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Nonconvex Optimization Is Combinatorial Optimization

Difficult nonconvex optimization problems contain a combinatorial number of local optima, making them extremely challenging for modern solvers. We present a novel nonconvex optimization algorithm that explicitly finds and exploits local structure in the objective function in order to decompose it into subproblems, exponentially reducing the size of the search space. Our algorithm’s use of decomposition, branch & bound, and caching solidifies the connection between nonconvex optimization and combinatorial optimization. We discuss preliminary experimental results on protein folding, Gaussian mixture models, and bundle adjustment.