A probabilistic framework for large classes of binary integer programming problems is constructed. The approach is given by a mean field annealing scheme where the annealing phase is substituted by the solution of a dual problem that gives a lower (upper) bound for the original minimization (maximization) integer task. This bound has an information theoretic interpretation by which a principled feasible solution generator is constructed. The method is tested in linear and quadratic knapsack problems for which is capable to find high quality solutions in running times that are orders of magnitude shorter than state of the art algorithms. Experimental evidence indicates that for the quadratic case, the mean field approximation improves with problem size for unstructured instances. This is reminiscent of the exact mean field limit found in several spin glass models.
Universidad Autónoma de Nuevo León, Facultad de Ingeniería Mecánica y Eléctrica, Posgrado en Ingeniería de Sistemas, AP 126, Cd. Universitaria, San Nicolás de los Garza, NL 66450, México, Nov 2017. / CONACYT- Centro de Investigación en Matemáticas (CIMAT), A.C., Fray Bartolomé de las Casas 314, Barrio La Estación, CP 20259, Aguascalientes, Ags, México, Nov 2017.