A technique for speeding up the solution of the Lagrangean dual

We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinear programming problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in the theoretical running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We use our method to the solution of the LP relaxation and the Langrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, thek-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems,K-polymatroid intersection, multiple item capacitated lot sizing problem, and stochastic programming. In all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them.     

Lagrangean decomposition: A model yielding stronger lagrangean bounds

Given a mixed-integer programming problem with two matrix constraints, it is possible to define a Lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems, each having one of the two matrices of the original problem as its constraints. There is one Lagrangean multiplier per variable. We prove that the optimal value of this new Lagrangean dual dominates the optimal value of the Lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement. We show on an example that the resulting bound improvement can be substantial. We show on a complex practical problem how Lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology. Research supported by the National Science Foundation under grant ECS-8508142.     

Ranking lower bounds for the bin-packing problem

In this paper, we evaluate different known lower bounds for the bin-packing problem including linear programming relaxation (LP), Lagrangean relaxation (LR), Lagrangean decomposition (LD) and the Martello–Toth (MT) [Martello, S., Toth, P., Knapsack Problems: Algorithms and Computer Implementations, Wiley, New York, 1990] lower bounds. We give conditions under which Lagrangean bounds are superior to the LP bound, show that Lagrangean decomposition (LD) yields the same bound as Lagrangean relaxation (LR) and prove that the MT lower bound is a Lagrangean bound evaluated at a finite set of Lagrange multipliers; hence, it is no better than the LR and LD lower bounds.     

Selection of Capacity Expansion Projects

The problem of determining a project selection schedule and a production-distribution-inventory schedule for each of a number of plants so as to meet the demands of multiregional markets at minimum discounted total cost during a discrete finite planning horizon is considered. We include the possibility of using inventory and/or imports to delay the expansion decision at each producing region in a transportation network. Through a problem reduction algorithm, the Lagrangean relaxation problem strengthened by the addition of a surrogate constraint becomes a 0–1 mixed integer knapsack problem. Its optimal solution, given a set of Lagrangean multipliers, can be obtained by solving at most two generally smaller 0–1 pure integer knapsack problems. The bound is usually very tight. At each iteration of the subgradient method, we generate a primal feasible solution from the Lagrangean solution. The computational results indicate that the procedure is effective in solving large problems to within acceptable error tolerances.     

Modeling demand management strategies for evacuations

Evacuations are massive operations that create heavy travel demand on road networks some of which are experiencing major congestions even with regular traffic demand. Congestion in traffic networks during evacuations, can be eased either by supply or demand management actions. This study focuses on modeling demand management strategies of optimal departure time, optimal destination choice and optimal zone evacuation scheduling (also known as staggered evacuation) under a given fixed evacuation time assumption. The analytical models are developed for a system optimal dynamic traffic assignment problem, so that their characteristics can be studied to produce insights to be used for large-scale solution algorithms. While the first two strategies were represented in a linear programming (LP) model, evacuation zone scheduling problem inevitable included integers and resulted in a mixed integer LP (MILP) one. The dual of the LP produced an optimal assignment principle, and the nature of the MILP formulations revealed clues about more efficient heuristics. The discussed properties of the models are also supported via numerical results from a hypothetical network example.     

A Dual Simplex Implementation of a Constraint Selection Algorithm for Linear Programming

The constraint selection approach to linear programming begins by solving a relaxed version of the problem using only a few of the original constraints. If the solution obtained to this relaxation satisfies the remaining constraints it is optimal for the original LP. Otherwise, additional constraints must be incorporated in a larger relaxation. The procedure successively generates larger subproblems until an optimal solution is obtained which satisfies all of the original constraints. Computational results for a dual simplex implementation of this technique indicate that solving several small subproblems in this manner is more computationally efficient than solving the original LP using the revised simplex method.     

Fixed Charge Transportation Problems: a new heuristic approach based on Lagrangean relaxation and?the?solving of core problems

In this paper the Fixed Charge Transportation Problem is considered. A new heuristic approach is proposed, based on the intensive use of Lagrangean relaxation techniques. The more novel aspects of this approach are new Lagrangean relaxation and decomposition methods, the consideration of several core problems, defined from the previously computed Lagrangean reduced costs, the heuristic selection of the most promising core problem and the final resort to enumeration by applying a branch and cut algorithm to the selected core problem. For problems with a small ratio of the average fixed cost to the average variable cost (lower than or equal to 25), the proposed method can obtain similar or better solutions than the state-of-art algorithms, such as the tabu search procedure and the parametric ghost image processes. For larger ratios (between 50 and 180), the quality of the obtained solutions could be considered to be halfway between both methods.     

Optimal infrastructure design and expansion of broadband wireless access networks

This paper investigates the important infrastructure design and expansion problem for broadband wireless access networks subject to user demand constraints and system capacity constraints. For the problem, an integer program is derived and a heuristic solution procedure is proposed based on Lagrangean relaxation. In the computational experiments, our Lagrangean relaxation based algorithm can solve this complex design and expansion problem quickly and near optimally. Based on the test results, it is suggested that the proposed algorithm may be practically used for the infrastructure design and expansion problem for broadband wireless access networks.     

