The complexity theory for black-box algorithms, introduced by
Droste et al. (2006), describes common limits on the efficiency of
a broad class of randomised search heuristics. There is an
obvious trade-off between the generality of the black-box model
and the strength of the bounds that can be proven in such a
model. In particular, the original black-box model allows
polynomial complexity for certain NP-complete problems and
provides for well-known benchmark problems relatively small lower
bounds, which are typically not met by popular search heuristics.

In this paper, we introduce a more restricted black-box model
which we claim captures the working principles of many randomised
search heuristics including simulated annealing, evolutionary
algorithms, randomised local search and others. The key concept
worked out is an unbiased variation operator. Considering this
class of algorithms, significantly better lower bounds on the
black-box complexity are proved, amongst them an
$\Omega(n\log n)$ bound for functions with unique optimum. Moreover,
a simple unimodal function and gap functions are considered.
We show that a simple (1+1) EA is able to match the runtime
bounds in several cases.

Some errors have been fixed. Section 4 has been thoroughly revised.

The complexity theory for black-box algorithms, introduced by
Droste et al. (2006), describes common limits on the efficiency of
a broad class of randomised search heuristics. There is an
obvious trade-off between the generality of the black-box model
and the strength of the bounds that can be proven in such a
model. In particular, the original black-box model allows
polynomial complexity for certain NP-complete problems and
provides for well-known benchmark problems relatively small lower
bounds, which are typically not met by popular search heuristics.

In this paper, we introduce a more restricted black-box model
which we claim captures the working principles of many randomised
search heuristics including simulated annealing, evolutionary
algorithms, randomised local search and others. The key concept
worked out is an unbiased variation operator. Considering this
class of algorithms, significantly better lower bounds on the
black-box complexity are proved, amongst them an
$\Omega(n\log n)$ bound for functions with unique optimum. Moreover,
a simple unimodal function and gap functions are considered.
We show that a simple (1+1) EA is able to match the runtime
bounds in several cases.