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Define the $n\times n$ symmetric matrix $M$ with elements $$M_{ij}=\pi_i \delta_{ij}-\pi_i\pi_j.$$ We seek to maximise the quadratic form $$f(v)=\sum_{ij} v_i M_{ij} v_j$$ where $v$ is a vector on the …

The damping factor is expressed in terms of the objective function that one seeks to minimise, as described in A new approach for determining damping factors in Levenberg-Marquardt algorithm for solvi …

In this particular case the word "principle" most likely became attached to Ekeland's variational theorem because of the title of the paper where he derived it:

If $(x,y)$ is the coordinate of the upper right corner of the rectangle, with the origin at the center of the circles (inner radius $r_1$, outer radius $r_2$), then the area of the rectangle is $A=2x( …

The equality of the arg max of the trace of the ratio and the ratio of determinants follows from the fact that each of these two maximisation problems has the same solution, given by the matrix $S$ co …

Set $x(0)=x_0$ and $x(T)=x_T$, with $x_0,x_T\in[m,M]$. I will first allow for excursions outside of this interval, and then later add the constraint that $m\leq x(t)\leq M$ for all $0\leq t\leq T$. M …

I would argue as follows: unitary invariance means you are free to choose a basis. Let me choose the basis where $\tilde\phi=\{1,0,0,\ldots,0,0\}$, then $H_J(\sigma)-\tilde\lambda\langle\tilde\phi|\si …

This is not an answer, but continuing the discussion in the comment box is a bit cumbersome. The point I want to make is to see if this would work for a simple test case, $f(t)=ix\sin\pi t$, with sadd …

For the requested examples see Rényi Divergence and Kullback-Leibler Divergence (2012). • Two continuous distributions: Equation 10 gives the Rényi divergence between two Gaussian distributions (mean …