Piecewise Linear Network Flow Problems Research Articles

A polynomial local optimality condition for the concave piecewise linear network flow problem

<abstract> <p>This paper studies the local optimality condition for the widely applied concave piecewise linear network flow problem (CPLNFP). Traditionally, for CPLNFP the complexity of checking the local optimality condition is exponentially related to the number of active arcs (i.e., arcs in which the flow is at the breakpoints). When the scale of CPLNFP is large, even local optimality is unverifiable and the corresponding local algorithms are inefficient. To overcome this shortcoming, a new local optimality condition is given. This new condition is based on the concavity and piecewise linearity of the cost function and makes full use of the network structure. It is proven that the complexity of the new condition is polynomial. Therefore, using the new condition to verify the local optimality is far superior to using the traditional condition, especially when there are many active arcs.</p> </abstract>

Adaptive dynamic cost updating procedure for solving fixed charge network flow problems

We approximate the objective function of the fixed charge network flow problem (FCNF) by a piecewise linear one, and construct a concave piecewise linear network flow problem (CPLNF). A proper choice of parameters in the CPLNF problem guarantees the equivalence between those two problems. We propose a heuristic algorithm for solving the FCNF problem, which requires solving a sequence of CPLNF problems. The algorithm employs the dynamic cost updating procedure (DCUP) to find a solution to the CPLNF problems. Preliminary numerical experiments show the effectiveness of the proposed algorithm. In particular, it provides a better solution than the dynamic slope scaling procedure in less CPU time.

A bilinear relaxation based algorithm for concave piecewise linear network flow problems

We present a continuous relaxation technique for the Concave Piecewise Linear Network Flow Problem (CPLNFP), which has a bilinear objective function and network constraints. We show that a global optimum of the resulting problem is a solution of CPLNFP. The theoretical results are generalized for a concave minimization problem with a separable objective function. An efficient and effective Dynamic Cost Updating Procedure (DCUP) is considered to find a local minimum of the relaxation problem, which converges in a finite number of iterations. We show that the CPLNFP is equivalent to a Network Flow Problem with Flow Dependent Cost Functions (NFPwFDCF), and we prove that the solution of the Dynamic Slope Scaling Procedure (DSSP) is an equilibrium solution of the NFPwFDCF. The numerical experiments show that the proposed algorithm can provide a better solution than DSSP using less amount of CPU time and iterations.

Dynamic slope scaling and trust interval techniques for solving concave piecewise linear network flow problems

In this paper, we propose an efficient heuristic approach for solving concave Piecewise Linear Network Flow Problems (PLNFP) in which the cost is separable and each arc cost is concave piecewise linear function of the total flow along the arcs. The problem is well known to be WNP-hard and exact global optimization algorithms are of limited use. In this paper, PLNFP is transformed into an equivalent Fixed-Charge Network Flow Problem (FCNFP) using an arc separation procedure, and we develop an algorithm combining the Dynamic Slope Scaling Procedure (DSSP) and Trust Interval techniques. Due to the fast growing size of the transformed FCNFP, exact branch-and-bound methods are not able to handle the problems of practical size. Based on the dynamic slope scaling algorithm developed for solving FCNFP in previous work, we add a trust interval technique based on solution tendencies to speed up the algorithm. The numerical experiments show that the proposed algorithm is quite efficient and its performance is very stable. © 2000 John Wiley & Sons, Inc.

Dynamic slope scaling and trust interval techniques for solving concave piecewise linear network flow problems

Dynamic slope scaling and trust interval techniques for solving concave piecewise linear network flow problems

A Dynamic Domain Contraction Algorithm for Nonconvex Piecewise Linear Network Flow Problems

We consider the Nonconvex Piecewise Linear Network Flow Problem (NPLNFP) which is known to be {\mathcal NP}-hard. Although exact methods such as branch and bound have been developed to solve the NPLNFP, their computational requirements increase exponentially with the size of the problem. Hence, an efficient heuristic approach is in need to solve large scale problems appearing in many practical applications including transportation, production-inventory management, supply chain, facility expansion and location decision, and logistics. In this paper, we present a new approach for solving the general NPLNFP in a continuous formulation by adapting a dynamic domain contraction. A Dynamic Domain Contraction (DDC) algorithm is presented and preliminary computational results on a wide range of test problems are reported. The results show that the proposed algorithm generates solutions within 0 to 0.94 % of optimality in all instances that the exact solutions are available from a branch and bound method.