Reoptimization in lagrangian methods for the quadratic knapsack problem

International audienceThe 0-1 quadratic knapsack problem consists in maximizing a quadratic objective function subject to a linear capacity constraint. To solve exactly large instances of this problem with a tree search algorithm (e.g. a branch and bound method), the knowledge of good lower and upper bounds is crucial for pruning the tree but also for fixing as many variables as possible in a preprocessing phase. The upper bounds used in the best known exact approaches are based on Lagrangian relaxation and decomposition. It appears that the computation of these Lagrangian dual bounds involves the resolution of numerous 0-1 linear knapsack subproblems. Thus, taking this huge number of solvings into account, we propose to embed reoptimization techniques for improving the efficiency of the preprocessing phase of the 0-1 quadratic knapsack resolution. Namely, reoptimization is introduced to accelerate each independent sequence of 0-1 linear knapsack problems induced by the Lagrangian relaxation as well as the Lagrangian decomposition. Numerous numerical experiments validate the relevance of our approach

Comparison of heuristic approaches for the multiple depot vehicle scheduling problem

Given a set of timetabled tasks, the multi-depot vehicle scheduling problemis a well-known problem that consists of determining least-cost schedulesfor vehicles assigned to several depots such that each task is accomplishedexactly once by a vehicle. In this paper, we propose to compare theperformance of five different heuristic approaches for this problem,namely, a heuristic \\mip solver, a Lagrangian heuristic, a columngeneration heuristic, a large neighborhood search heuristic using columngeneration for neighborhood evaluation, and a tabu search heuristic. Thefirst three methods are adaptations of existing methods, while the last twoare novel approaches for this problem. Computational results on randomlygenerated instances show that the column generation heuristic performs thebest when enough computational time is available and stability is required,while the large neighborhood search method is the best alternative whenlooking for a compromise between computational time and solution quality.tabu search;column generation;vehicle scheduling;heuristics;Lagrangian heuristic;large neighborhood search;multiple depot

05031 Abstracts Collection -- Algorithms for Optimization with Incomplete Information

From 16.01.05 to 21.01.05, the Dagstuhl Seminar 05031 ``Algorithms for Optimization with Incomplete Information\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Comparison of heuristic approaches for the multiple depot vehicle scheduling problem

Given a set of timetabled tasks, the multi-depot vehicle scheduling problem is a well-known problem that consists of determining least-cost schedules for vehicles assigned to several depots such that each task is accomplished exactly once by a vehicle. In this paper, we propose to compare the performance of five different heuristic approaches for this problem, namely, a heuristic \\mip solver, a Lagrangian heuristic, a column generation heuristic, a large neighborhood search heuristic using column generation for neighborhood evaluation, and a tabu search heuristic. The first three methods are adaptations of existing methods, while the last two are novel approaches for this problem. Computational results on randomly generated instances show that the column generation heuristic performs the best when enough computational time is available and stability is required, while the large neighborhood search method is the best alternative when looking for a compromise between computational time and solution quality

Changing Bases: Multistage Optimization for Matroids and Matchings

This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge is to continually maintain near-optimal solutions to the underlying optimization problems, without creating too much churn in the solution itself. We model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions (one for each time step); while we can change the solution from step to step, we incur an additional cost for every such change. We study the multistage matroid maintenance problem, where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements. The online version of this problem generalizes online paging. E.g., given a graph, we need to maintain a spanning tree $T_t$ at each step: we pay $c_t(T_t)$ for the cost of the tree at time $t$, and also $| T_t\setminus T_{t-1} |$ for the number of edges changed at this step. Our main result is an $O(\log m \log r)$-approximation, where $m$ is the number of elements/edges and $r$ is the rank of the matroid. We also give an $O(\log m)$ approximation for the offline version of the problem. These bounds hold when the acquisition costs are non-uniform, in which caseboth these results are the best possible unless P=NP. We also study the perfect matching version of the problem, where we must maintain a perfect matching at each step under changing cost functions and costs for adding new elements. Surprisingly, the hardness drastically increases: for any constant $\epsilon>0$, there is no $O(n^{1-\epsilon})$-approximation to the multistage matching maintenance problem, even in the offline case

Ellipsoidal relaxations of the stable set problem:theory and algorithms

A new exact approach to the stable set problem is presented, which attempts to avoid the pitfalls of existing approaches based on linear and semidefinite programming. The method begins by constructing an ellipsoid that contains the stable set polytope and has the property that the upper bound obtained by optimising over it is equal to the Lovasz theta number. This ellipsoid can then be used to construct useful convex relaxations of the stable set problem, which can be embedded within a branch-and-bound framework. Extensive computational results are given, which indicate the potential of the approach

Fiscal sustainability in a new Keynesian model

Recent work on optimal monetary and fiscal policy in New Keynesian models suggests that it is optimal to allow steady-state debt to follow a random walk. In this paper we consider the nature of the time inconsistency involved in such a policy and its implication for discretionary policymaking. We show that governments are tempted, given inflationary expectations, to utilize their monetary and fiscal instruments in the initial period to change the ultimate debt burden they need to service. We demonstrate that this temptation is only eliminated if following shocks, the new steady-state debt is equal to the original (efficient) debt level even though there is no explicit debt target in the government's objective function. Analytically and in a series of numerical simulations we show which instrument is used to stabilize the debt depends crucially on the degree of nominal inertia and the size of the debt stock. We also show that the welfare consequences of introducing debt are negligible for precommitment policies, but can be significant for discretionary policy. Finally, we assess the credibility of commitment policy by considering a quasi-commitment policy, which allows for different probabilities of reneging on past promises

Path Optimization for the Resource-Constrained Searcher

Naval Research LogisticsWe formulate and solve a discrete-time path-optimization problem where a single searcher, operating in a discretized 3-dimensional airspace, looks for a moving target in a finite set of cells. The searcher is constrained by maximum limits on the consumption of several resources such as time, fuel, and risk along any path. We develop a special- ized branch-and-bound algorithm for this problem that utilizes several network reduction procedures as well as a new bounding technique based on Lagrangian relaxation and net- work expansion. The resulting algorithm outperforms a state-of-the-art algorithm for solving time-constrained problems and also is the first algorithm to solve multi-constrained problems

A network airline revenue management framework based on decomposition by origins and destinations

We propose a framework for solving airline revenue management problems on large networks, where the main concern is to allocate the flight leg capacities to customer requests under fixed class fares. This framework is based on a mathematical programming model that decomposes the network into origin-destination pairs so that each pair can be treated as a single flight leg problem. We first discuss that the proposed framework is quite generic in the sense that not only several well-known models from the literature fit into this framework but also many single flight leg models can be easily extended to a network setting through the prescribed construction. Then, we analyze the structure of the overall mathematical programming model and establish its relationship with other models frequently used in practice. The application of the proposed framework is illustrated through two examples based on static and dynamic single-leg models, respectively. These illustrative examples are then benchmarked against several existing methods on a set of real-life network problems