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     

Computational aspects of simplex and MBU-simplex algorithms using different anti-cycling pivot rules

Abstract

Several variations of index selection rules for simplex-type algorithms for linear programming, like the Last-In-First-Out or the Most-Often-Selected-Variable are rules not only theoretically finite, but also provide significant flexibility in choosing a pivot element. Based on an implementation of the primal simplex and the monotonic build-up (MBU) simplex method, the practical benefit of the flexibility of these anti-cycling pivot rules is evaluated using public benchmark LP test sets. Our results also provide numerical evidence that the MBU-simplex algorithm is a viable alternative to the traditional simplex algorithm.     

Creating Advanced Bases For Large Scale Linear Programs Exploiting Embedded Network Structure

In this paper, we investigate how an embedded pure network structure arising in many linear programming (LP) problems can be exploited to create improved sparse simplex solution algorithms. The original coefficient matrix is partitioned into network and non-network parts. For this partitioning, a decomposition technique can be applied. The embedded network flow problem can be solved to optimality using a fast network flow algorithm. We investigate two alternative decompositions namely, Lagrangean and Benders. In the Lagrangean approach, the optimal solution of a network flow problem and in Benders the combined solution of the master and the subproblem are used to compute good (near optimal and near feasible) solutions for a given LP problem. In both cases, we terminate the decomposition algorithms after a preset number of passes and active variables identified by this procedure are then used to create an advanced basis for the original LP problem. We present comparisons with unit basis and a well established crash procedure. We find that the computational results of applying these techniques to a selection of Netlib models are promising enough to encourage further research in this area.     

Finding all Solutions of Systems of Nonlinear Equations Using the Dual Simplex Method

An efficient algorithm is proposed for finding all solutions of nonlinear equations using linear programming (LP). This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations in a given region. In the conventional LP test, the system of nonlinear equations is transformed into an LP problem, to which the simplex method is applied. However, although the LP test is very powerful, it requires many pivotings for each region. In this paper, we use the dual simplex method in the LP test, which makes the average number of pivotings per region much smaller (less than one, for example) and makes the algorithm very efficient. By numerical examples, it is shown that the proposed algorithm can find all solutions of systems of 200 nonlinear equations in practical computation time.     

Towards a practical parallelisation of the simplex method

The simplex method is frequently the most efficient method of solving linear programming (LP) problems. This paper reviews previous attempts to parallelise the simplex method in relation to efficient serial simplex techniques and the nature of practical LP problems. For the major challenge of solving general large sparse LP problems, there has been no parallelisation of the simplex method that offers significantly improved performance over a good serial implementation. However, there has been some success in developing parallel solvers for LPs that are dense or have particular structural properties. As an outcome of the review, this paper identifies scope for future work towards the goal of developing parallel implementations of the simplex method that are of practical value.     

Markov decision processes for infinite horizon problems solved with the cosine simplex method

This article presents a new method of linear programming (LP) for solving Markov decision processes (MDPs) based on the simplex method (SM). SM has shown to be the most efficient method in many practical problems; unfortunately, classical SM has an exponential complexity. Therefore, new SMs have emerged for obtaining optimal solutions in the most efficient way. The cosine simplex method (CSM) is one of them. CSM is based on the Karush Kuhn Tucker conditions, and is able to efficiently solve general LP problems. This work presents a new method named the Markov Cosine Simplex Method (MCSM) for solving MDP problems, which is an extension of CSM. In this article, the efficiency of MCSM is compared to the traditional revised simplex method (RSM); experimental results show that MCSM is far more efficient than RSM.     

Intelligent gradient search in linear programming

The simplex algorithm is still the best known and most frequently used way to solve LP problems. Khachian has suggested a method to solve these problems in polynomial time. The average behaviour of his method, however, is still inferior to the modern simplex based LP codes. A new gradient based approach which also has polynomial worst-case behaviour has been suggested by Karmarkar. This method was modified, programmed and compared with other available LP codes. It is shown that the numerical efficiency of Karmarkar's method compares favourably with other LP codes, particularly for problems with high numbers of variables and few constraints.     

一种简易的单纯形表

通过对单纯形法的分析和研究,提出了一种简易的单纯形表,并利用矩形法则进行计算而得到一种改进的单纯形法.结果表明该法简单易行,并减少了计算量和存储量.     

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.     

Network simplex algorithm for the general equal flow problem

In this paper the general equal flow problem is considered. This is a minimum cost network flow problem with additional side constraints requiring the flow of arcs in some given sets of arcs to take on the same value. This model can be applied to approach water resource system management problems or multiperiod logistic problems in general involving policy restrictions which require some arcs to carry the same amount of flow through the given study period. Although the bases of the general equal flow problem are no longer spanning trees, it is possible to recognize a similar structure that allows us to take advantage of the practical computational capabilities of network models. After characterizing the bases of the problem as good (r+1)-forests, a simplex primal algorithm is developed that exploits the network structure of the problem and requires only slight modifications of the well-known network simplex algorithm.     

A specialized network simplex algorithm for the constrained maximum flow problem

The constrained maximum flow problem is to send the maximum flow from a source to a sink in a directed capacitated network where each arc has a cost and the total cost of the flow cannot exceed a budget. This problem is similar to some variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications in practice. The constrained maximum flow problem itself has important applications, such as in logistics, telecommunications and computer networks. In this research, we present an efficient specialized network simplex algorithm that significantly outperforms the two widely used LP solvers: CPLEX and lp_solve. We report CPU times of an average of 27 times faster than CPLEX (with its dual simplex algorithm), the closest competitor of our algorithm.     

