In this paper we propose a new hybrid heuristic for solving 0-1 mixed integer programs based on the principle of variable neighbourhood decomposition search. It combines variable neighbourhood search with a general-purpose CPLEX MIP solver. We perform systematic hard variable fixing (or diving) following the variable neighbourhood search rules. The variables to be fixed are chosen according to their distance from the corresponding linear relaxation solution values. If there is an improvement, variable neighbourhood descent branching is performed as the local search in the whole solution space. Numerical experiments have proven that exploiting boundary effects in this way considerably improves solution quality. With our approach, we have managed to improve the best known published results for 8 out of 29 instances from a well-known class of very difficult MIP problems. Moreover, computational results show that our method outperforms the CPLEX MIP solver, as well as three other recent most successful MIP solution methods.

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This paper concerns an optimization algorithm for unconstrained non-convex problems where the objective function has sparse connections between the unknowns. The algorithm is based on applying a dissipation preserving numerical integrator, the Itoh-Abe discrete gradient scheme, to the gradient flow of an objective function, guaranteeing energy decrease regardless of step size. We introduce the algorithm, prove a convergence rate estimate for non-convex problems with Lipschitz continuous gradients, and show an improved convergence rate if the objective function has sparse connections between unknowns. The algorithm is presented in serial and parallel versions. Numerical tests show its use in Euler's elastica regularized imaging problems and its convergence rate and compare the execution time of the method to that of the iPiano algorithm and the gradient descent and Heavy-ball algorithms.

In recent years many so-called matheuristics have been proposed for solving Mixed Integer Programming (MIP) problems. Though most of them are very efficient, they do not all theoretically converge to an optimal solution. In this paper we suggest two matheuristics, based on the variable neighbourhood decomposition search (VNDS), and we prove their convergence. Our approach is computationally competitive with the current state-of-the-art heuristics, and on a standard benchmark of 59 0-1 MIP instances, our best heuristic achieves similar solution quality to that of a recently published VNDS heuristic for 0-1 MIPs within a shorter execution time.

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