Nonlinear Multicommodity Network Flow Problem Research Articles

Algorithms for an Integer Multicommodity Network Flow Problem with Node Reliability Considerations

In this paper, we consider the problem of sending a set of multiple commodities from their origin to destination nodes via intermediate hubs. Each hub node is associated with a reliability function, which depends on the total flow that crosses that hub. The probability that each commodity is successfully relayed from its origin to its destination is given by the product of hub reliabilities on the commodity's path. The problem we consider seeks to find minimum-cost commodity paths such that each commodity reaches its destination with a sufficiently large probability. We first formulate the problem as a nonlinear multicommodity network-flow problem and prove that it is strongly NP-hard. We then present two linearization techniques for this formulation, and propose a pair of lower- and upper-bounding formulations, which can then be used within an exact cutting-plane algorithm to solve the problem. Finally, we analyze the computational effectiveness of our proposed strategies on a set of randomly generated instances.

A separable nonlinear model for the multicommodity flow problem

This article studies the convex nonlinear multicommodity network flow problem with separable increasing cost. A numerical implementation of the primal-dual interior-point method is designed to solve the problem. At each iteration of the interior-point method, the corresponding linear problem is solved, if it is expressed by the normal equations, by using the approximate inverse algorithm (AINV) alone or combined with the preconditioned conjugate gradient algorithm. If the linear system is expressed as an augmented indefinite system, it is solved by using the AINV algorithm combined with an indefinite preconditioned conjugate gradient algorithm. Numerical experiments are conducted for networks of different dimensions and numbers of commodities and for some nonlinear costs. The computational results show the effectiveness of the interior-point method for this class of network problems.

A Parallel Algorithm for NMNF Problems with a Large Number of Capacity Constraints

In this paper, we propose a converting technique based method to solve nonlinear multi-commodity network flow (NMNF) problems with a large number of capacity constraints and discuss the associated implementation. We have combined this method with a successive quadratic programming (SQP) method and a parallel dual-type (PDt) method possessing decomposition effects. We have tested our method in solving a kind of lattice-type network system examples of NMNF problems. The simulation results show that the proposed algorithm is efficient for solving NMNF problems and successfully handles a large number of coupling capacity constraints. Furthermore, the computational efficiency of the proposed algorithm is more significant while the numbers of capacity constraints are increased.

Lagrangian Relaxation via Ballstep Subgradient Methods

We exhibit useful properties of ballstep subgradient methods for convex optimization using level controls for estimating the optimal value. Augmented with simple averaging schemes, they asymptotically find objective and constraint subgradients involved in optimality conditions. When applied to Lagrangian relaxation of convex programs, they find both primal and dual solutions, and have practicable stopping criteria. Up until now, similar results have only been known for proximal bundle methods, and for subgradient methods with divergent series stepsizes, whose convergence can be slow. Encouraging numerical results are presented for large-scale nonlinear multicommodity network flow problems.

An efficient graph technique based dual-type algorithm for NMNF problems with large capacity constraints

In this paper, we propose a Graph Method Decomposition Technique based dual-type (GbDt) algorithm to solve the Nonlinear Multi-commodity Network Flow (NMNF) problems with large number of capacity constraints. The GbDt algorithm is imbedded with two techniques, the first is the Graph Method Decomposition (GmD) technique which can decouple the large dimension projection problem induced by the large number of capacity constraints into several independent small dimension projection problems, the second is the Modified Successive Projection and Truncated (MSPT) technique can solve the disjoint projection problems efficiently. We have combined these two techniques with a Successive Quadratic Programming (SQP) method and a Dual Projected Pseudo-Quasi-Newton (DPPQN) method possessing decomposition effects. With the decomposition, our algorithm is computationally efficient. We have tested our method in solving numerous examples of NMNF problems. The simulation results show that the proposed algorithm is efficient in solving the NMNF problems and successfully handle the large number of coupling capacity constraints. Furthermore, the computational efficiency of the proposed algorithm is more significant while the numbers of commodity are increased.

Inverse Nonlinear Multicommodity Flow Optimization by Column Generation

We present a solution algorithm for an inverse nonlinear multicommodity network flow problem. This problem is to find link cost adjustments that make a given target link flow solution optimal in a nonlinear multicommodity network flow problem, and that are optimal with respect to a specified objective. The solution procedure uses column generation. We present computational results for instances where the nonlinear multicommodity network flow problems are small and medium scale traffic equilibrium problems, and where system optimal link flows are targeted. The computational results show that the solution procedure is a viable approach for solving medium-scale instances of the inverse traffic equilibrium problem.

Simplicial with truncated Dantzig–Wolfe decomposition for nonlinear multicommodity network flow problems with side constraints

The simplicial decomposition (SD) subproblem for a nonlinear multicommodity network flow problem is simply its linear approximation. Instead of solving the subproblem optimally, this paper demonstrates that performing one iteration of Dantzig–Wolfe decomposition is generally sufficient for SD to efficiently converge to an optimal solution.

