An Investigation of Interior-Point Algorithms for the Linear Transportation Problem

Weakly admissible transformations are introduced for solving algebraic assignment and transportation problems, which cover so important classes as problems with sum objectives, bottleneck objectives, lexicographical objectives and others. A transformation of the cost matrix is called weakly admissible, if there are two constants α and β in the underlying semigroup that for all feasible solutions the composition of α and the objective value with respect to the original cost coefficients is equal to the composition of β and the objective value with respect to the transformed cost coefficients. The elements α and β can be determined by shortest path algorithms. An optimal solution for the algebraic assignment problem can be found after at most n weakly admissible transformations, therefore the proposed method yields an O(n 3) algorithm for algebraic assignment problems.

Variants of simplex-based methodologies are generally used to solve underlying linear programming (LP) problems. An implementation of the dual affine (DA) algorithm (a variant of N. Karmarkar's (1984) interior point method) is described in detail and some computational results are presented. This algorithm is particularly suitable for problems with a large number of constraints, and is applicable to linear and nonlinear optimization problems. In contrast with the simplex method, the number of iterations required by the DA algorithm to solve large-scale problems is relatively small. The DA algorithm has been implemented considering the sparsity of the constraint matrix. The normal equation that is required to be solved in every iteration is solved using a preconditioned conjugate gradient method. An application of the technique to a hydro-scheduling is presented; the largest problem is solved over nine times faster than an efficient simplex (MINOS) code. A new heuristic basis recovery procedure is implemented to provide primal and dual optimal basic solutions which are not generally available if interior point methods are used. The tested examples indicate that this new approach requires less than 10% of the original iterations of the simplex method to find the optimal basis. >

This paper proposes a nonlinear optimization interior point (IP) method for the determination of maximum loadability in a power system. Details of the implementation of pure primal-dual and predictor-corrector primal-dual IP algorithms are presented. It is shown that most of the computational effort of the algorithm is taken by the formation and factorization of the augmented Hessian matrix of the IP algorithm. The size of this matrix can be as large as ten times the number of buses in the system. Comparisons of the two IP implementations with large scale power systems with as many as 4000 buses are presented. It is shown that the IP algorithm constitutes an effective method for the determination of the maximum loadability in a power system.

Validation of measurement data using an interior point SQP

: A globally convergent primal-dual predictor corrector interior point algorithm for linear programming has been developed.

This article describes an approach for the dynamic optimal power flow problem using interior point method (IPM) and Benders decomposition considering both active and reactive constraints. An efficient predictor-corrector primal-dual interior-point algorithm is used to solve the linearized dynamic OPF problem. The inclusion of active and reactive security constraints will assure a subsequent feasible solution for the dynamic OPF problem. The dynamic OPF problem is decomposed into a master problem and subproblems for checking the feasibility of the active and reactive constraints. The master problem is formulated and solved without active and reactive constraints to avoid complexity. The obtained dynamic OPF schedule from the master problem is applied to the active and reactive subproblems to minimize the violations. In case the current mix of scheduled units cannot remove the violations, additional constraints will be introduced in the master problem for rescheduling dynamic OPF. The proposed approach has been evaluated on an IEEE 118-bus test system. The results obtained with the proposed approach are presented and compared favorably with results of other stochastic techniques.

A novel sine transform based preconditioned conjugate gradient (PCG) method for signal restoration is presented. If the bandwidth 2? +1 in the model matrix is a constant, then the asymptotic complexity for the new algorithm is O(N) per iterative step, where Nis the order of the model. This complexity is reduced by an order of magnitude in comparison with the known fast Fourier transform (FFT) technique proposed recently in the literature. Moreover, if ? has a size similar to N, then the new PCG method is accomplished by the fast sine transform (FST) and the fast Hartley transform (FHT). The computational and storage cost per preconditioning step for the new PCG method is reduced by 50% as compared to the FFT approach. To stabilize the computation, a special Tikhonov regularization is introduced. Numerical experimentations show that the new PCG method is of a speedy convergence rate. The new PCG methods maintain less computational complexity per iterative step and have a better convergence rate than the other known PCG methods.

