Dual Heuristics Research Articles

Machine Learning-Guided Dual Heuristics and New Lower Bounds for the Refueling and Maintenance Planning Problem of Nuclear Power Plants

This paper studies the hybridization of Mixed Integer Programming (MIP) with dual heuristics and machine learning techniques, to provide dual bounds for a large scale optimization problem from an industrial application. The case study is the EURO/ROADEF Challenge 2010, to optimize the refueling and maintenance planning of nuclear power plants. Several MIP relaxations are presented to provide dual bounds computing smaller MIPs than the original problem. It is proven how to get dual bounds with scenario decomposition in the different 2-stage programming MILP formulations, with a selection of scenario guided by machine learning techniques. Several sets of dual bounds are computable, improving significantly the former best dual bounds of the literature and justifying the quality of the best primal solution known.

Two-Dimensional Phase Unwrapping via Balanced Spanning Forests

Phase unwrapping is the process of recovering a continuous phase signal from an original signal wrapped in the ([Formula: see text]] interval. It is a critical step of coherent signal processing, with applications such as synthetic aperture radar, acoustic imaging, magnetic resonance, X-ray crystallography, and seismic processing. In the field of computational optics, this problem is classically treated as a norm-minimization problem, in which one seeks to minimize the differences between the gradients of the original wrapped signal and those of the continuous unwrapped signal. When the [Formula: see text]–norm is considered, the number of differences should be minimized, leading to a difficult combinatorial optimization problem. We propose an approximate model for the [Formula: see text]–norm phase unwrapping problem in two dimensions (2D), in which the singularities of the wrapped phase image are associated with a graph where the vertices have −1 or [Formula: see text] polarities. The objective is to find a minimum-cost balanced spanning forest where the sum of the polarities is equal to zero in each tree. We introduce a set of primal and dual heuristics, a branch-and-cut algorithm, and a hybrid metaheuristic to efficiently find exact or heuristic solutions. These approaches move us one step closer to optimal solutions for 2D [Formula: see text]–norm phase unwrapping; such solutions were previously viewed, in the signal processing literature, as highly desirable but not achievable.

On the effectiveness of primal and dual heuristics for the transportation problem

On the effectiveness of primal and dual heuristics for the transportation problem

A dual heuristic for mixed integer programming

In linear programming based branch-and-bound algorithms, many heuristics have been developed to improve primal solutions, but on the dual side we rely solely on cutting planes to improve dual bounds. We design a dual heuristic which incorporates relaxation algorithms within a branch-and-bound framework to improve dual bounds. We study the effect of solving various relaxations with dual heuristics by conducting a series of computational tests on the multi-dimensional knapsack problem.

New Lower Bounds and Exact Method for the m-PVRP

This paper presents new lower bounding procedures and an exact method for the m-peripatetic vehicle routing problem (m-PVRP) based on polyhedral and column generation approaches. The branch-and-cut algorithms use three types of valid cuts on the edge-based formulation. The column-generation-based lower bounding procedure is applied on the dual set partitioning formulation and is composed of dual heuristics that estimate good dual variable values and therefore high lower bounds. Computational results on instances from the literature show that these new lower bounds reach on average 97.5% to 99% of the best known upper bounds and optimality is proven for a third of the instances.

Column Generation for Discrete-Rate Multi-User and Multi-Carrier Power Control

We consider a constrained multi-carrier power allocation problem in interference-limited multi-user systems with a finite set of transmission rates. The Lagrange relaxation is a common technique for decomposing such problems into independently solvable per-subcarrier problems. Deviating from this approach our main contribution is the proposal of a novel spectrum management framework based on a Nonlinear Dantzig-Wolfe problem decomposition. It allows for suboptimal initialization and suboptimal power allocation methods with low complexity. While we show that the combinatorial per-subcarrier problems have polynomial complexity in the number of users, we find that such suboptimal methods are indispensable in large systems. Thus we give an overview of various basic dual heuristics and provide simulation results on a set of thousand digital subscriber line (DSL) networks which show the superior performance of our framework compared to previous power control algorithms.

A Lagrangean Relaxation Based Algorithm for Solving Set Partitioning Problems

In this paper, we discuss a solver that is developed to solve set partitioning problems. The methods used include problem reduction techniques, Lagrangean relaxation and primal and dual heuristics. The optimal solution is found using a branch and bound approach. In this paper, we discuss these techniques. Furthermore, we present the results of several computational experiments and compare the performance of our solver with the well-known mathematical optimization solver Cplex.

