A parametric algorithm for convex cost network flow and related problems

The convex cost network flow problem is to determine the minimum cost flow in a network when cost of flow over each arc is given by a piecewise linear convex function. In this paper, we develop a parametric algorithm for the convex cost network flow problem. We define the concept of optimum basis structure for the convex cost network flow problem. The optimum basis structure is then used to parametrize v, the flow to be transsshipped from source to sink. The resulting algorithm successively augments the flow on the shortest paths from source to sink which are implicitly enumerated by the algorithm. The algorithm is shown to be polynomially bounded. Computational results are presented to demonstrate the efficiency of the algorithm in solving large size problems. We also show how this algorithm can be used to (i) obtain the project cost curve of a CPM network with convex time-cost tradeoff functions; (ii) determine maximum flow in a network with concave gain functions; (iii) determine optimum capacity expansion of a network having convex arc capacity expansion costs.     

A conic quadratic formulation for a class of convex congestion functions in network flow problems

In this paper we consider a multicommodity network flow problem with flow routing and discrete capacity expansion decisions. The problem involves trading off congestion and capacity assignment (or expansion) costs. In particular, we consider congestion costs involving convex, increasing power functions of flows on the arcs. We first observe that under certain conditions the congestion cost can be formulated as a convex function of the capacity level and the flow. Then, we show that the problem can be efficiently formulated by using conic quadratic inequalities. As most of the research on this problem is devoted to heuristic approaches, this study differs in showing that the problem can be solved to optimum by branch-and-bound solvers implementing the second-order cone programming (SOCP) algorithms. Computational experiments on the test problems from the literature show that the continuous relaxation of the formulation gives a tight lower bound and leads to optimal or near optimal integer solutions within reasonable CPU times.     

Generate upper boundary vectors meeting the demand and budget for a p-commodity network with unreliable nodes

The system capacity for a single-commodity flow network is the maximum flow from the source to the sink. This paper discusses the system capacity problem for a p-commodity limited-flow network with unreliable nodes. In such a network, arcs and nodes all have several possible capacities and may fail. Different types of commodity, which are transmitted through the same network simultaneously, competes the capacities of arcs and nodes. In particular, the consumed capacity by different types of commodity varies from arcs and nodes. We first define the system capacity as a vector and then a performance index, the probability that the upper bound of the system capacity is a given pattern subject to the budget constraint, is proposed. Such a performance index can be easily computed in terms of upper boundary vectors meeting the demand and budget. A simple algorithm based on minimal cuts is thus presented to generate all upper boundary vectors. The manager can apply this performance index to measure the system capacity level for a supply-demand system.     

Optimizing cost flows by edge cost and capacity upgrade

We examine a network upgrade problem for cost flows. A budget can be distributed among the arcs of the network. An investment on each single arc can be used either to decrease the arc flow cost, or to increase the arc capacity, or both. The goal is to maximize the flow through the network while not exceeding bounds on the budget and on the total flow cost.

The problems are NP-hard even on series-parallel graphs. We provide an approximation algorithm on series-parallel graphs which, for arbitrary δ,>0, produces a solution which exceeds the bounds on the budget and the flow cost by factors of at most 1+δ and 1+, respectively, while the amount of flow is at least that of an optimum solution. The running time of the algorithm is polynomial in the input size and 1/(δ). In addition we give an approximation algorithm on general graphs applicable to problem instances with small arc capacities.     

Maximum network flows with concave gains

This paper deals with a generalized maximum flow problem with concave gains, which is a nonlinear network optimization problem. Optimality conditions and an algorithm for this problem are presented. The optimality conditions are extended from the corresponding results for the linear gain case. The algorithm is based on the scaled piecewise linear approximation and on the fat path algorithm described by Goldberg, Plotkin and Tardos for linear gain cases. The proposed algorithm solves a problem with piecewise linear concave gains faster than the naive solution by adding parallel arcs. Supported by a Grant-in-Aid for Scientific Research (No. 13780351 and No.14380188) from The Ministry of Education, Culture, Sports, Science and Technology of Japan.     

On solving continuous-time dynamic network flows

Temporal dynamics is a crucial feature of network flow problems occurring in many practical applications. Important characteristics of real-world networks such as arc capacities, transit times, transit and storage costs, demands and supplies etc. are subject to fluctuations over time. Consequently, also flow on arcs can change over time which leads to so-called dynamic network flows. While time is a continuous entity by nature, discrete-time models are often used for modeling dynamic network flows as the resulting problems are in general much easier to handle computationally. In this paper, we study a general class of dynamic network flow problems in the continuous-time model, where the input functions are assumed to be piecewise linear or piecewise constant. We give two discrete approximations of the problem by dividing the considered time range into intervals where all parameters are constant or linear. We then present two algorithms that compute, or at least converge to optimum solutions. Finally, we give an empirical analysis of the performance of both algorithms.     