The focus of this paper is to present the results of our investigation and evaluation of various shared-memory parallelizations of the data association problem in multitarget tracking. The multitarget tracking algorithm developed was for a sparse air traffic surveillance problem, and is based on an Interacting Multiple Model (IMM) state estimator embedded into the (2D) assignment framework. The IMM estimator imposes a computational burden in terms of both space and time complexity, since more than one filter model is used to calculate state estimates, covariances, and likelihood functions. In fact, contrary to conventional wisdom, for sparse multitarget tracking problems, we show that the assignment (or data association) problem is not the major computational bottleneck. Instead, the interface to the assignment problem, namely, computing the rather numerous gating tests and IMM state estimates, covariance calculations, and likelihood function evaluations (used as cost coefficients in the assignment problem), is the major source of the workload. Using a measurement database based on two FAA air traffic control radars, we show that a coarse-grained (dynamic) parallelization across the numerous tracks found in a multitarget tracking problem is robust, scalable, and demonstrates superior computational performance to previously proposed fine-grained (static) parallelizations within the IMM.

This paper presents a current-based optimal power flow (OPF) with predictor-corrector interior point algorithm (PCIPA). The distinctive features of this OPF formulation are: i) the elements of Hessian matrix are all real, ii) the objective function and constraints are quadratic functions and such quadratic properties are explored in the development of a robust nonlinear OPF solution procedure. This paper also presents the proposed method can further decomposes into two optimal sub-problems (active and reactive). The computational results on power systems of IEEE 9 to 118 buses have shown that the proposed algorithms are very effective.

In this paper, a transportation problem with an objective function as the sum of a linear and linear fractional function is considered. We present an algorithm to solve the above transportation problem. In addition, we consider two special cases of the problem, where the transportation flow is either restricted or enhanced. The first case deals with when one wishes to keep reserve stocks at the sources for emergencies, thereby restricting the total flow to a known specified level. The second case deals with when extra demand in the market compels some of the factories to increase their production, thereby enhancing the total flow to meet the customer demands fully. For the above special cases, we formulate a related linear plus linear fractional transportation problem. We present a numerical example to illustrate the proposed algorithm for the different cases. We show that the solution obtained is a local minimum occurring at an extreme point of the convex set of feasible solutions.

Logistics, supply chain management, and resource allocation are just a few of the many real-world contexts in which the Multi-Objective Linear Transportation Problem presents itself as a formidable optimization issue. In this paper, we seek to solve a multi-objective, linear and non-linear transportation problem by suggesting the most efficient solution possible. The unique Maximum divide and minimum allotment (MDMA) approach was proposed to find a workable answer to a particular transportation issue. It also calculates the solution to the problem using a number of different transportation methods and compares the result to MDMA methods. It is determined that the MDMA approach offers a practical algorithm and solution for the designated transportation problem.

We propose a new set of preconditioners for the iterative solution, via a preconditioned conjugate gradient (PCG) method, of the KKT systems that must be solved at each iteration of an interior point (IP) algorithm for the solution of linear min cost flow (MCF) problems. These preconditioners are based on the idea of extracting a proper triangulated subgraph of the original graph which strictly contains a spanning tree. We define a new class of triangulated graphs, called brother-connected trees (BCTs), and discuss some fast heuristics for finding BCTs of "large" weight. Computational experience shows that the new preconditioners can complement tree preconditioners, outperforming them both in iterations count and in running time on some classes of graphs.

Groundwater flow simulation has become one of the top international issues in new generation of environmental applications. When managing large-scale groundwater flow problems, the intensive computational ability and large amounts of memory space required for modeling are the main bottlenecks for researchers. In order to solve three-dimensional large-scale groundwater flow problems more rapidly, the Open MP was adopted to parallelize the preconditioned conjugate gradient (PCG) algorithm in this paper. And this paper carried out a numerical experiment of the three-dimensional groundwater flow model on a computer with four cores. Based on the numerical experiment, it is found that the execution time of the original serial PCG program is about 1.74 to 2.86 times of the parallel PCG program executed with different number of threads. The experimental results also demonstrate that the PCG solver based on Open MP is an effective way for solving large-scale three-dimensional groundwater flow problem.

Gauss-Seidel preconditioned conjugate gradient (GS-PCG) method based on element-by-element (EBE) is derived and the calculation process of finite element method (FEM) is presented in this paper. The corresponding FEM program are developed and implemented on the magnetic field calculations which produced by straight conductor carrying current and 3-phases induction motor respectively. The validity and accuracy of the method have been verified by comparing the result with the analytical solution. Compared with Jacobi preconditioned conjugate gradient (J-PCG) method based on EBE-FEM, less computational time and cost are needed for the proposed method with same accuracy. The proposed method can be applied to electromagnetic field analysis of transformer and motor, which may refer large scale numerical computation.

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential.