Lagrangian relaxation and pegging test for the clique partitioning problem

The clique partitioning problem is an NP-hard combinatorial optimization problem with applications to data analysis such as clustering. Though a binary integer linear programming formulation has been known for years, one needs to deal with a huge number of variables and constraints when solving a large instance. In this paper, we propose a size reduction algorithm which is based on the Lagrangian relaxation and the pegging test, and verify its validity through numerical experiments. We modify the conventional subgradient method in order to manage the high dimensionality of the Lagrangian multipliers, and also make an improvement on the ordinary pegging test by taking advantage of the structural property of the clique partitioning problem.     

A Lagrangian relaxation approach to solve the second phase of the exam scheduling problem

The problem of scheduling final exams at a large university can be viewed as a three phase process. The first phase consists of grouping the exams into sets called exam blocks. The second phase deals with the assignment of exam blocks to exam days and the third phase consists of arranging the exam days and also arranging the blocks within days.In this paper, we present new integer programming formulations for the second phase of the scheduling problem. We present an integer program with a single objective of minimizing the number of students with two or more exams per day. We then present a Lagrangian relaxation based solution procedure to solve this problem. Further, we present a bicriterion integer programming formulation to minimize the number of students with two exams per day and the number of students with three exams per day. Finally, we present some computational experience using randomly generated problems as well as real world data obtained from the State University of New York at Buffalo.     

A Lagrangian relaxation approach to large-scale flow interception problems

The paper presents a tight Lagrangian bound and an efficient dual heuristic for the flow interception problem. The proposed Lagrangian relaxation decomposes the problem into two subproblems that are easy to solve. Information from one of the subproblems is used within a dual heuristic to construct feasible solutions and is used to generate valid cuts that strengthen the relaxation. Both the heuristic and the relaxation are integrated into a cutting plane method where the Lagrangian bound is calculated using a subgradient algorithm. In the course of the algorithm, a valid cut is added and integrated efficiently in the second subproblem and is updated whenever the heuristic solution improves. The algorithm is tested on randomly generated test problems with up to 500 vertices, 12,483 paths, and 43 facilities. The algorithm finds a proven optimal solution in more than 75% of the cases, while the feasible solution is on average within 0.06% from the upper bound.     

A nonconvex quadratic optimization approach to the maximum edge weight clique problem

The maximum edge weight clique (MEWC) problem, defined on a simple edge-weighted graph, is to find a subset of vertices inducing a complete subgraph with the maximum total sum of edge weights. We propose a quadratic optimization formulation for the MEWC problem and study characteristics of this formulation in terms of local and global optimality. We establish the correspondence between local maxima of the proposed formulation and maximal cliques of the underlying graph, implying that the characteristic vector of a MEWC in the graph is a global optimizer of the continuous problem. In addition, we present an exact algorithm to solve the MEWC problem. The algorithm is a combinatorial branch-and-bound procedure that takes advantage of a new upper bound as well as an efficient construction heuristic based on the proposed quadratic formulation. Results of computational experiments on some benchmark instances are also presented.     

Lagrangian relaxation and its application to the unit-commitment-economic-dispatch problem

This study is concerned with the optimal scheduling of an electricitypower system consisting of both hydro and thermal units. UsingLagrangian relaxation, the original (primal) problem may bewritten in a dual formulation; the problem then admits decompositioninto more tractable sub-problems. Furthermore, the primal solutioncan be approximated closely by the dual solution, by using theduality gap as a termination criterion. A heuristic has beenused to construct nearly optimal solutions to the primal problembased on the information provided by the dual problem. Thispaper highlights three main points: improved computational times,the economic significance of the Lagrange multipliers, and theimplicit parallelism of this algorithm.     

