We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with worst case estimates for the rate at which the residuals vanish. Strong and linear convergence are obtained in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and … Read more
We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a black-box algorithm of Zhang et al. that approximates an $\epsilon$-stationary point of any directionally differentiable Lipschitz objective using $O(\epsilon^{-4})$ calls to … Read more
We consider the bilevel knapsack problem with interdiction constraints, a fundamental bilevel integer programming problem which generalizes the 0-1 knapsack problem. In this problem, there are two knapsacks and \(n\) items. The objective is to select some items to pack into the first knapsack such that the maximum profit attainable from packing some of the … Read more
A critical measure of model quality for a mixed-integer program (MIP) is the difference, or gap, between its optimal objective value and that of its linear programming relaxation. In some cases, the right-hand side is not known exactly; however, there is no consensus metric for evaluating a MIP model when considering multiple right-hand sides. In … Read more
This paper focuses on algorithms for multi-stage stochastic linear programming (MSLP). We propose an ensemble method named the “compromise policy”, which not only reduces the variance of the function approximation but also reduces the bias of the estimated optimal value. It provides a tight lower bound estimate with a confidence interval. By exploiting parallel computing, … Read more
A new projective exact penalty function method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing its value at the projection of an infeasible point on the feasible set with the distance to the projection. The … Read more
For vehicle routing problems, strong dual bounds on the optimal value are needed to develop scalable exact algorithms, as well as to evaluate the performance of heuristics. In this work, we propose an iterative algorithm to compute dual bounds motivated by connections between decision diagrams (DDs) and dynamic programming (DP) models used for pricing in … Read more
This work analyzes the solution trajectory of gradient-based algorithms via a novel basis function decomposition. We show that, although solution trajectories of gradient-based algorithms may vary depending on the learning task, they behave almost monotonically when projected onto an appropriate orthonormal function basis. Such projection gives rise to a basis function decomposition of the solution … Read more
The objective of this paper is to develop a scalable binary linear programming model for finding the optimal aggregation of communes into spatially contiguous administrative territorial units (ATUs) constrained on certain balancing criteria. The requirement for the ATUs to be contiguous represents the main computational bottleneck and, therefore, it prevents one from using such models … Read more
The capacitated vehicle routing problem with stochastic demands (CVRPSD) is a variant of the deterministic capacitated vehicle routing problem where customer demands are random variables. While the most successful formulations for several deterministic vehicle-routing problem variants are based on a set-partitioning formulation, adapting such formulations for the CVRPSD under mild assumptions on the demands remains … Read more