Nonlinear Network Flow Problems Research Articles

Staircase compatibility and its applications in scheduling and piecewise linearization

We introduce the Clique Problem with Multiple-Choice constraints (CPMC) and characterize a case where it is possible to give an efficient description of the convex hull of its feasible solutions. This special case, which we name staircase compatibility, generalizes common properties in several applications and allows for a linear description of the integer feasible solutions to (CPMC) with a totally unimodular constraint matrix of polynomial size. We derive two such totally unimodular reformulations for the problem: one that is obtained by a strengthening of the compatibility constraints and one that is based on a representation as a dual network flow problem. Furthermore, we show a natural way to derive integral solutions from fractional solutions to the problem by determining integral extreme points generating this fractional solution. We also evaluate our reformulations from a computational point of view by applying them to two different real-world problem settings. The first one is a problem in railway timetabling, where we try to adapt a given timetable slightly such that energy costs from operating the trains are reduced. The second one is the piecewise linearization of non-linear network flow problems, illustrated at the example of gas networks. In both cases, we are able to reduce the solution times significantly by passing to the theoretically stronger formulations of the problem.

Solving short-term electric hydrothermal scheduling problems by exponential multiplier methods†

The short-term electric hydrothermal scheduling (STEHS) problem consists in optimizing the production of hydro and thermal electric generation units over a short time period (up to one week long). The problem described in this work can be modelled as a nonlinear network flow problem with linear and nonlinear side constraints. The minimization of this kind of problem can be performed by exploiting the efficiency of network flow techniques. It lies in minimizing approximately a series of augmented Lagrangian functions including only the side constraints, subject to balance constraints in the nodes and capacity bounds. One of the drawbacks of the multiplier methods with quadratic penalty function is that the augmented Lagrangian is not twice differentiable when it is applied to problems with inequality constraints. This article overcomes this difficulty by using the exponential multiplier method. In order to improve the performance some parameters are tuned. The efficiency of this method over STEHS test problems is illustrated by comparing its CPU-times with those of the quadratic multiplier method and with those of the general purpose codes MINOS, SNOPT, and KNITRO. Numerical results are promising. †The research was partially supported by grant MCYT DPI 2005-09117-C02-01.

Approximate Subgradient Methods for Nonlinearly Constrained Network Flow Problems

The minimization of nonlinearly constrained network flow problems can be performed by using approximate subgradient methods. The idea is to solve this kind of problem by means of primal-dual methods, given that the minimization of nonlinear network flow problems can be done efficiently exploiting the network structure. In this work, it is proposed to solve the dual problem by using e-subgradient methods, as the dual function is estimated by minimizing approximately a Lagrangian function, which includes the side constraints (nonnetwork constraints) and is subject only to the network constraints. Some well-known subgradient methods are modified in order to be used as e-subgradient methods and the convergence properties of these new methods are analyzed. Numerical results appear very promising and effective for this kind of problems

An efficient method for nonlinearly constrained networks

Many nonlinear network flow problems (in addition to the balance constraints in the nodes and capacity constraints on the arc flows) have nonlinear side constraints, which specify a flow relationship between several of the arcs in the network flow model. The short-term hydrothermal coordination of electric power generation is an example of this type. In this work we solve this kind of problem using an approach in which the efficiency of the well-known techniques for network flow can be preserved. It lies in relaxing the side constraints in an augmented Lagrangian function, and minimizing a sequence of these functions subject only to the network constraints for different estimates of the Lagrange multipliers of the side constraints. This method gives rise to an algorithm, which combines first- and superlinear-order multiplier methods to estimate these multipliers. When the number of free variables is very high we can obtain a superlinear-order estimate by means of the limited memory BFGS method fitted to our problem. An extensive computational comparison with other methods has been performed. The numerical results reported indicate that the algorithm described may be employed advantageously to solve large-scale network flow problems with nonlinear side constraints.

An implementation of Newton-like methods on nonlinearly constrained networks

The minimization of a nonlinear function with linear and nonlinear constraints and simple bounds can be performed by minimizing an augmented Lagrangian function, including only the nonlinear constraints. This procedure is particularly interesting when the linear constraints are flow conservation equations, as there exist efficient techniques for solving nonlinear network problems. It is then necessary to estimate their multipliers, and variable reduction techniques can be used to carry out the successive minimizations. This work analyzes the possibility of estimating the multipliers of the nonlinear constraints using Newton-like methods. Also, an algorithm is designed to solve nonlinear network problems with nonlinear inequality side constraints, which combines first and superlinear-order multiplier methods. The computational performance of this method is compared with that of MINOS 5.5. Scope and purpose Real problems with the structure of a nonlinear network flow problem with nonlinear side constraints exist and have a high dimensionality (e.g. the short-term hydrothermal coordination of electric power generation). They often need to be solved and it is important to find the procedure that will solve them with the highest efficiency. In previous works this author has analyzed the viability of estimating the multipliers of the nonlinear constraints from the Kuhn–Tucker system in order to improve the inexpensive Hestenes–Powell update. However, the conclusion was that there is no significant advantage for multipliers estimated through the Kuhn–Tucker system. The purpose of this paper is mainly to use the convergence properties of the Newton-like methods in order to develop a new estimator of the nonlinear constraint multipliers that improves the efficiency and the robustness of this augmented Lagrangian method.