Dual Heuristics on the Exact Solution of Large Steiner Problems

Abstract Abstract We present dual heuristics for the directed cut formulation of the Steiner problem in graphs. These heuristics usually give tight lower and upper bounds, and are enough to quickly solve two thirds of the instances from the literature. For harder instances, we propose two exact algorithms using those heuristics: branch-and-ascent, an implicit enumeration without LP solving; and a branch-and-cut that starts from bases provided by dual heuristics, which may be called afterwards to improve convergence. These algorithms have a good practical performance and solved several open instances, including the 1320 series and very large and degenerated problems from VLSI layout.

A dynamic subgradient-based branch-and-bound procedure for set covering : Egon Balas and Maria C. Carrera Operations Research Vol. 44, No. 6, November–December 1996, pp. 875–890

We discuss a branch and bound algorithm for set covering, whose centerpiece is a new integrated upper bounding/lower bounding procedure called dynamic subgradient optimization DYNSGRAD. This new procedure, applied to a Lagrangean dual at every node of the search tree, combines the standard subgradient method with primal and dual heuristics that interact to change the Lagrange multipliers and tighten the upper and lower bounds, fix variables, and periodically restate the Lagrangean itself. Extensive computational testing is reported. As a stand-alone heuristic, DYNSGRAD performs significantly better than other procedures in terms of the quality of solutions obtainable with a certain computational effort. When incorporated into a branch-and-bound algorithm, DYNSGRAD considerably advances the state of the art in solving set covering problems.

A Dynamic Subgradient-Based Branch-and-Bound Procedure for Set Covering

We discuss a branch and bound algorithm for set covering, whose centerpiece is a new integrated upper bounding/lower bounding procedure called dynamic subgradient optimization (DYNSGRAD). This new procedure, applied to a Lagrangean dual at every node of the search tree, combines the standard subgradient method with primal and dual heuristics that interact to change the Lagrange multipliers and tighten the upper and lower bounds, fix variables, and periodically restate the Lagrangean itself. Extensive computational testing is reported. As a stand-alone heuristic, DYNSGRAD performs significantly better than other procedures in terms of the quality of solutions obtainable with a certain computational effort. When incorporated into a branch-and-bound algorithm, DYNSGRAD considerably advances the state of the art in solving set covering problems.

Fast primal and dual heuristics for the p-median location problem

A greedy-type heuristic is presented for the p-median problem and computational results for problems having up to 400 vertices are shown. We describe a dual heuristic procedure for the p-median based on Erlenkotter's heuristic (1978) for the uncapacitated facility location problem. A primal-dual heuristic that generates a primal solution based on a good dual solution is also presented. For both procedures, computational experience relative to the same set of large size test problems is reported. Finally, the results obtained from all procedures are compared with the bounds, and corresponding computing times, provided by other available heuristics for the p-median location problem.

Optimal Solution of Set Covering/Partitioning Problems Using Dual Heuristics

We present an algorithm for a mixed set covering/partitioning model that includes as special cases the well-known set covering problem and set partitioning problem. The novel feature of our algorithm is the use of continuous heuristics applied to the dual of the linear programming relaxation to provide lower bounds for a branch and bound algorithm. The heuristics are continuous adaptations of the well-known greedy and 3-opt methods that have been applied to a variety of combinatorial optimization problems. Our algorithm has outperformed the current best set covering algorithm of Balas and Ho (1980) by about a factor of 3, and appears to improve on the best existing set partitioning algorithm by more than an order of magnitude.

Optimal Solution of Set Covering/Partitioning Problems Using Dual Heuristics

We present an algorithm for a mixed set covering/partitioning model that includes as special cases the well-known set covering problem and set partitioning problem. The novel feature of our algorit...

Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems

Recently it has been shown how various results on the behaviour of exact and approximate primal algorithms for certain discrete location problems can be obtained by studying the more general problem of maximising a submodular set function. Unfortunately, this more general model tells us little about an important aspect of the discrete location problems, their linear programming relaxations and the possible use of dual solutions in obtaining bounds. Here we examine a slightly more specialised model that again generalises the location problems of interest, but now also includes the continuous aspects of the problems missing from the earlier model, namely the problem: max{w(y): ∑j=1najyj ≤ b, 0 ≤ yj < 1} of maximising a real-valued nondecreasing piecewise linear, concave submodular function subject to a knapsack constraint. We show that a continuous greedy heuristic always attains at least (1 − e−1) × 100% of the optimal value, and that for the discrete problem an adapted greedy heuristic always attains 35% of the optimal value. Specialised to the capacitated and uncapacitated location problems these results permit us to strengthen earlier results, and to obtain new results on dual greedy heuristics.