Optimization with matrix exponential constraint

Suppose I'm optimizing for an unknown $x\in\mathbb{R}^k.$ I have a linear operator $A(\cdot)$ that maps $x$ to an $n\times n$ symmetric matrix, i.e., $A:\mathbb{R}^k\rightarrow\mathbb{R}^{n\times n}.$

I'd like to solve a problem of the form $$\begin{array}{rl} \min_{x\in\mathbb{R}^k} & f(x)\\ \textrm{s.t.} & x\in\mathcal C\\ & \mathrm{exp}(A(x))\cdot v=w, \end{array}$$ where $f:\mathbb{R}^k\rightarrow\mathbb{R}$ is convex, $\mathcal C\subseteq\mathbb{R}^k$ is some convex set and $v,w\in\mathbb{R}^n$ are constant vectors.

Is there any chance this problem can be transformed into something convex? If not (or if so), what would be a good optimization technique/algorithm for problems of this form?

• Your problem looks interesting. Could you please tell me where this problem comes from? I would like to spend some time looking into it.
– user17640
Sep 22 '15 at 19:21
• Any help with this problem would be much appreciated! This problem comes from some numerical challenges in differential geometry. At a high level, you can think of $A$ as defining an ODE $y'=Ay$, and the constraint links $y(1)=exp(A)v$ to $y(0)=v$. Please feel free to email me if you have ideas or thoughts about how to solve this problem. Sep 24 '15 at 20:55

No, this is not possible to cast as a convex problem, as the feasible set generically is nonconvex. Consider, e.g., the case $A = \begin{bmatrix} x & 0\\0 & 2x\end{bmatrix}$, $v = \begin{bmatrix}2\\-1\end{bmatrix}$ and $w=1/2$. This data generates the constraint $2e^x-e^{2x}=1/2$ which has two distinct feasible points ($-.123$ and $.534$). This is of course to be expected, as anything but affine equalities are nonconvex.