  Lower temperature leads to a higher probability of visiting low-energy
states. This intuitive belief underlies most physics-inspired strategies for
addressing hard optimization problems. For instance, the popular simulated
annealing (SA) dynamics is expected to approach a ground state if the
temperature is lowered appropriately. Here we demonstrate that this belief is
not always justified. Specifically, we employ the cavity method to analyze the
minimum strong defensive alliance problem and discover a bifurcation in the
solution space, induced by an inflection point in the entropy--energy profile.
While easily accessible configurations are associated with the
lower-free-energy branch, the low-energy configurations are associated with the
higher-free-energy branch within the same temperature range. There is a
discontinuous phase transition between the high-energy configurations and the
ground states, which generally cannot be followed by SA. We introduce an
energy-clamping strategy to obtain superior solutions by following the
higher-free-energy branch, overcoming the limitations of SA.