An Evolution Program for Non-Linear Transportation Problems

In this paper we describe main features of a Strongly Feasible Evolution Program (SFEP) designed to solve non-linear network flow problems. The program can handle non-linearities both in the constraints and in the objective function. The solutions procedure is based on a recombination operator in which all parents in a small mating pool have equal chance of contributing their genetic material to offspring. When offspring is created with better fitness value than that of the worst parent, the worst parent is discarded from the mating pool while the offspring is placed in it. The main contributions are in the massive parallel initialization procedure which creates only feasible solutions with simple heuristic rules that increase chances of creating solutions with good fitness values for the initial mating pool, and the gene therapy procedure which fixes “defective genes” ensuring that the offspring resulting from recombination is always feasible. Both procedures utilize the properties of network flows. The algorithm is capable of handling mixed integer problems with non-linearities in both constraints and the objective function. Tests were conducted on a number of previously published transportation problems with 49 and 100 decision variables, which constitute a subset of network flow problems. Convergence to equal or better solutions was achieved with often less than one tenth of the previous computational efforts.

Asynchronous Iterative Algorithms with Flexible Communication for Nonlinear Network Flow Problems

The strictly convex network flow problem is considered. The dual of this problem is unconstrained, differentiable, and well suited for solution via distributed or parallel iterative methods. A new class of asynchronous iterative methods is proposed: the asynchronous iterations with flexible communication. Communication to other processors of the value of the components of the iteration vector resulting from intermediary steps of computation is the main feature of this new class of methods. Convergence is speeded up when such partial updates are used. A convergence result is given. Preliminary computational results are presented and analyzed.

Multi-area economic dispatch with tie line constraints

This paper presents a new method for solving the multi-area power system economic dispatch (MAED) problem with tie line constraints. This formulation extends the traditional economic dispatch methods used in study applications such as unit commitment to include area demand constraints, area reserve constraints, and tie line capacity constraints between the modeled areas. The MAED is formulated as a capacitated nonlinear network flow problem which is solved using a high-speed network load flow code. The MAED determines the amount of power that can be economically generated in one area and transferred to another area to displace generation in the second area. The network flow model does not compute the actual power transfer between areas. The paper demonstrates that a significant number of the problem constraints may be formulated as network-type, and as a result, the method provides a robust, high-speed solution method for surprisingly large class of electric utility problems including the ability to analyze potential purchases and sales in a multi-area power system environment.

DUAL-BASED NEWTON METHODS FOR NONLINEAR MINIMUM COST NETWORK FLOW PROBLEMS

Nonlinear network optimization is of great importance not only in theory but also in practical applications. The range of its applications covers a variety of problems which arise in transportation systems, water distribution systems, resistive electrical networks, and so on. There are various methods to solve nonlinear network flow problems, and many of them belong to the class of descent methods which successively generate search directions and perform line searches. In this paper we propose an algorithm, based on the Newton method, which exploits the network structure of the problems. The algorithm directly solves the dual problem which, under appropriate conditions, can be formulated as an unconstrained convex minimization problem with a continuously differentiable objective function. We give a global convergence theorem of the algorithm and present practical strategies for computing search directions and finding steplengths. Some computational results for test problems of up to 4900 nodes and 14490 arcs show the practical efficiency of the proposed algorithm.

Development-Scheme Selection for River Valleys

This paper presents a new approach to the problem of selecting a development scheme for a river valley from preliminary surveys on candidate sites. Rather than proceeding by elimination, this approach identifies the scheme which minimizes the investment and operating costs. This minimization problem is decomposed using the Benders technique into a master problem covering the site selection and sizing aspects and a subproblem covering the production aspect. The former is solved by mixed-integer linear programming whereas the latter is transformed by a Lagrangian relaxation into a nonlinear network flow problem which is solved by a variant of the Frank-Wolfe method. Numerical results are reported.

Development-Scheme Selection for River Valleys

This paper presents a new approach to the problem of selecting a development scheme for a river valley from preliminary surveys on candidate sites. Rather than proceeding by elimination, this approach identifies the scheme which minimizes the investment and operating costs. This minimization problem is decomposed using the Benders technique into a master problem covering the site selection and sizing aspects and a subproblem covering the production aspect. The former is solved by mixed-integer linear programming whereas the latter is transformed by a Lagrangian relaxation into a nonlinear network flow problem which is solved by a variant of the Frank-Wolfe method. Numerical results are reported.

Optimal Short Term Operation Planning of a Large Hydrothermal Power System Based on a Nonlinear Network Flow Concept

This paper presents a procedure for solving the short term generation scheduling problem for a large hydrothermal system that includes transmission limitations. The integrated system is divided into a hydro and a thermal subsystem. A reduced gradient algorithm is employed for the solution of the hydro subproblem. This algorithm is specialized to efficiently solve nonlinear network flow problems with additional constraints of non-netwrk type. The thermal subsystem is solved using a fast unit commitment and dispatch algorithm. A case study with the Swedish system is discussed.

A hybrid algorithm for solving convex separable network flow problems

Based on computational experiments with different approaches to convex separable network flow problems a hybrid algorithm is developed and implemented. Phase one of the algorithm uses a rapidly converging series of piecewise linear secant approximations in order to determine a good solution within some distance of the optimum. Starting from this solution, a feasible direction method, based on reduced Newton directions, is used in the second phase of the algorithm to determine the optimal solution. Since nonlinear network flow problems tend to be degenerate, special emphasis is put on the construction of a basis that yields a strictly positive step length at the beginning of phase two of the hybrid algorithm.