A hybrid algorithm for large-scale non-separable nonlinear multicommodity flow problems

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

In structural optimization design, obtaining the optimal solution of the objective function is the key to optimal design, and one‐dimensional search is one of the important methods for function optimization. The Golden Section method is the main method of one‐dimensional search, which has better convergence and stability. Based on the solution of the Golden Section method, this paper proposes an efficient one‐dimensional search algorithm, which has the advantages of fast convergence and good stability. An objective function calculation formula is introduced to compare and analyse this method with the Golden Section method, Newton method, and Fibonacci method. It is concluded that when the accuracy is set to 0.1, the new algorithm needs 3 iterations to obtain the target value. The Golden Section method takes 11 iterations, and the Fibonacci method requires 11 iterations. The Newton method cannot obtain the target value. When the accuracy is set to 0.01, the number of iterations of the new method is still the least. The optimized design of the T‐section beam is introduced for engineering application research. When the accuracy is set to 0.1, the new method needs 3 iterations to obtain the target value and the Golden Section method requires 13 iterations. When the accuracy is set to 0.01, the new method requires 4 iterations and the Golden Section method requires 18 iterations. The new method has significant advantages in the one‐dimensional search optimization problem.

Restricted Simplicial Decomposition with Side Constraints is a price decomposition method designed for large‐scale nonlinear problems with a set of linear constraints with special structure and an additional set of linear side constraints. The resultant algorithm iterates by solving a linear subproblem subject to the set of structured constraints and a small nonlinear master problem whose feasible region is defined by the intersection of a simplex and the set of side constraints. The number of variables of the master problem is controlled by a parameter named r. The conditions required for the parameter r and the rules for dropping extreme points of the simplex are established for global convergence of the algorithm. Computational results are presented for nonlinear network problems with side constraints.

The problem of optimally (re)allocating Navy personnel to combat units is compounded by several considerations: availability of trained personnel, staffing of positions by occupation groups or ranks, and maintaining an acceptable level of readiness. In this paper we model this problem as a nonlinear nondifferentiable optimization problem. A reformulation of the nonlinear optimization problem as a network flow problem is then developed. The formulation results in a network flow problem with side constraints. An additional, nonnetwork, variable measures the readiness level. This new formulation permits the use of network optimization tools in order to solve effectively very large problems. We then develop two numerical methods for solving this problem. One method is based on a heuristic that solves (approximately) the nondifferentiable problem. The second method is based on a Linear-Quadratic Penalty (LQP) algorithm, and it exploits the embedded network structure by placing the side constraints into the objective function. The resulting nonlinear network program is solved using a simplicial decomposition of the network constraint set. Numerical results indicate the viability of this approach on problems with up to 36,000 arcs and 17,000 nodes with 3,700 side constraints.

The scheduling of gasoline-blending operations is an important problem in the oil refining industry. This problem not only exhibits the combinatorial nature that is intrinsic to scheduling problems, but also non-convex nonlinear behavior, due to the blending of various materials with different quality properties. In this work, a global optimization algorithm is proposed to solve a previously published continuous-time mixed-integer nonlinear scheduling model for gasoline blending. The model includes blend recipe optimization, the distribution problem, and several important operational features and constraints. The algorithm employs piecewise McCormick relaxation (PMCR) and normalized multiparametric disaggregation technique (NMDT) to compute estimates of the global optimum. These techniques partition the domain of one of the variables in a bilinear term and generate convex relaxations for each partition. By increasing the number of partitions and reducing the domain of the variables, the algorithm is able to refine the estimates of the global solution. The algorithm is compared to two commercial global solvers and two heuristic methods by solving four examples from the literature. Results show that the proposed global optimization algorithm performs on par with commercial solvers but is not as fast as heuristic approaches.

Bilevel learning for large-scale flexible flow shop scheduling

Reduced costs obtained from the optimal simplex tableau are not necessarily correct under degeneracy in network flow problems with side constraints. In one application, simplex reduced costs were shown not to be good when applied to actual operations, while reoptimization to find all true reduced costs was too time-consuming. An efficient algorithm to obtain approximate reduced costs was developed in the research to offset these disadvantages. The algorithm combines the uses of Lagrangian relaxation, a minimum-cost network flow algorithm, and a shortest path algorithm. In theory, the approximate reduced costs prove to be a better lower bound than are simplex reduced costs. Numerical experiments have been performed to show its potential usefulness in practice. © 1996 John Wiley & Sons, Inc.

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

Highly nonlinear advection–dispersion-reaction equations govern numerous transport phenomena. Robust, accurate, and efficient algorithms to solve these equations hold the key to the success of applying numerical models to field problems. This paper presents the development and verification of a computational algorithm to approximate the highly nonlinear transport equations of reactive chemical transport and multiphase flow. The algorithm was developed based on the Lagrangian-Eulerian decoupling method with an adaptive ZOOMing and Peak/valley Capture (LEZOOMPC) scheme. It consists of both backward and forward node tracking, rough element determination, peak/valley capturing, and adaptive local grid refinement. A second-order tracking was implemented to accurately and efficiently track all fictitious particles. Shanks’ method was introduced to deal with slowly converging case. The accuracy and efficiency of the algorithm were verified with the Burger equation for a variety of cases. The robustness of the algorithm to achieve convergent solutions was demonstrated by highly nonlinear reactive contaminant transport and multiphase flow problems.

In virtual circuit networks, all of the packets in a session are transmitted over exactly one path established between the origin and the destination. The problem of choosing a path for each origin-destination pair so as to minimize the average number of packets in the network is considered here. The usual formulation of this problem as a nonlinear multicommodity flow problem with integer decision variables is considered. A distributed algorithm is developed to solve the optimization problem, using Lagrangean relaxation as the basic approach. A multiplier update technique is developed which facilitates the solution of the nonlinear integer programming problem using distributed computation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Partially explicit splitting scheme with explicit–implicit-null method for nonlinear multiscale flow problems

Large-scale nonlinear optimal power flow (OPF) problems have been efficiently solved lately by extensions from linear programming to nonlinear programming of the primal-dual logarithmic barrier interior-point method and its predictor-corrector variant. Motivated by the impressive performance of the nonlinear predictor-corrector extension, in this paper we extend from linear programming to nonlinear OPF the efficient multiple centrality corrections (MCC) technique that was developed by Gondzio. The numerical performance of the proposed MCC algorithm is evaluated on a set of power networks ranging in size from 118 buses to 2098 buses. Extensive computational results demonstrate that the MCC technique is fast and robust, and outperforms the successful predictor-corrector technique.

We consider a class of time-dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers important cases such as Fokker–Planck type equations and aggregation models, which have been studied intensively in the past decades. In this paper, we design a high order discontinuous Galerkin method for such problems. If the interaction potential is not involved, or the interaction is defined by a smooth kernel, our semi-discrete scheme admits an entropy inequality on the discrete level. Furthermore, by applying the positivity-preserving limiter, our fully discretized scheme produces non-negative solutions for all cases under a time step constraint. Our method also applies to two dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics.

A method of obtaining solutions to nonlinear flow problems by integrating the rate of change of the solution with respect to a suitable parameter is described. The advantage of this method is that the nonlinearity of the problem is confined to a first order equation where it causes little difficulty. The Falkner-Skan boundary-layer problem and the problem of flow about a nonlifting airfoil at transonic speeds are used to illustrate the method.

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.