Newton Interior-point Method Research Articles

A trust-region based an active-set interior-point algorithm for fuzzy continuous Static Games

<abstract><p>In this paper, a novel treatment for fuzzy continuous static games (FCSGs) is introduced. This treatment is based on the fact that, as well as having a fuzzy number, the fuzziness is applied to the control vectors to deal with high vagueness and imprecision in a continuous static game. The concept of the $ \alpha $-level set used for converting the FCSGs to a deterministic problem $ \alpha $-FCSGs. An active-set strategy is used with Newton's interior point method and a trust-region strategy to insure global convergence for deterministic $ \alpha $-FCSGs problems from any starting point. A reduced Hessian technique is used to overcome the difficulty of having an infeasible trust-region subproblem. The active-set interior-point trust-region algorithm has new features; it is easy to implement and has rapid convergence. Preliminary numerical results are reported.</p></abstract>

An interior-point trust-region algorithm to solve a nonlinear bilevel programming problem

<abstract><p>In this paper, a nonlinear bilevel programming (NBLP) problem is transformed into an equivalent smooth single objective nonlinear programming (SONP) problem utilized slack variable with a Karush-Kuhn-Tucker (KKT) condition. To solve the equivalent smooth SONP problem effectively, an interior-point Newton's method with Das scaling matrix is used. This method is locally method and to guarantee convergence from any starting point, a trust-region strategy is used. The proposed algorithm is proved to be stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem.</p> <p>A global convergence theory of the proposed algorithm is introduced and applications to mathematical programs with equilibrium constraints are given to clarify the effectiveness of the proposed approach.</p></abstract>

Jacobian Free Methods for Coupling Transport with Chemistry in Heterogenous Porous Media

Reactive transport plays an important role in various subsurface applications, including carbon dioxide sequestration, nuclear waste storage, biogeochemistry and the simulation of hydro–thermal reservoirs. The model couples a set of partial differential equations, describing the transport of chemical species, to nonlinear algebraic or differential equations, describing the chemical reactions. Solution methods for the resulting large nonlinear system can be either fully coupled or can iterate between transport and chemistry. This paper extends previous work by the authors where an approach based on the Newton–Krylov method applied to a reduced system has been developed. The main feature of the approach is to solve the nonlinear system in a fully coupled manner while keeping transport and chemistry modules separate. Here we extend the method in two directions. First, we take into account mineral precipitation and dissolution reactions by using an interior point Newton method, so as to avoid the usual combinatorial approach. Second, we study two-dimensional heterogeneous geometries. We show how the method can make use of an existing transport solver, used as a black box. We detail the methods and algorithms for the individual modules, and for the coupling step. We show the performance of the method on synthetic examples.

Multi-objective optimal load flow problem with interior-point trust-region strategy

In this paper, an interior-point trust-region algorithm to generate a Pareto optimal solution for a multi-objective optimal load flow (MOLF) problem is introduced. A weighted Tchebychev approach is used in this paper to transform (MOLF) problem to a single objective optimization problem.In the algorithm, an interior-point Newton method is used with Coleman-Li scaling matrix and a trust-region globalization strategy to insure global convergence. The trust region technique is suitable for multi-objective optimal load flow problem such that its objective functions may be ill-defined or having a non-convex Pareto-optimal front. A projected Hessian technique is used in the algorithm to handling the difficulty of having an infeasible trust region subproblems.The proposed algorithm is carried out on the standard IEEE 30-bus 6-generator test systems to assert the efficacy of the algorithm used to solve the multi-objective (MOLF) problem. Our results have been compared to those reported in the literature. The comparison demonstrates the superiority of the proposed approach and confirm its potential to generate the Pareto optimal solution for (MOLF) problem.

Time domain localization technique with sparsity constraint for imaging acoustic sources

This paper addresses source localization technique in time domain for broadband acoustic sources. The objective is to accurately and quickly detect the position and amplitude of noise sources in workplaces in order to propose adequate noise control options and prevent workers hearing loss or safety risk. First, the generalized cross correlation associated with a spherical microphone array is used to generate an initial noise source map. Then a linear inverse problem is defined to improve this initial map. Commonly, the linear inverse problem is solved with an l2-regularization. In this study, two sparsity constraints are used to solve the inverse problem, the orthogonal matching pursuit and the truncated Newton interior-point method. Synthetic data are used to highlight the performances of the technique. High resolution imaging is achieved for various acoustic sources configurations. Moreover, the amplitudes of the acoustic sources are correctly estimated. A comparison of computation times shows that the technique is compatible with quasi real-time generation of noise source maps. Finally, the technique is tested with real data.

Key Technology of Real-Time Road Navigation Method Based on Intelligent Data Research.

The effect of traffic flow prediction plays an important role in routing selection. Traditional traffic flow forecasting methods mainly include linear, nonlinear, neural network, and Time Series Analysis method. However, all of them have some shortcomings. This paper analyzes the existing algorithms on traffic flow prediction and characteristics of city traffic flow and proposes a road traffic flow prediction method based on transfer probability. This method first analyzes the transfer probability of upstream of the target road and then makes the prediction of the traffic flow at the next time by using the traffic flow equation. Newton Interior-Point Method is used to obtain the optimal value of parameters. Finally, it uses the proposed model to predict the traffic flow at the next time. By comparing the existing prediction methods, the proposed model has proven to have good performance. It can fast get the optimal value of parameters faster and has higher prediction accuracy, which can be used to make real-time traffic flow prediction.

Complex two‐dimensional TNIPM for l1 norm‐based sparse optimization to collocated MIMO radar

In collocated multiple‐input multiple‐output (MIMO) radar, because of the sparse nature of the received signal in the three dimensions of range, angle, and Doppler, accurate estimates of range/angle/Doppler parameters can be achieved using a sparse signal recovery. In this paper, we develop a complex two‐dimensional truncated Newton interior point method (2D TNIPM) for l1‐norm‐based sparse optimization. Because of the 2D sparse representation of received signal in collocated MIMO radar systems, the performance of proposed algorithm is investigated in order to estimate the target position and velocity. Simulation results show that the 2D TNIPM requires much lower computations compared to the 1D one. Also, it outperforms some other 2D algorithms in the estimation of range, angle, and Doppler parameters under low signal‐to‐noise ratios. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Parallel collocation solution of index-1 BVP-DAEs arising from constrained optimal control problems

The indirect solution of constrained optimal control problems gives rise to two-point boundary value problems (BVPs) that involve index-1 differential-algebraic equations (DAEs) and inequality constraints. This paper presents a parallel collocation algorithm for the solution of these inequality constrained index-1 BVP-DAEs. The numerical algorithm is based on approximating the DAEs using piecewise polynomials on a nonuniform mesh. The collocation method is realized by requiring that the BVP-DAE be satisfied at Lobatto points within each interval of the mesh. A Newton interior-point method is used to solve the collocation equations, and maintain feasibility of the inequality constraints. The implementation of the algorithm involves: (i) parallel evaluation of the collocation equations; (ii) parallel evaluation of the system Jacobian; and (iii) parallel solution of a boarded almost block diagonal (BABD) system to obtain the Newton search direction. Numerical examples show that the parallel implementation provides significant speedup when compared to a sequential version of the algorithm.

P-splines quantile regression estimation in varying coefficient models

Quantile regression, as a generalization of median regression, has been widely used in statistical modeling. To allow for analyzing complex data situations, several flexible regression models have been introduced. Among these are the varying coefficient models, that differ from a classical linear regression model by the fact that the regression coefficients are no longer constant but functions that vary with the value taken by another variable, such as for example, time. In this paper, we study quantile regression in varying coefficient models for longitudinal data. The quantile function is modeled as a function of the covariates and the main task is to estimate the unknown regression coefficient functions. We approximate each coefficient function by means of P-splines. Theoretical properties of the estimators, such as rate of convergence and an asymptotic distribution are established. The estimation methodology requests solving an optimization problem that also involves a smoothing parameter. For a special case the optimization problem can be transformed into a linear programming problem for which then a Frisch–Newton interior point method is used, leading to a computationally fast and efficient procedure. Several data-driven choices of the smoothing parameters are briefly discussed, and their performances are illustrated in a simulation study. Some real data analysis demonstrates the use of the developed method.

Parallel indirect solution of optimal control problems

SUMMARY This paper presents an algorithm for the indirect solution of optimal control problems that contain mixed state and control variable inequality constraints. The necessary conditions for optimality lead to an inequality constrained two-point BVP with index-1 differential-algebraic equations (BVP-DAEs). These BVP-DAEs are solved using a multiple shooting method where the DAEs are approximated using a single-step linearly implicit Runge–Kutta (Rosenbrock–Wanner) method. An interior-point Newton method is used to solve the residual equations associated with the multiple shooting discretization. The elements of the residual equations, and the Jacobian of the residual equations, are constructed in parallel. The search direction for the interior-point method is computed by solving a sparse bordered almost block diagonal (BABD) linear system. Here, a parallel-structured orthogonal factorization algorithm is used to solve the BABD system. Examples are presented to illustrate the efficiency of the parallel algorithm. It is shown that an American National Standards Institute C implementation of the parallel algorithm achieves significant speedup with the increase in the number of processors used. Copyright © 2013 John Wiley & Sons, Ltd.

Transmitted Waveform Design Based on Iterative Methods

In intelligent radar, it is an important problem for the transmitted waveform to adapt to the environment in which radar works. In this paper, we propose mutual information model of adaptive waveform design, which can convert the problem of adaptive waveform design into the problem of optimization. We consider two situations of no clutter and clutter and use Newton method and interior point method to solve the optimization problem. Then we can draw the design criterion for the transmitted waveform in cognitive radar and get a greater mutual information from the simulation results. Finally, the whole paper is summarized.

An Affine Scaling Interior Point Filter Line-Search Algorithm for Linear Inequality Constrained Minimization

We propose and analyze a new affine scaling interior point method in association with line-search filter technique for solving the linear inequality constrained optimization. We extend an existing filter of Wächter and Biegler [9] for nonlinear equality constrained problems to linear inequality constraint problems. The two main ingredients of the method are a filter-line-search algorithm which is used to determine step size and an affine-scaling interior point Newton method which is used to generate a search direction. The global convergence of the proposed algorithm is established under some reasonable conditions. Further, the method is shown to be locally Q-superlinearly convergent under the strong second order sufficiency condition. Numerical tests are presented that confirm the robustness and efficiency of the approach.

Total Transfer Capability Assessment Incorporating Corrective Controls for Transient Stability using TSCOPF

Advancements have been made towards inclusion of both static and dynamic security into transfer capability calculation. However, to the authors' knowledge, work on considering corrective controls into the calculation has not been reported yet. Therefore, we propose a Total Transfer Capability (TTC) assessment considering transient stability corrective controls. The method is based on the Newton interior point method for nonlinear programming and transfer capability is approached as a maximization of power transfer with both static and transient stability constraints are incorporated into our Transient Stability Constrained Optimal Power Flow (TSCOPF) formulation. An interconnected power system is simulated to be subjected to a severe unbalanced 3-phase 4-line to ground fault and following the fault, generator and load are shed in a pre-defined sequence to mimic actual corrective controls. In a deregulated electricity market, both generator companies and large load customers are encouraged to actively participate in maintaining power system stability as corrective controls upon agreement of compensation for being shed following a disturbance. Implementation of this proposal on the actual power system operation should be carried out through combining it with the existing transient stabilization controller system. Utilization of these corrective controls results in increasing TTC as suggested in our numerical simulation. As Lagrange multipliers can also describe sensitivity of both inequality and equality constraints to the objective function, then selection of which generator or load to be shed can be carried out on the basis of values of Lagrange multipliers of its respective generator's rotor angle stability and active power balance equation. Hence, the proposal in this paper can be utilized by system operator to assess the maximum TTC for specific loads and network conditions.