A heuristic lagrangean algorithm for the capacitated plant location problem

Lagrangean techniques have been widely applied to the uncapacitated plant location problem, and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account. In our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one. Several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems, the heuristics consisting of a two phase procedure. The first (location phase) defines a set of multipliers from the analysis of the dual LP relaxation, and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems. Several heuristics have been studied for this second phase, based either on a decomposition of knapsack type subproblems through a definition of a set of penalties, or of looking into the duality gap and trying to reduce it. Computational experience is reported.     

A comprehensive simplex-like algorithm for network optimization and perturbation analysis

The family of network optimization problems includes the following prototype models: assignment, critical path, max flow, shortest path, and transportation. Although it is long known that these problems can be modeled as linear programs (LP), this is generally not done. Due to the relative inefficiency and complexity of the simplex methods (primal, dual, and other variations) for network models, these problems are usually treated by one of over 100 specialized algorithms. This leads to several difficulties. The solution algorithms are not unified and each algorithm uses a different strategy to exploit the special structure of a specific problem. Furthermore, small variations in the problem, such as the introduction of side constraints, destroys the special structure and requires modifying andjor restarting the algorithm. Also, these algorithms obtain solution efficiency at the expense of managerial insight, as the final solutions from these algorithms do not have sufficient information to perform postoptimality analysis.

Another approach is to adapt the simplex to network optimization problems through network simplex. This provides unification of the various problems but maintains all the inefficiencies of simplex, as well as, most of the network inflexibility to handle changes such as side constraints. Even ordinary sensitivity analysis (OSA), long available in the tabular simplex, has been only recently transferred to network simplex.

This paper provides a single unified algorithm for all five network models. The proposed solution algorithm is a variant of the self-dual simplex with a warm start. This algorithm makes available the full power of LP perturbation analysis (PA) extended to handle optimal degeneracy. In contrast to OSA, the proposed PA provides ranges for which the current optimal strategy remains optimal, for simultaneous dependent or independent changes from the nominal values in costs, arc capacities, or suppliesJdemands. The proposed solution algorithm also facilitates incorporation of network structural changes and side constraints. It has the advantage of being computationally practical, easy for managers to understand and use, and provides useful PA information in all cases. Computer implementation issues are discussed and illustrative numerical examples are provided in the Appendix     

Multicategory Classification by Support Vector Machines

We examine the problem of how to discriminate between objects of three or more classes. Specifically, we investigate how two-class discrimination methods can be extended to the multiclass case. We show how the linear programming (LP) approaches based on the work of Mangasarian and quadratic programming (QP) approaches based on Vapnik's Support Vector Machine (SVM) can be combined to yield two new approaches to the multiclass problem. In LP multiclass discrimination, a single linear program is used to construct a piecewise-linear classification function. In our proposed multiclass SVM method, a single quadratic program is used to construct a piecewise-nonlinear classification function. Each piece of this function can take the form of a polynomial, a radial basis function, or even a neural network. For the k > 2-class problems, the SVM method as originally proposed required the construction of a two-class SVM to separate each class from the remaining classes. Similarily, k two-class linear programs can be used for the multiclass problem. We performed an empirical study of the original LP method, the proposed k LP method, the proposed single QP method and the original k QP methods. We discuss the advantages and disadvantages of each approach.     

A warm-start dual simplex solution algorithm for the minimum flow networks with postoptimality analyses

This paper presents an algorithm for finding the minimum flow in general (s, t) networks with m directed arcs. The minimum flow problem (MFP) arises in many transportation and communication systems. Here, we construct a linear programming (LP) formulation of MFP and develop a simplex-type algorithm to find a minflow. Unlike other simplex-like algorithms, the algorithm developed here starts with an incomplete Basic Variable Set (BVS) initially and then fills-up the BVS completely while pushing toward an optimal vertex. If this results in pushing too far into infeasibility, the next step pulls the solution back to feasibility. Both phases use the Gauss-Jordan pivoting transformation used in the standard simplex and dual simplex algorithms.

The proposed approach has some common features with Dantzig's self-dual simplex algorithm. We have avoided, however, the need for extra variables (slack and surplus) for equality constraints, as well as an artificial constraint for the self-dual algorithm for initial phase and the dual simplex, respectively. The proposed Push phase takes at most 2m − 1 iterations to complete a tree (this augmentation may not be feasible). An infeasible path to the optimal solution contains at most 2m − 1 iterations; therefore in theory, the algorithm needs a sequence of at most 4m − 2 iterations.

Furthermore, the algorithm developed here makes available the full power of LP perturbation analysis and facilitates introducing network structural changes and side constraints also. It can also detect clerical errors in data entry which may make the problem infeasible or unbounded. It is assumed that the reader is familiar with LP terminology.     

