We demonstrate an analog neuro-optimization hardware, suitable for solving a large number of combinatorial optimization problems, based on a crossbar circuit with 4096 passively-integrated analog-grade memristors. The proposed hardware supports a variety of metaheuristic techniques for improving optimization performance, such as stochastic and chaotic simulated annealing, and novel "exponential" annealing. The hardware operation is successfully tested by experimentally solving weighted graph partitioning, maximum clique, vertex cover, and independent set problems, and observing good agreement with simulation results.