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# Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm

*Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics*, PMLR 5:456-463, 2009.

#### Abstract

An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus low-rank quadratic approximation to the function. The quadratic approximation is constructed using a limited-memory quasi-Newton update. The method is suitable for large-scale problems where evaluation of the function is substantially more expensive than projection onto the constraint set. Numerical experiments on one-norm regularized test problems indicate that the proposed method is competitive with state-of-the-art methods such as bound-constrained L-BFGS and orthant-wise descent. We further show that the method generalizes to a wide class of problems, and substantially improves on state-of-the-art methods for problems such as learning the structure of Gaussian graphical models and Markov random fields.