  We consider the problem of finding nearly optimal solutions of optimization
problems with random objective functions. Two concrete problems we consider are
(a) optimizing the Hamiltonian of a spherical or Ising $p$-spin glass model,
and (b) finding a large independent set in a sparse Erd\H{o}s-R\'{e}nyi graph.
The following families of algorithms are considered: (a) low-degree polynomials
of the input; (b) low-depth Boolean circuits; (c) the Langevin dynamics
algorithm. We show that these families of algorithms fail to produce nearly
optimal solutions with high probability. For the case of Boolean circuits, our
results improve the state-of-the-art bounds known in circuit complexity theory
(although we consider the search problem as opposed to the decision problem).
  Our proof uses the fact that these models are known to exhibit a variant of
the overlap gap property (OGP) of near-optimal solutions. Specifically, for
both models, every two solutions whose objectives are above a certain threshold
are either close or far from each other. The crux of our proof is that the
classes of algorithms we consider exhibit a form of stability. We show by an
interpolation argument that stable algorithms cannot overcome the OGP barrier.
  The stability of Langevin dynamics is an immediate consequence of the
well-posedness of stochastic differential equations. The stability of
low-degree polynomials and Boolean circuits is established using tools from
Gaussian and Boolean analysis -- namely hypercontractivity and total influence,
as well as a novel lower bound for random walks avoiding certain subsets. In
the case of Boolean circuits, the result also makes use of
Linal-Mansour-Nisan's classical theorem. Our techniques apply more broadly to
low influence functions and may apply more generally.
