On the approximation of separable non-convex optimization programs to an arbitrary numerical precision
Résumé
We consider the problem of minimizing the sum of a series of univariate (possibly non-convex) functions on a polyhedral domain. We introduce an iterative method with optimality guarantees to approximate this problem to an arbitrary numerical precision. At every iteration, our method replaces the objective by a lower bounding piecewise linear approximation to compute a dual bound. A primal bound is computed by evaluating the cost function on the solution provided by the approximation. If the difference between these two values is deemed as not satisfactory, the approximation is locally tightened and the process repeated. By keeping the scope of the update local, the computational burden is only slightly increased from iteration to iteration. The convergence of the method is assured under very mild assumptions, and no NLP nor MINLP solver/oracle is required to ever be invoked to do so. As a consequence, our method presents very nice scalability properties and is little sensitive to the desired precision. We provide a formal proof of the convergence of our method, and assess its efficiency in approximating the non-linear variants of three problems: the transportation problem, the capacitated facility location problem, and the multi-commodity network design problem. Our results indicate that the overall performance of our method is superior to five state-of-the-art mixed-integer nonlinear solvers by a significant margin, and scales better than a naive variant of the method that avoids performing successive iterations in exchange of solving a much larger mixed-integer linear program.
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