Tagged Questions
0
votes
0answers
23 views
Convexifying Functions
I have the following question:
Imagine you have a function $F(x,y(x),y'(x)), F:\mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $ and this function is not convex.
Then you can ...
1
vote
1answer
50 views
Hessian of a function that takes matrix arguments
I have a function that that takes a matrix and returns a scalar, $f : \mathbb{R}^{m\times n} \rightarrow \mathbb{R}$. I know how to calculate the derivative of this function with respect to the matrix ...
2
votes
1answer
70 views
Strictly Convex Function and Well-Separated Minimum
Suppose $\Theta \subset \mathbb{R}^d$ is a convex set, and $f:\Theta \rightarrow \mathbb{R}$ is a strictly convex function that has a minimum at $\theta_0\in\Theta$. Is it true then that $\forall ...
0
votes
1answer
63 views
Maximization of sum of two functions
Is there any relationship similar to the following.
Let $X$ be the maximum of functions $f_1(x)+f_2(x)$.
Let $X_1$ be a maximum of $f_1(x)$ and let $X_2$ be a maximum of $f_2(x)$.
Is there any ...
3
votes
2answers
63 views
Constrained maximization problem
I need help with the following optimization problem
$$
\max\;\alpha\ln(x(1-y^2))+(1-\alpha)\ln(z)
$$
where the maximization is with respect to $x,y,z$, subject to
\begin{align}
\alpha ...
1
vote
0answers
103 views
How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$
I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral
\begin{equation}
\begin{aligned}
I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
0
votes
1answer
140 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 ...
3
votes
0answers
142 views
Global optimum of sum of convex functions
Take two real differentiable convex functions, $f_1$ and $f_2$, defined on the unit interval $[0; 1]$. I want to find the global optimum of: $\min_{x \in [0;1]} af_1(x)+bf_2(x)$, for given $a, b \in ...
0
votes
2answers
92 views
Simple example where strong convexity is necessary over strict (or even regular) convexity.
I am trying to get a hold on exactly what strong convexity gives you over strict (or regular) convexity.
Yes there are simple functions which demonstrate the difference between these ideas, but what ...
1
vote
2answers
119 views
Entropy expression optimization with Langrange multipliers
I have recently encountered variants of the following expression:
\begin{equation}
S = H(a,b,c,d)-H(a+b,c+d)
\end{equation}
where $H$ is the Shannon entropy function, that is $H(X)=\sum_{x\in X}-x\log ...
2
votes
1answer
400 views
Maximum likelihood covariance estimation of Gaussian
I was reading these notes on matrix calculus
http://research.microsoft.com/en-us/um/people/minka/papers/matrix/minka-matrix.pdf
and I could not figure out how to go from equation (30) to (31).
Any ...
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 ...
