  Optimizing highly complex cost/energy functions over discrete variables is at
the heart of many open problems across different scientific disciplines and
industries. A major obstacle is the emergence of many-body effects among
certain subsets of variables in hard instances leading to critical slowing down
or collective freezing for known stochastic local search strategies. An
exponential computational effort is generally required to unfreeze such
variables and explore other unseen regions of the configuration space. Here, we
introduce a quantum-inspired family of nonlocal Nonequilibrium Monte Carlo
(NMC) algorithms by developing an adaptive gradient-free strategy that can
efficiently learn key instance-wise geometrical features of the cost function.
That information is employed on-the-fly to construct spatially inhomogeneous
thermal fluctuations for collectively unfreezing variables at various length
scales, circumventing costly exploration versus exploitation trade-offs. We
apply our algorithm to two of the most challenging combinatorial optimization
problems: random k-satisfiability (k-SAT) near the computational phase
transitions and Quadratic Assignment Problems (QAP). We observe significant
speedup and robustness over both specialized deterministic solvers and generic
stochastic solvers. In particular, for 90% of random 4-SAT instances we find
solutions that are inaccessible for the best specialized deterministic
algorithm known as Survey Propagation (SP) with an order of magnitude
improvement in the quality of solutions for the hardest 10% instances. We also
demonstrate two orders of magnitude improvement in time-to-solution over the
state-of-the-art generic stochastic solver known as Adaptive Parallel Tempering
(APT).
