  We apply a generalized Kibble-Zurek out-of-equilibrium scaling ansatz to
simulated annealing when approaching the spin-glass transition at temperature
$T=0$ of the two-dimensional Ising model with random $J= \pm 1$ couplings.
Analyzing the spin-glass order parameter and the excess energy as functions of
the system size and the annealing velocity in Monte Carlo simulations with
Metropolis dynamics, we find scaling where the energy relaxes slower than the
spin-glass order parameter, i.e., there are two different dynamic exponents.
The values of the exponents relating the relaxation time scales to the system
length, $\tau \sim L^z$, are $z=8.28 \pm 0.03$ for the relaxation of the order
parameter and $z=10.31 \pm 0.04$ for the energy relaxation. We argue that the
behavior with dual time scales arises as a consequence of the entropy-driven
ordering mechanism within droplet theory. We point out that the dynamic
exponents found here for $T \to 0$ simulated annealing are different from the
temperature-dependent equilibrium dynamic exponent $z_{\rm eq}(T)$, for which
previous studies have found a divergent behavior; $z_{\rm eq}(T\to 0) \to
\infty$. Thus, our study shows that, within Metropolis dynamics, it is easier
to relax the system to one of its degenerate ground states than to migrate at
low temperatures between regions of the configuration space surrounding
different ground states. In a more general context of optimization, our study
provides an example of robust dense-region solutions for which the excess
energy (the conventional cost function) may not be the best measure of success.