The zero-one knapsack problem with equality constraint

The zero-one knapsack problem is a linear zero-one programming problem with a single inequality constraint. This problem has been extensively studied and many applications and efficient algorithms have been published. In this paper we consider a similar problem, one with an equality instead of the inequality constraint. By replacing the equality by two inequalities one of which is placed in the economic function, a Lagrangean relaxation of the problem is obtained. The relation between the relaxed problem and the original problem is examined and it is shown how the optimal value of the relaxed problem varies with increasing values of the Lagrangean multiplier. Using these results an algorithm for solving the problem is proposed.The paper concludes with a discussion of computational experience.     

Multi-Item, Multi-Facility Supply Chain Planning: Models, Complexities, and Algorithms

We propose a planning model for products manufactured across multiple manufacturing facilities sharing similar production capabilities. The need for cross-facility capacity management is most evident in high-tech industries that have capital-intensive equipment and a short technology life cycle. We propose a multicommodity flow network model where each commodity represents a product and the network structure represents manufacturing facilities in the supply chain capable of producing the products. We analyze in depth the product-level (single-commodity, multi-facility) subproblem when the capacity constraints are relaxed. We prove that even the general-cost version of this uncapacitated subproblem is NP-complete. We show that there exists an optimization algorithm that is polynomial in the number of facilities, but exponential in the number of periods. We further show that under special cost structures the shortest-path algorithm could achieve optimality. We analyze cases when the optimal solution does not correspond to a source-to-sink path, thus the shortest path algorithm would fail. To solve the overall (multicommodity) planning problem we develop a Lagrangean decomposition scheme, which separates the planning decisions into a resource subproblem, and a number of product-level subproblems. The Lagrangean multipliers are updated iteratively using a subgradient search algorithm. Through extensive computational testing, we show that the shortest path algorithm serves as an effective heuristic for the product-level subproblem (a mixed integer program), yielding high quality solutions with only a fraction (roughly 2%) of the computer time.     

A Hybrid Bounding Procedure for the Workload Allocation Problem on Parallel Unrelated Machines with Setups

A nonpreemptive single stage manufacturing process with parallel, unrelated machines and multiple job types with setups (PUMS) is considered. We propose a hybrid approximation procedure where a Lagrangean relaxation dual and a Lagrangean decomposition dual are solved one after the other to generate a good lower bound on the optimal makespan value. Computational results are reported.     

Lagrangean decomposition for integer nonlinear programming with linear constraints

We present a Lagrangean decomposition to study integer nonlinear programming problems. Solving the dual Lagrangean relaxation we have to obtain at each iteration the solution of a nonlinear programming with continuous variables and an integer linear programming. Decreasing iteratively the primal—dual gap we propose two algorithms to treat the integer nonlinear programming.This work was partially supported by CNPq and FINEP.     

On the block upper-triangularity of undiscounted multi-chain markov decision problems

We show that the LP formulation for an undiscounted multi-chain Markov decision problem can be put in a block upper-triangular form by a polynomial time procedure. Each minimal block (after an appropriate dynamic revision) gives rise to a single-chain Markov decision problem which can be treated independently. An optimal solution to each single-chain problem can be connected by auxiliary dual programs to obtain an optimal solution to a multi-chain problem.     

A branch-and-price algorithm for the capacitated facility location problem

The capacitated facility location problem (CFLP) is a well-known combinatorial optimization problem with applications in distribution and production planning. It consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs. A number of solution approaches based on Lagrangean relaxation and subgradient optimization has been proposed for this problem. Subgradient optimization does not provide a primal (fractional) optimal solution to the corresponding master problem. However, in order to compute optimal solutions to large or difficult problem instances by means of a branch-and-bound procedure information about such a primal fractional solution can be advantageous. In this paper, a (stabilized) column generation method is, therefore, employed in order to solve a corresponding master problem exactly. The column generation procedure is then employed within a branch-and-price algorithm for computing optimal solutions to the CFLP. Computational results are reported for a set of larger and difficult problem instances.     

Heuristic and stochastic methods in optimization

The shifting bottleneck (SB) heuristic is among the most successful approximation methods for solving the job shop problem. It is essentially a machine based decomposition procedure where a series of one machine sequencing problems (OMSPs) are solved. However, such a procedure has been reported to be highly ineffective for the flow shop problems. In particular, we show that for the 2-machine flow shop problem, the SB heuristic will deliver the optimal solution in only a small number of instances. We examine the reason behind the failure of the machine based decomposition method for the flow shop. An optimal machine based decomposition procedure is formulated for the 2-machine flow shop, the time complexity of which is worse than that of the celebrated Johnson’s rule. The contribution of the present study lies in showing that the same machine based decomposition procedures which are so successful in solving complex job shops can also be suitably modified to optimally solve the simpler flow shops.