We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, * but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new algorithm is an inner loop of optimized linearization steps, similar to those taken in the classical Frank-Wolfe method. Numerical results demonstrate that our new algorithm empirically outperforms the standard implementation of progressive hedging for obtaining bounds in SMIP.

We propose methods for improving the relaxations obtained by the normalized multiparametric disaggregation technique (NMDT). These relaxations constitute a key component for some methods for solving nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problems. It is shown that these relaxations can be more efficiently formulated by significantly reducing the number of auxiliary variables (in particular, binary variables) and constraints. Moreover, a novel algorithm for solving MIQCQP problems is proposed. It can be applied using either its original NMDT or the proposed reformulation. Computational experiments are performed using both benchmark instances from the literature and randomly generated instances. The numerical results suggest that the proposed techniques can improve the quality of the relaxations.

In this paper we study some second order conditions that may be added to the first order nessessary optimality condition 0 ∈ ∂pf (x) (with ∂pf denoting the proximal subdifferential) in order to obtain a sufficient condition for a strict local minimum for extended-real-valued, nonsmooth functions. Three different types of second order conditions are investigated, all based on a different second order subdifferential. Namely, the subhessian, the graphical derivative, and the (contingent) coderivative to the proximal subdifferential.

The feasibility pump is a recent, highly successful heuristic for general mixed integer linear programming problems. We show that the feasibility pump heuristic can be interpreted as a discrete version of the proximal point algorithm. In doing so, we extend and generalize some of the fundamental results in this area to provide new supporting theory. We show that feasibility pump algorithms implicitly minimize a weighted combination of the objective and a term which penalizes lack of integrality. This function has many local minima, some of which correspond to feasible integral solutions; the feasibility pump's use of random restarts can be viewed as seeking to escape these local minima when they are not feasible integral solutions. This interpretation suggests alternative ways of incorporating restarts, one of which is the application of cutting planes. Numerical experiments with cutting planes show encouraging results on standard test libraries. Introduction. In spite of the continuous improvement of both commercialand open source solvers, numerous mixed integer programming (MIP) problems of practical importance remain intractable. In practice, where rigorous algorithms fail, heuristics often succeed to provide feasible solutions of good quality. Apart from their self-evident value, good feasible solutions are also useful in speeding up the search of branch and cut algorithms. General purpose heuristics include [2,3,4,13,14,15,16,19,20,21,22,23]. We direct the reader to [6] for a recent survey.A heuristic that has attracted significant interest in recent years is the feasibility pump (FP) [17]. The FP starts from the linear program (LP) optimum and computes two trajectories of points, one integer and the other LP feasible, by iteratively applying rounding and projection operations. Every LP feasible point is rounded to an integer point and every integer point is projected back onto the LP feasible region using the l 1 norm. As both the rounding and the projection are minimal distance operations (with respect to the l 1 norm), the distance between pairs of points on the two trajectories decreases monotonically until the method cycles, unable to further decrease the distance. If the distance is reduced to zero before cycling, then a feasible point has been obtained. Otherwise a random move is used to restart the method from a new point. In the original FP [17], the only bias of the method toward points of good objective value is the starting point, so the random restarts threaten to escape the good quality area. Thus, the feasibility pump uses a sophisticated restart scheme, attempting to displace the current point in an economic way by "minor perturbations," and performs a "major restart" only in the presence of further failures.

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