  The (1+1)-evolution strategy (ES) with success-based step-size adaptation is
analyzed on a general convex quadratic function and its monotone
transformation, that is, $f(x) = g((x - x^*)^\mathrm{T} H (x - x^*))$, where
$g:\mathbb{R}\to\mathbb{R}$ is a strictly increasing function, $H$ is a
positive-definite symmetric matrix, and $x^* \in \mathbb{R}^d$ is the optimal
solution of $f$. The convergence rate, that is, the decrease rate of the
distance from a search point $m_t$ to the optimal solution $x^*$, is proven to
be in $O(\exp( - L / \mathrm{Tr}(H) ))$, where $L$ is the smallest eigenvalue
of $H$ and $\mathrm{Tr}(H)$ is the trace of $H$. This result generalizes the
known rate of $O(\exp(- 1/d ))$ for the case of $H = I_{d}$ ($I_d$ is the
identity matrix of dimension $d$) and $O(\exp(- 1/ (d\cdot\xi) ))$ for the case
of $H = \mathrm{diag}(\xi \cdot I_{d/2}, I_{d/2})$. To the best of our
knowledge, this is the first study in which the convergence rate of the
(1+1)-ES is derived explicitly and rigorously on a general convex quadratic
function, which depicts the impact of the distribution of the eigenvalues in
the Hessian $H$ on the optimization and not only the impact of the condition
number of $H$.