A Lagrangian relaxation approach to multi-period inventory/distribution planning

We consider a multi-period inventory/distribution planning problem (MPIDP) in a one-warehouse multiretailer distribution system where a fleet of heterogeneous vehicles delivers products from a warehouse to several retailers. The objective of the MPIDP is to minimise transportation costs for product delivery and inventory holding costs at retailers over the planning horizon. In this research, the problem is formulated as a mixed integer linear programme and solved by a Lagrangian relaxation approach. A subgradient optimisation method is employed to obtain lower bounds. We develop a Lagrangian heuristic algorithm to find a good feasible solution of the MPIDP. Computational experiments on randomly generated test problems showed that the suggested algorithm gave relatively good solutions in a reasonable amount of computation time.     

A novel Lagrangian relaxation approach for a hybrid flowshop scheduling problem in the steelmaking-continuous casting process

One of the largest bottlenecks in iron and steel production is the steelmaking-continuous casting (SCC) process, which consists of steel-making, refining and continuous casting. The SCC scheduling is a complex hybrid flowshop (HFS) scheduling problem with the following features: job grouping and precedence constraints, no idle time within the same group of jobs and setup time constraints on the casters. This paper first models the scheduling problem as a mixed-integer programming (MIP) problem with the objective of minimizing the total weighted earliness/tardiness penalties and job waiting. Next, a Lagrangian relaxation (LR) approach relaxing the machine capacity constraints is presented to solve the MIP problem, which decomposes the relaxed problem into two tractable subproblems by separating the continuous variables from the integer ones. Additionally, two methods, i.e., the boundedness detection method and time horizon method, are explored to handle the unboundedness of the decomposed subproblems in iterations. Furthermore, an improved subgradient level algorithm with global convergence is developed to solve the Lagrangian dual (LD) problem. The computational results and comparisons demonstrate that the proposed LR approach outperforms the conventional LR approaches in terms of solution quality, with a significantly shorter running time being observed.     

A Lagrangian relaxation approach to the mixed-product assembly line sequencing problem: A case study of a door-lock company in Taiwan

In mixed-product assembly line sequencing, the production resources required for the assembly lines should be scheduled to minimize the overall cost and meet customer demand. In this paper, we study an assembly line sequencing problem for the door-lock industry in Taiwan and develop an integer programming formulation with realistic constraints. The complex solution space makes the resulting program difficult to solve using commercial optimization packages. Therefore, a heuristic based on the Lagrangian relaxation principle is developed to solve this problem efficiently. We evaluate the efficiency of the developed Lagrangian relaxation heuristic by comparing its solutions with those obtained using a commercial optimization package: the computational results show that the developed heuristic solves the real-world problem faster than the optimization package by almost 15 times in CPU time at a comparable solution quality.     

Partial Lagrangian relaxation for the unbalanced orthogonal Procrustes problem

Based on a novel reformulation of the feasible region, we propose and analyze a partial Lagrangian relaxation approach for the unbalanced orthogonal Procrustes problem (UOP). With a properly selected Lagrangian multiplier, the Lagrangian relaxation (LR) is equivalent to the recent matrix lifting semidefinite programming relaxation (MSDR), which has much more variables and constraints. Numerical results show that (LR) is solved more efficiently than (MSDR). Moreover, based on the special structure of (LR), we successfully employ the well-known Frank–Wolfe algorithm to efficiently solve very large instances of (LR). The rate of the convergence is shown to be independent of the row-dimension of the matrix variable of (UOP). Finally, motivated by (LR), we propose a Lagrangian heuristic for (UOP). Numerical results show that it can efficiently find the global optimal solutions of some randomly generated instances of (UOP).     

A matheuristic based on Lagrangian relaxation for the multi-activity shift scheduling problem

The multi-activity shift scheduling problem involves assigning a sequence of activities to a set of employees. In this paper, we consider the variant where the employees have different qualifications and each activity must be performed in a specified time window; i.e., we specify the earliest start period and the latest finish period. We propose a matheuristic in which Lagrangian relaxation is used to identify a subset of promising shifts, and a restricted set covering problem is solved to find a feasible solution. Each shift is represented by a context-free grammar. Computational tests are carried out on two sets of instances from the literature. For the first set, the matheuristic finds a solution with an optimality gap less than 0.01% for 70% of the instances and improves the best-known solution for 16% of them; for the second set, the matheuristic reaches the best-known solutions for 55% of the instances and finds better solutions for 37.5% of them.     