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.     

Optimal expansion of an existing network

Given an existing network, a list of arcs which could be added to the network, the arc costs and capacities, and an available budget, the problem considered in this paper is one of choosing which arcs to add to the network in order to maximize the maximum flow from a sources to a sinkt, subject to the budgetary constraint. This problem appears in a large number of practical situations which arise in connection with the expansion of electricity or gas supply, telephone, road or rail networks. The paper describes an efficient tree-search algorithm using bounds calculated by a dynamic programming procedure which are very effective in limiting the solution space explicitly searched. Computational results for a number of medium sized problems are described and computing times are seen to be very reasonable.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

An algorithm for optimizing network flow capacity under economies of scale

The problem of optimally allocating a fixed budget to the various arcs of a single-source, single-sink network for the purpose of maximizing network flow capacity is considered. The initial vector of arc capacities is given, and the cost function, associated with each arc, for incrementing capacity is concave; therefore, the feasible region is nonconvex. The problem is approached by Benders' decomposition procedure, and a finite algorithm is developed for solving the nonconvex relaxed master problems. A numerical example of optimizing network flow capacity, under economies of scale, is included.This research was supported by the National Science Foundation, Grant No. GK-32791.     

New formulation and relaxation to solve a concave-cost network flow problem

In this paper we study a minimum cost, multicommodity network flow problem in which the total cost is piecewise linear, concave of the total flow along the arcs. Specifically, the problem can be defined as follows. Given a directed network, a set of pairs of communicating nodes and a set of available capacity ranges and their corresponding variable and fixed cost components for each arc, the problem is to select for each arc a range and identify a path for each commodity between its source and destination nodes so as to minimize the total costs. We also extend the problem to the case of piecewise nonlinear, concave cost function. New mathematical programming formulations of the problems are presented. Efficient solution procedures based on Lagrangean relaxations of the problems are developed. Extensive computational results across a variety of networks are reported. These results indicate that the solution procedures are effective for a wide range of traffic loads and different cost structures. They also show that this work represents an improvement over previous work made by other authors. This improvement is the result of the introduction of the new formulations of the problems and their relaxations.     

Spatial price equilibrium,distribution center location and successive over-relaxation

A spatial price equilibrium problem is modeled which allows piecewise linear convex flow costs, and a capacity limit on the trade flow between each supply/demand pair of regions. Alternatively, the model determines the locations of intermediate distribution centers in a market economy composed of separate regions, each with approximately linear supply and demand functions. Equilibrium prices, regional supply and demand quantities, and commodity flows are determined endogenously. The model has a quadratic programming formulation which is then reduced by exploiting the structure. The reduced model is particularly well suited to solution using successive over-relaxation.     

A CAPACITY EXPANSION PROBLEM WITH BUDGET CONSTRAINT AND BOTTLENECK LIMITATION

1 IntroductionIn [2] tl1e authors considered a type of coustrained maximim capacity path problem whichcan be described briefly as: kuowing the costs for expallding one unit of capacity along differentedge8 of a l1etwork aud the availabIe budget, how to iucrease the caparities of the edges so thattlle capasity between any pair of nodes in the lletwork can be raised unifornily to the maximumextent? As the total cost is a summation of the expansion costs on all edges, this problem i8related to mi…     

Concave cost minimization on networks

The paper deals with the problem of finding a minimum cost multicommodity flow on an uncapacitated network with concave link costs. Problems of this type are the optimal design of a network in the presence of scale economies and the telpack problem.Two different definitions of local optimality are given and compared both from the point of view of the computational complexity and from the point of view of the goodness of the solution they may provide.A vertex following algorithm to find a local optimum is proposed. The computational complexity of each iteration of the algorithm is O(n³), where n is the number of nodes of the network, and is independent of the differentiability of the objective function.Experimental results obtained from a set of test problems of size ranging from 11 nodes and 23 arcs to 48 nodes and 174 arcs, with number of commodities up to 5, are given.     