Modified Primal Path-Following Scheme for the Monotone Variational Inequality Problem

Recently, several interior-point methods have been developed under a scaled Lipschitz condition which may be strong in general. In this paper, we develop a modified path-following scheme which does not require the above condition. Its global convergence is proved under only the assumptions of monotonicity and differentiability of the mapping. The scheme is adapted to the network equilibrium problem (a nonlinear multicommodity network flow problem) with a simplicial decomposition technique.

Using an interior point method for the master problem in a decomposition approach

We address some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concem the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The particular structure is the presence of GUB constraints and the natural partitioning of the constraint matrix into blocks built of cuts generated by different subproblems. The method can be used in a fairly general case, i.e., in any decomposition approach whenever the master is solved by an interior point method in which the normal equations are used to compute orthogonal projections. Computational results demonstrate its advantages for one particular decomposition approach: Analytic Center Cutting Plane Method (ACCPM) is applied to solve large scale nonlinear multicommodity network flow problems (up to 5000 arcs and 10000 commodities)

A computationally efficient method for nonlinear multicommodity network flow problems

In this paper, we present a new method for solving nonlinear multicommodity network flow problems with convex objective functions. This method combines a well-known projected Jacobi method and a new dual projected pseudo-quasi-Newton (DPPQN) method which solves multicommodity flow quadratic subproblems induced in the projected Jacobi method. The DPPQN method is a dual Newton-type method that differs very much from the conventional Lagrangian Newton method; our method fully exploits the structural advantages of network-type linear equality constraints to obtain a constant sparse approximate Hessian matrix with a decoupling structure and includes a novel finite-iteration successive projection and (truncated) seal algorithm to resolve the difficulty caused by coupling capacity constraints. The DPPQN method also consists of two decomposition effects, the commodity decomposition effect and the arc decomposition effect, which resolve the potential numerical difficulties caused by large dimensions. We show the convergence of our method including the convergence of the finite-iteration successive projection and (truncated) seal algorithm. Compared with the Frank–Wolfe with PARTAN algorithm in which a price-directive decomposition method is used to solve linearized multicommodity flow problems, our method is dramatically faster in terms of the CPU time on a Sparc-10 workstation at solving numerous nonlinear multicommodity network flow examples. © 1997 John Wiley & Sons, Inc.

Crew pairing at Air France

In the airline industry, crew schedules consist of a number of pairings. These are round trips originating and terminating at the same crew home base composed of legal work days, called duties, separated by rest periods. The purpose of the airline crew pairing problem is to generate a set of minimal cost crew pairings covering all flight legs. The set of pairings must satisfy all the rules in the work convention and all the appropriate air traffic regulations. The resulting constraints can affect duty construction, may restrict each pairing, or be imposed on the overall crew schedule. The pairing problem is formulated as an integer, nonlinear multi-commodity network flow problem with additional resource variables. Nonlinearities occur in the objective function as well as in a large subset of constraints. A branch-and-bound algorithm based on an extension of the Dantzig-Wolfe decomposition principle is used to solve this model. The master problem becomes a Set Partitioning type model, as in the classical formulation, while pairings are generated using resource constrained shortest path subproblems. This primal approach implicitly considers all feasible pairings and also provides the optimality gap value on a feasible solution. A nice feature of this decomposition process is that it isolates all nonlinear aspects of the proposed multi-commodity model in the subproblems which are solved by means of a specialized dynamic programming algorithm. We present the application and implementation of this approach at Air France. It is one of the first implementations of an optimal approach for a large airline carrier. We have chosen a subproblem network representation where the duties rather than the legs are on the arcs. This ensures feasibility relative to duty restrictions by definition. As opposed to Lavoie, Minoux and Odier (1988), the nonlinear cost function is modeled without approximations. The computational experiments were conducted using actual Air France medium haul data. Even if the branch-and-bound trees were not fully explored in all cases, the gaps certify that the computed solutions are within a fraction of one percentage point of the optimality. Our results illustrate that our approach produced substantial improvements over solutions derived by the expert system in use at Air France. Their magnitude led to the eventual implementation of the approach.

Two-Metric Projection Methods for Constrained Optimization

This paper is concerned with the problem min $\{ f(x)\mid x \in X\} $ where X is a convex subset of a linear space H, and f is a smooth real-valued function on H. We propose the class of methods $x_{k + 1} = P(x_k - \alpha _k g_k )$, where P denotes projection on X with respect to a Hilbert space norm $\| \cdot \|$ , $g_k $ denotes the Frechet derivative of f at $x_k $ with respect to another Hilbert space norm $\| \cdot \|_k $ on H, and $\alpha _k $ is a positive scalar stepsize. We thus remove an important restriction in the original proposal of Goldstein [1] and Levitin and Poljak [2], where the norms $\| \cdot \|$ and $\| \cdot \|_k $ must be the same. It is therefore possible to match the norm $\| \cdot \|$ with the structure of X so that the projection operation is simplified while at the same time reserving the option to choose $\| \cdot \|_k $on the basis of approximations to the Hessian of f so as to attain a typically superlinear rate of convergence. The resulting methods are particularly attractive for large-scale problems with specially structured constraint sets such as optimal control and nonlinear multi-commodity network flow problems. The latter class of problems is discussed in some detail.