On the augmented Lagrangian approach to Signorini elastic contact problem

Summary. The Signorini problem describes the contact of a linearly elastic body with a rigid frictionless foundation. It is transformed into a saddle point problem of some augmented Lagrangian functional and then discretized by finite element methods. Optimal error estimates are obtained for general smooth domains which are not necessarily convex. The key ingredient in the analysis is a discrete inf-sup condition which guarantees the existence of the saddle point. Received January 29, 1999 / Revised version received May 2, 2000 / Published online December 19, 2000     

A Lagrangian approximation to the water-wave problem

We describe the derivation by a variational approach of the Camassa-Holm model for periodic shallow water waves.     

Solving the maximum clique problem using a tabu search approach

We describe two variants of a tabu search heuristic, a deterministic one and a probabilistic one, for the maximum clique problem. This heuristic may be viewed as a natural alternative implementation of tabu search for this problem when compared to existing ones. We also present a new random graph generator, the -generator, which produces graphs with larger clique sizes than comparable ones obtained by classical random graph generating techniques. Computational results on a large set of test problems randomly generated with this new generator are reported and compared with those of other approximate methods.The authors are grateful to the Quebec Government (Fonds F.C.A.R.) and to the Canadian Natural Sciences and Engineering Research Council (grant 0GP0038816) for financial support.     

On a continuous approach for the maximum weighted clique problem

This paper is focused on computational study of continuous approach for the maximum weighted clique problem. The problem is formulated as a continuous optimization problem with a nonconvex quadratic constraint given by the difference of two convex functions (d.c. function). The proposed approach consists of two main ingredients: a local search algorithm, which provides us with crucial points; and a procedure which is based on global optimality condition and which allows us to escape from such points. The efficiency of the proposed algorithm is illustrated by computational results.     

A Lagrangian relaxation approach to simultaneous strategic and tactical planning in supply chain design

We study a multi-echelon joint inventory-location model that simultaneously determines the location of warehouses and inventory policies at the warehouses and retailers. The model is formulated as a nonlinear mixed-integer program, and is solved using a Lagrangian relaxation-based approach. The efficiency of the algorithm and benefits of integration are evaluated through a computational study.     

An alternative framework to Lagrangian relaxation approach for job shop scheduling

A new Lagrangian relaxation (LR) approach is developed for job shop scheduling problems. In the approach, operation precedence constraints rather than machine capacity constraints are relaxed. The relaxed problem is decomposed into single or parallel machine scheduling subproblems. These subproblems, which are NP-complete in general, are approximately solved by using fast heuristic algorithms. The dual problem is solved by using a recently developed “surrogate subgradient method” that allows approximate optimization of the subproblems. Since the algorithms for subproblems do not depend on the time horizon of the scheduling problems and are very fast, our new LR approach is efficient, particularly for large problems with long time horizons. For these problems, the machine decomposition-based LR approach requires much less memory and computation time as compared to a part decomposition-based approach as demonstrated by numerical testing.     

An application of Lagrangian relaxation to a capacity planning problem under uncertainty

A supply chain network-planning problem is presented as a two-stage resource allocation model with 0-1 discrete variables. In contrast to the deterministic mathematical programming approach, we use scenarios, to represent the uncertainties in demand. This formulation leads to a very large scale mixed integer-programming problem which is intractable. We apply Lagrangian relaxation and its corresponding decomposition of the initial problem in a novel way, whereby the Lagrangian relaxation is reinterpreted as a column generator and the integer feasible solutions are used to approximate the given problem. This approach addresses two closely related problems of scenario analysis and two-stage stochastic programs. Computational solutions for large data instances of these problems are carried out successfully and their solutions analysed and reported. The model and the solution system have been applied to study supply chain capacity investment and planning.