  Motivated by the grid search method and Bayesian optimization, we introduce
the concept of contractibility and its applications in model-based
optimization. First, a basic framework of contraction methods is established to
construct a nonempty closed set sequence that contracts from the initial domain
to the set of global minimizers. Then, from the perspective of whether the
contraction can be carried out effectively, relevant conditions are introduced
to divide all continuous optimization problems into three categories: (i)
logarithmic time contractible, (ii) polynomial time contractible, or (iii)
noncontractible. For every problem from the first two categories, there exists
a contraction sequence that converges to the set of all global minimizers with
linear convergence; for any problem from the last category, we discuss possible
troubles caused by contraction. Finally, a practical algorithm is proposed with
high probability bounds for convergence rate and complexity. It is shown that
the contractibility contributes to practical applications and can also be seen
as a complement to smoothness for distinguishing the optimization problems that
are easy to solve.
