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Aug 16, 2019 · Check the Hessian matrix of the function. If the matrix is: Positive-definite then your function is strictly convex. Positive semi-definite then your function is convex. A matrix is positive definite when all the eigenvalues are positive and semi-definite if all the eigenvalues are positive or zero-valued.
- How to check convexity? - Mathematics Stack Exchange
The book "Convex Optimization" by Boyd, available free...
- convex analysis - How to check the convexity of a function ...
With functions of one variable, you would check for...
- Proper way to check convexity or concavity for a function
I am working with a function I would like to check if it is...
- How to check convexity? - Mathematics Stack Exchange
The book "Convex Optimization" by Boyd, available free online here, describes methods to check. The standard definition is if f(θx + (1 − θ)y) ≤ θf(x) + (1 − θ)f(y) for 0≤θ≤1 and the domain of x,y is also convex. So if you could prove that for your function, you would know it's convex.
Let \(I\) be an interval and let \(f, g: I \rightarrow \mathbb{R}\) be convex functions. Prove that \(cf\), \(f + g\), and \(\max \{f, g\}\) are convex functions on \(I\), where \(c \geq 0\) is a constant. Find two convex functions \(f\) and \(g\) on an interval \(I\) such that \(f \cdot g\) is not convex. Answer. Add texts here.
Jun 3, 2022 · With functions of one variable, you would check for convexity by looking at the second derivative. Suppose you have $f(x)$: the function is convex on an interval $I$ if and only if $f''(x) \geq 0 \quad \forall x \in I$.
- Definition of Convexity of A Function
- Geometric Interpretation of Convexity
- Sufficient Conditions For Convexity/Concavity
- Properties of Convex Functions
- Solved Problems
Consider a function y = f (x), which is assumed to be continuous on the interval [a, b]. The function y = f (x) is called convex downward (or concave upward) if for any two points x1 and x2 in [a, b], the following inequality holds: If this inequality is strict for any x1, x2 ∈ [a, b], such that x1 ≠ x2, then the function f (x) is called strictly c...
The introduced concept of convexity has a simple geometric interpretation. If a function is convex downward (Figure ), the midpoint of each chord lies above the corresponding point of the graph of the function or coincides with this point. Similarly, if a function is convex upward (Figure ), the midpoint of each chord is located below the correspon...
Suppose that the first derivative of a function exists in a closed interval and the second derivative exists in an open interval Then the following sufficient conditions for convexity/concavity are valid: 1. If for all then the function is convex downward (or concave upward) on the interval 2. If for all then the function is convex upward (or conca...
We list some properties of convex functions assuming that all functions are defined and continuous on the interval 1. If the functions and are convex downward (upward), then any linear combination where , are positive real numbers is also convex downward (upward). 2. If the function is convex downward, and the function is convex downward and non-de...
Solution. Consider an arbitrary combination of these values, such as the following: The graph of such a function is located in the upper half-plane, and the function is strictly decreasing (since ). Given that the function is convex downward. Its schematic view is shown in Figure in the first column and second row. It is clear that the total number...
Dec 6, 2018 · I am working with a function I would like to check if it is convex or concave. The function is the next: f(x1,x2) = max{x61,ex1+3x22, 3x21 −x1x2 +x42 − log (x2 + 2)} f (x 1, x 2) = max {x 1 6, e x 1 + 3 x 2 2, 3 x 1 2 − x 1 x 2 + x 2 4 − log (x 2 + 2)}
Find the parts of the graph where the function is convex or concave, and find the point(s) of inflexion. [3 marks] f(x) = x(x^2 + 1) = x^3 + x gives. f''(x) = 6x. f''(x) = 0, when x = 0. f''(x) \textcolor{red}{< 0} when x<0. Here we have a concave section. f''(x) \textcolor{purple}{> 0} when x>0. Here we have a convex section.
