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Discrete optimization is a central problem in mathematical optimization witha broad range of applications, among which binary optimization and sparseoptimization are two common ones. However, these problems are NP-hard and thusdifficult to solve in general. Combinatorial search methods such asbranch-and-bound and exhaustive search find the global optimal solution but areconfined to small-sized problems, while coordinate descent methods such ascoordinate gradient descent are efficient but often suffer from poor localminima. In this paper, we consider a hybrid method that combines theeffectiveness of combinatorial search and the efficiency of coordinate descent.Specifically, we consider random strategy or/and greedy strategy to select asubset of coordinates as the working set, and then perform global combinatorialsearch over the working set based on the original objective function. Inaddition, we provide some optimality analysis and convergence analysis for theproposed method. Our method finds stronger stationary points than existingmethods. Finally, we demonstrate the efficacy of our method on some sparseoptimization and binary optimization applications. As a result, our methodachieves state-of-the-art performance in terms of accuracy. For example, ourmethod generally outperforms the well-known orthogonal matching pursuit methodin sparse optimization.  Keywords: Sparsity Optimization, Binary Optimization, Coordinate Descent,Combinatorial Search, Discrete Optimization, Convergence Analysis.

A Hybrid Method of Combinatorial Search and Coordinate Descent for Discrete Optimization