New efficient shortest path simplex algorithm: pseudo permanent labels instead of permanent labels

We introduce a new network simplex pivot rule for the shortest path simplex algorithm. This new pivot rule chooses a subset of non-basic arcs to simultaneously enter into the basis. We call this operation a multiple pivot. We show that a shortest path simplex algorithm with this pivot rule performs O(n) multiple pivots and runs in O(nm) time. Our pivot rule is based on the new concept of a pseudo permanently labeled node, and it can be adapted to design a new label-correcting algorithm that runs in O(nm). Moreover, this concept lets us introduce new rules to identify negative cycles. Finally, we compare the network simplex algorithm with multiple pivots with other previously proposed efficient network simplex algorithm in a computational experiment.     

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

The dual simplex algorithm has become a strong contender in solving large scale LP problems. One key problem of any dual simplex algorithm is to obtain a dual feasible basis as a starting point. We give an overview of methods which have been proposed in the literature and present new stable and efficient ways to combine them within a state-of-the-art optimization system for solving real world linear and mixed integer programs. Furthermore, we address implementation aspects and the connection between dual feasibility and LP-preprocessing. Computational results are given for a large set of large scale LP problems, which show our dual simplex implementation to be superior to the best existing research and open-source codes and competitive to the leading commercial code on many of our most difficult problem instances.     

ON AN EFFICIENT IMPLEMENTATION OF THE FACE ALGORITHM FOR LINEAR PROGRAMMING＊

In this paper, we consider the solution of the standard linear programming [Lt＇）. A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The face algorithm proposed by Pan [19] targets at the optimal face by iterating from face to face, along an orthogonal projection of the negative objective gradient onto a relevant null space. The algorithm exhibits a favorable numerical performance by comparing the simplex method. In this paper, we further investigate the face algorithm by proposing an improved implementation. In exact arithmetic computation, the new algorithm generates the same sequence as Pan＇s face algorithm, but uses less computational costs per iteration, and enjoys favorable properties for sparse problems.     

Parallel distributed-memory simplex for large-scale stochastic LP problems

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates.     

Linear programming with multiple choice constraints for single chain undiscounted Markov decision problems

This paper develops an efficient LP algorithm for solving single chain undiscounted Markov decision problems. The algorithm imposes, in the framework of the simplex method, the multiple choice constraints that exactly one basic variable be chosen from each Markov state. It is proved that the algorithm converges to an optimal solution in a finite number of steps.     

A cycle augmentation algorithm for minimum cost multicommodity flows on a ring

Minimum cost multicommodity flows are a useful model for bandwidth allocation problems. These problems are arising more frequently as regional service providers wish to carry their traffic over some national core network. We describe a simple and practical combinatorial algorithm to find a minimum cost multicommodity flow in a ring network. Apart from 1 and 2-commodity flow problems, this seems to be the only such “combinatorial augmentation algorithm” for a version of exact mincost multicommodity flow. The solution it produces is always half-integral, and by increasing the capacity of each link by one, we may also find an integral routing of no greater cost. The “pivots” in the algorithm are determined by choosing an >0, increasing and decreasing sets of variables, and adjusting these variables up or down accordingly by . In this sense, it generalizes the cycle cancelling algorithms for (single source) mincost flow. Although the algorithm is easily stated, proof of its correctness and polynomially bounded running time are more complex.     

Studies of lexicography in the generalized network simplex method

This paper introduces an analytical approach for studying lexicography in generalized network problems. The equations obtained can help us to understand and to extend the existing theory. First, it is verified that all nonzero elements have the same sign in each row vector of a basis inverse for a generalized network (GN) problem with positive multipliers. However, this property does not necessarily hold when there exist negative multipliers. Second, we developed a strategy to select the dropping arc in the GN simplex algorithm when addressing GN problems with positive andnegative multipliers. This strategy is also based on lexicography and requires performing some comparisons. However, the values to be compared are already known since they can be obtained as a by-product of the calculations necessary to compute the basis representation of the entering arc. Consequently, the computational effort per pivot step isO(n) in the worst case. This worst case effort is the same as that required by the strongly convergent rules for selecting the dropping arc in the method of strong convergence.     

On dual minimum cost flow algorithms

We describe a new dual algorithm for the minimum cost flow problem. It can be regarded as a variation of the best known strongly polynomial minimum cost flow algorithm, due to Orlin. Indeed we obtain the same running time of O(m log m(m+n log n)), where n and m denote the number of vertices and the number of edges. However, in contrast to Orlin's algorithm we work directly with the capacitated network (rather than transforming it to a transshipment problem). Thus our algorithm is applicable to more general problems (like submodular flow) and is likely to be more efficient in practice.  Our algorithm can be interpreted as a cut cancelling algorithm, improving the best known strongly polynomial bound for this important class of algorithms by a factor of m. On the other hand, our algorithm can be considered as a variant of the dual network simplex algorithm. Although dual network simplex algorithms are reportedly quite efficient in practice, the best worst-case running time known so far exceeds the running time of our algorithm by a factor of n.