Superlinear convergence of affine scaling interior point newton method for linear inequality constrained minimization without strict complementarity

In this paper we extend and improve the classical affine scaling interior-point Newton method for solving nonlinear optimization subject to linear inequality constraints in the absence of the strict complementarity assumption. Introducing a computationally efficient technique and employing an identification function for the definition of the new affine scaling matrix, we propose and analyze a new affine scaling interior-point Newton method which improves the Coleman and Li affine scaling matrix in [2] for solving the linear inequality constrained optimization. Local superlinear and quadratical convergence of the proposed algorithm is established under the strong second order sufficiency condition without assuming strict complementarity of the solution.

A nonmonotone inexact Newton method

In this paper, we describe a variant of the inexact Newton method for solving nonlinear systems of equations. We define a nonmonotone inexact Newton step and a nonmonotone backtracking strategy. For this nonmonotone inexact Newton scheme, we present the convergence theorems. Finally, we show how we can apply these strategies to the inexact Newton interior-point method and we present some numerical examples.

An inexact Newton method combined with Hestenes multipliers’ scheme for the solution of Karush–Kuhn–Tucker systems

In this work a Newton interior-point method for the solution of Karush–Kuhn–Tucker systems is presented.A crucial feature of this iterative method is the solution, at each iteration, of the inner subproblem. This subproblem is a linear-quadratic programming problem, that can solved approximately by an inner iterative method such as the Hestenes multipliers’ method.A deep analysis on the choices of the parameters of the method (perturbation and damping parameters) has been done.The global convergence of the Newton interior-point method is proved when it is viewed as an inexact Newton method for the solution of nonlinear systems with restriction on the sign of some variables.The Newton interior-point method is numerically evaluated on large scale test problems arising from elliptic optimal control problems which show the effectiveness of the approach.

Primal–dual Newton interior point methods in shape and topology optimization

AbstractWe consider non‐linear minimization problems with both equality and inequality constraints on the state variables and design parameters as they typically arise in shape and topology optimization. In particular, the state variables are subject to a partial differential equation or systems of partial differential equations describing the operating behaviour of the device or system to be optimized. For the numerical solution of the appropriately discretized problems we emphasize the use of all‐in‐one approaches where the numerical solution of the discretized state equations is an integral part of the optimization routine. Such an approach is given by primal–dual Newton interior point methods which we present combined with a suitable steplength selection and a watchdog strategy for convergence monitoring. As applications, we deal with the topology optimization of electric drives for high power electromotors and with the shape optimization of biotemplated microcellular biomorphic ceramics based on homogenization modelling. Copyright © 2004 John Wiley & Sons, Ltd.

Local Convergence of the Interior-Point Newton Method for General Nonlinear Programming

In this paper, a formulation for an interior-point Newton method of general nonlinear programming problems is presented. The formulation uses the Coleman-Li scaling matrix. The local convergence and the q-quadratic rate of convergence for the method are established under the standard assumptions of the Newton method for general nonlinear programming.

A solution of optimal power flow with multicontingency transient stability constraints

This paper presents a formulation of the multicontingency transient stability constrained optimal power flow (MC-TSCOPF) problem and proposes a method to solve it. In the MC-TSCOPF formulation, this paper introduces a modified formulation for integrating transient stability model into conventional OPF, which reduces the calculation load considerably. In our MC-TSCOPF solution, the primal-dual Newton interior point method (IPM) for nonlinear programming (NLP) is adopted. Computation results on the IEEJ WEST10 model system demonstrate the effectiveness of the presented MC-TSCOPF formulation and the efficiency of the proposed solution approach. Moreover, based on quite convincing simulation results, some phenomena occurred when considering multicontingency are elaborated.

Local convergence properties of primal-dual interior point methods based on the shifted barrier KKT conditions for nonlinear optimization

In this paper, we consider the shifted barrier KKT conditions for nonlinear optimization. We propose a primal-dual interior point method based on these conditions. By choosing suitable parameters used in our method, we prove local and q-quadratic convergence of the Newton interior point method, and local and q-superlinear convergence of the quasi-Newton interior point method.