Tagged Questions
4
votes
1answer
182 views
Can one define the derivative of a function using tangent cones? Does such a notion already exist?
I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
0
votes
0answers
17 views
need help regarding multivariate function
What are the steps should I follow to proof the following multivariate function (function of two variables) is strictly concave, continuous, and differentiable?
For example:
$$
f(v,q) = -0.0042 * ...
4
votes
1answer
156 views
The composition of two convex functions is convex
Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
1
vote
1answer
48 views
A consequence of convexity
Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$.
Studying the behaviour of the difference quotient, it is clear that
$$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$
...
0
votes
1answer
139 views
Global Min-Max Optimization
When is
\begin{equation}
\min_X \max_Y f(X,Y)
\end{equation}
globally solvable? (i.e. we can find global solution for the optimization problem?)
I am not looking for reformulations.
Is it only when ...
6
votes
0answers
184 views
Behaviour at infinity of a function in terms of first and second derivatives
In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$:
there exists a constant $C>0$ and a ...
2
votes
0answers
172 views
Quasiconcavity for the sum of specific quasiconcave functions
I want to show that a function $ψ(a_1,a_0)$, which is separable (additively decomposed) in two quasiconcave functions, is also quasiconcave (QC). I know that the sum of QC functions is not generally ...
0
votes
1answer
280 views
How to prove $f(x,y)=\sqrt {xy}$ is concave?
How can I prove (preferably elegantly) that $f(x,y)=\sqrt {xy}$ where $x≥0$ and $y≥0$ is concave in both $x$ and $y$?
0
votes
0answers
50 views
Relaxing strict convexity
Suppose $F: \mathbb{R^2} \rightarrow \mathbb{R}$ is convex. If $F$ is moreover strictly convex, then
$$ \frac{1}{2} F(x_1,y_1) + \frac{1}{2} F(x_2, y_2) - F\left( \frac{x_1 + x_2}{2}, \frac{y_1 + ...
1
vote
0answers
118 views
Convexity of a Set
Consider the following function,
$$
f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right)
$$
where $a, b, c, m$ and $n$ are positive constants.
I want to show $f(x, y)$ is ...
2
votes
1answer
172 views
How to compute the subdifferential of a variational represention of the trace norm?
Let $f : {\cal S}_+^n \mapsto \mathbb{R}$ be a function defined as
$f(Q) := {\rm tr} WQ^{-1}W + {\rm tr} Q$
where $W \in {\cal S}_{+}^{n}$ is a symmetric positive definite matrix.
How to compute a ...
