Pré-Publication, Document De Travail Année : 2025

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 tolerance. At every iteration, our method replaces the objective by a piecewise linear relaxation to compute a dual bound. Since the polyhedral domain in our method remains unchanged, a primal bound is computed by evaluating the cost function on the solution provided by the relaxation. If the difference between these two values is deemed as not satisfactory, the relaxation is locally tightened with an objective-driven refinement procedure, that computes an optimal domain partitioning 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 tolerance. We provide a formal proof of the convergence of our method, and assess its efficiency in approximating the non-linear variants of five problems: the transportation problem, the uncapacitated facility location problem, the multicommodity flow problem, the multi-commodity network design problem, and the continuous knapsack problem. Our results indicate that the overall performance of our method is competitive to three state-of-the-art mixed-integer nonlinear solvers, often performing better. It also 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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Dates et versions

hal-03336022 , version 1 (06-09-2021)
hal-03336022 , version 2 (08-09-2021)
hal-03336022 , version 3 (21-01-2025)

Identifiants

  • HAL Id : hal-03336022 , version 3

Citer

Claudio Contardo, Sandra Ulrich Ngueveu. On the approximation of separable non-convex optimization programs to an arbitrary numerical precision. 2025. ⟨hal-03336022v3⟩
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