Dive Danna

Branch-and-price is a powerful framework to solve hard combinatorial problems. It is an interesting alternative to general purpose mixed integer programming as column generation usually produces at the root node tight lower bounds (when minimizing) that are further improved when branching. Branching also helps to generate integer solutions, however branch-and-price can be quite weak at producing good integer solutions rapidly because the solution of the relaxed master problem is rarely integer-valued. In this paper, we propose a general cooperation scheme between branch-and-price and local search to help branch-and-price finding good integer solutions earlier. This cooperation scheme extends to branch-and-price the use of heuristics in branch-and-bound and it also generalizes three previously known accelerations of branch-and-price. We show on the vehicle routing problem with time windows (Solomon benchmark) that it consistently improves the ability of branch-and-price to generate good integer solutions ea rly while retaining the ability of branch-and-price to produce good lower bounds.

In this paper, we describe the implementation of some heuristics for convex mixed integer nonlinear programs. The work focuses on three families of heuristics that have been successfully used for mixed integer linear programs: diving heuristics, the Feasibility Pump, and Relaxation Induced Neighborhood Search (RINS). We show how these heuristics can be adapted in the context of mixed integer nonlinear programming. We present results from computational experiments on a set of instances that show how the heuristics implemented help finding feasible solutions faster than the traditional branch-and-bound algorithm and how they help in reducing the total solution time of the branch-and-bound algorithm.

As mixed integer programming (MIP) problems become easier to solve in pratice, they are used in a growing number of applications where producing a unique optimal solution is often not enough to answer the underlying business problem. Examples include problems where some optimization criteria or some constraints are difficult to model, or where multiple solutions are wanted for quick solution repair in case of data changes. In this paper, we address the problem of effectively generating multiple solutions for the same model, concentrating on optimal and near-optimal solutions. We first define the problem formally, study its complexity, and present three different algorithms to solve it. The main algorithm we introduce, the one-tree algorithm, is a modification of the standard branch-and-bound algorithm. Our second algorithm is based on MIP heuristics. The third algorithm generalizes a previous approach that generates solutions sequentially. We then show with extensive computational experiments that the one-tree algorithm significantly outperforms previously known algorithms in terms of the speed to generate multiple solutions, while providing an acceptable level of diversity in the solutions produced.