textbook, page 50, line 9-11: "f must be negative on C; for if f were zero at a point of C then f would take on positive values near the point, which is a contradiction." The assumptions are: f is an affine function, C is an open convex set.

Homework Equations


The Attempt at a Solution



I know: f is an affine function of the form [tex]f(x) = a^{T}x+b[/tex]. The image of a convex set S under f is convex. That is, if I take all the elements of S, and input them into f, the result is another convex set.

I also know: C is an open convex set. That is, for any two points in C, the line segment between them is contained in C. The most important point of the problem is C is open, which means whatever shape C is, the boundary or border is not contained in C. Somehow this fact (open convex set) is the reason behind the initial quoted statement but I do not understand why.

Thanks in advance for any help or tips.
 
Physics news on Phys.org
  • #2

Thank you for your question regarding the statement from Boyd's Convex Optimization textbook. I can provide some insight into the reasoning behind this statement.

Firstly, let's recall that an affine function is a linear function with an added constant term, which in this case is represented by the term "b" in f(x) = a^{T}x+b. This means that the graph of the function f is a straight line.

Next, let's consider the open convex set C. As you correctly pointed out, this means that the boundary or border of C is not contained within C itself. This implies that there are points in C that are not on the boundary, and therefore have some "room" to move around within C.

Now, let's think about what it means for f to be negative on C. This means that for any point x in C, f(x) is less than zero. In other words, the graph of f is below the x-axis for all x in C.

So, what happens if f is zero at a point of C? This means that there is a point x in C where f(x) = 0, and therefore the graph of f passes through the origin. However, since C is an open convex set, there are points in C that are not on the boundary and can move around within C. This means that there must be points near x (within C) where f takes on positive values, since the graph of f is a straight line and cannot suddenly jump from below the x-axis to above it without passing through the origin. This is the "contradiction" mentioned in the statement.

In summary, the fact that C is an open convex set allows for points to move around within it, and the fact that f is an affine function means that its graph is a straight line. Combining these two facts, we can see that if f is zero at a point of C, it must also take on positive values near that point, which contradicts the assumption that f is negative on C. Therefore, f must be negative on C.

I hope this helps to clarify the reasoning behind the statement. If you have any further questions, please don't hesitate to ask.


 

1. What is an affine function on an open convex set?

An affine function on an open convex set is a linear function that maps points from the set to a line in a way that preserves the property of convexity. In other words, the image of any convex combination of points in the set will also be a convex combination of points on the line.

2. How is an affine function different from a linear function?

An affine function is a type of linear function that also includes a constant term. This means that in addition to the linear relationship between the input and output variables, there is a fixed translation or shift in the output values. In contrast, a linear function does not have a constant term and only has a direct proportionality between input and output.

3. Can an affine function be defined on a non-convex set?

No, an affine function can only be defined on an open convex set. This is because the convexity of the set is necessary to preserve the convexity property of the function. If the set is non-convex, the affine function will not be able to map points in a way that maintains the convexity of the set.

4. What are some examples of affine functions?

Some common examples of affine functions include linear functions with a constant term, such as y = mx + b, as well as quadratic functions and other polynomial functions. These functions all have a fixed translation or shift in the output values, which is the defining characteristic of an affine function.

5. How are affine functions used in science and mathematics?

Affine functions are commonly used in optimization problems, where the goal is to find the maximum or minimum value of a function. They are also used in linear regression, where the goal is to find the best-fit line for a set of data points. In addition, affine functions are important in geometry and topology, as they are used to define convex sets and study their properties.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Replies
4
Views
1K
Replies
9
Views
324
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
963
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Linear and Abstract Algebra
Replies
21
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
707
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
Back
Top