Network flow interdiction on planar graphs

The network flow interdiction problem asks to reduce the value of a maximum flow in a given network as much as possible by removing arcs and vertices of the network constrained to a fixed budget. Although the network flow interdiction problem is strongly NP-complete on general networks, pseudo-polynomial algorithms were found for planar networks with a single source and a single sink and without the possibility to remove vertices. In this work, we introduce pseudo-polynomial algorithms that overcome various restrictions of previous methods. In particular, we propose a planarity-preserving transformation that enables incorporation of vertex removals and vertex capacities in pseudo-polynomial interdiction algorithms for planar graphs. Additionally, a new approach is introduced that allows us to determine in pseudo-polynomial time the minimum interdiction budget needed to remove arcs and vertices of a given network such that the demands of the sink node cannot be completely satisfied anymore. The algorithm works on planar networks with multiple sources and sinks satisfying that the sum of the supplies at the sources equals the sum of the demands at the sinks. A simple extension of the proposed method allows us to broaden its applicability to solve network flow interdiction problems on planar networks with a single source and sink having no restrictions on the demand and supply. The proposed method can therefore solve a wider class of flow interdiction problems in pseudo-polynomial time than previous pseudo-polynomial algorithms and is the first pseudo-polynomial algorithm that can solve non-trivial planar flow interdiction problems with multiple sources and sinks. Furthermore, we show that the k-densest subgraph problem on planar graphs can be reduced to a network flow interdiction problem on a planar graph with multiple sources and sinks and polynomially bounded input numbers.     

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. Research was partially supported by NSF and Air Force grants.     

Dynamic graph generation for the shortest path problem in time expanded networks

In discrete optimization problems the progress of objects over time is frequently modeled by shortest path problems in time expanded networks, but longer time spans or finer time discretizations quickly lead to problem sizes that are intractable in practice. In convex relaxations the arising shortest paths often lie in a narrow corridor inside these networks. Motivated by this observation, we develop a general dynamic graph generation framework in order to control the networks’ sizes even for infinite time horizons. It can be applied whenever objects need to be routed through a traffic or production network with coupling capacity constraints and with a preference for early paths. Without sacrificing any information compared to the full model, it includes a few additional time steps on top of the latest arcs currently in use. This “frontier” of the graphs can be extended automatically as required by solution processes such as column generation or Lagrangian relaxation. The corresponding algorithm is efficiently implementable and linear in the arcs of the non-time-expanded network with a factor depending on the basic time offsets of these arcs. We give some bounds on the required additional size in important special cases and illustrate the benefits of this technique on real world instances of a large scale train timetabling problem.     

Approximation of convex functions by projections of polyhedra

A method for approximate solution of minimization problems for multivariable convex functions with convex constraints is proposed. The main idea consists in approximation of the objective function and constraints by piecewise linear functions and subsequent reduction of the initial convex programming problem to a problem of linear programming. We present algorithms constructing approximating polygons for some classes of single variable convex functions. The many-dimensional problem is reduced to a one-dimensional one by an inductive procedure. The efficiency of the method is illustrated by numerical examples.     

MC-based Algorithm for a telecommunication network under node and budget constraints

It is an important issue to design some performance indexes in order to measure the performance for a telecommunication network. Network analysis is an available approach to solve the performance problem for a real-life system. We construct a two-commodity stochastic-flow network with unreliable nodes (arcs and nodes all have several possible capacities and may fail) to model the telecommunication network. In which, all types of commodity are transmitted through the same network simultaneously and compete the capacities. This paper defines the system capacity as a 2-tuple vector, and then proposes a performance index, the probability that the upper bound of the system capacity equals a demand vector subject to the budget constraint. An upper boundary point is a vector representing the capacities of arcs and nodes, and is the maximal vector exactly meeting the demand vector. A simple algorithm based on minimal cuts (or named MC-based algorithm) is then presented to generate all upper boundary points in order to evaluate the performance index. The storage and computational time complexity of this algorithm are also analyzed. The performance evaluation for the multicommodity case can be extended easily.     

The out-of-kilter algorithm applied to traffic congestion

The out-of-kilter algorithm finds a minimum cost assignment of flows to a network defined in terms of one-way arcs, each with upper and lower bounds on flow, and a cost. It is a mathematical programming algorithm which exploits the network structure of the data. The objective function, being the sum taken over all the arcs of the products, cost×flow, is linear. The algorithm is applied in a new way to minimise a series of linear functions in a heuristic method to reduce the value of a non-convex quadratic function which is a measure of traffic congestion. The coefficients in these linear functions are chosen in a way which avoids one of the pitfalls occurring when Beale's method is applied to such a quadratic function. The method does not guarantee optimality but has produced optimal results with networks small enough for an integer linear programming method to be feasible.