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Is $\text{Trace}(e^{XA+A^TX})$ a convex function of $X$? $X$ is diagonal and positive definite, $A$ is symmetric negative definite definite.

And by the way, what is the best way to solve a problem of the form:

$\text{min}.$ $\text{Trace}(e^{XA+A^TX})$

s.t. $X$ diagonal, $X_{ii}>0$ and $1^T X 1<x$

where $X$ is the variable and $A$ is symmetric negative definite, $x>0$, $x \in R$

Pedro
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questioner
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1 Answers1

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Let $D$ (resp. $D^+$) be the set of diagonal matrices (resp. diagonal positive matrices). Let $f:X\in D^+\rightarrow tr(e^{XA+AX})$. We seek $\min_X f(X)$ under the condition $tr(X)=\sum_i x_{i,i}=x$.

Fortunately the first derivative of $f$ is easy to calculate. For every $H\in D$, $f'_X(H)=tr(e^{XA+AX}(HA+AH))$ ; the Lagrange condition of our problem can be written: there is $\lambda\in\mathbb{R}$ s.t., for every $H\in D$,$f'_X(H)+\lambda tr(H)=0$ or $tr((Ae^{XA+AX}+e^{XA+AX}A+\lambda I)H)=0$. That is equivalent to the following $n$ equalities:

(*) for every $i$, $(Ae^{XA+AX}+e^{XA+AX}A+\lambda I)_{i,i}=0$.

Remark 1. If $tr(X)<x$, then put $\lambda=0$ in the previous formula.

Remark 2. Unfortunately the second derivative of $f$ is difficult to calculate.

EDIT 1: @ questioner , Tim Notke (a highschool basketball coach) said « Hard work beats talent when talent fails to work hard ».

If we search the critical points, then necessarily $tr(X)=x$ and even $\lambda >0$. Indeed $XA+AX$ symmetric implies that $e^{XA+AX}$ is symmetric $>0$. Thus $spectrum(Ae^{XA+AX})\subset \mathbb{R}^{*-}$ and $tr(Ae^{XA+AX})<0$. Finally, the relations (*) imply that $\lambda>0$ and therefore $tr(X)=x$.

EDIT 2. Let $S_n$ be the set of real symmetric matrices. $X\in S_n\rightarrow e^X$ is not a matrix convex function. Yet $X\in S_n\rightarrow \log(tr(e^X))$ is a spectral matrix function and is convex. cf. convexity of matrix "soft-max" (log trace of matrix exponential)

Thus $g:X\in S_n\rightarrow tr(e^X)$ is also convex. Here we consider $f:X\in S_n\rightarrow g(XA+AX)$ and for every symmetric $H$, $f''_X(H,H)=g''_{XA+AX}(HA+AH,HA+AH)\geq 0$. Finally, for every symmetric $A$, $f$ is convex over $S_n$.

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  • So, what about convexity?! What can be used with the sole first derivate of f?! – questioner Jan 04 '15 at 06:26
  • Careful there. He offers a partial solution. That sounds a bit ungrateful. – Michael Grant Jan 04 '15 at 07:01
  • @loup: this sounds a bit vague. What do you mean exactly by $f'_X(H)$? why does X is still appearing in it? (thank you for the quote anyway.) Furthermore, the constraint is inequality not equality, which is why, I think that you have to KKT instead of Lagrange Condition. – questioner Jan 05 '15 at 23:24
  • @ questioner , 1. the derivative $f'_X$ is a linear function over $D$ (see a book about differential calculus). Note that (star) is equivalent to $\nabla (f+\lambda tr)(X)=0$. 2. Since $f$ and $tr(X)$ are convex, the required minimum -if it exists- is reached on the edge, that is for $tr(X)=x$. 3. The necessary part of KKT condition is a direct consequence of Lagrange's theory (one line proof). 4. (star) is difficult to solve numerically. Perhaps it is better to use the convexity of $f$. You can ask Michael Grant ; he is a specialist in the subject. –  Jan 06 '15 at 09:31