General Nonlinear Problems Research Articles

An augmented Lagrangian affine scaling method for nonlinear programming

In this paper, we propose an augmented Lagrangian affine scaling (ALAS) algorithm for general nonlinear programming, for which a quadratic approximation to the augmented Lagrangian is minimized at each iteration. Different from the classical sequential quadratic programming (SQP), the linearization of nonlinear constraints is put into the penalty term of this quadratic approximation, which results in smooth objective of the subproblem and avoids possible inconsistency among the linearized constraints and trust region constraint. By applying affine scaling techniques to handle the strict bound constraints, an active set type affine scaling trust region subproblem is proposed. Through special strategies of updating Lagrange multipliers and adjusting penalty parameters, this subproblem is able to reduce the objective function value and feasibility errors in an adaptive well-balanced way. Global convergence of the ALAS algorithm is established under mild assumptions. Furthermore, boundedness of the penalty parameter is analyzed under certain conditions. Preliminary numerical experiments of ALAS are performed for solving a set of general constrained nonlinear optimization problems in CUTEr problem library.

On least trimmed squares neural networks

In this paper, least trimmed squares (LTS) estimators, frequently used in robust (or resistant) linear parametric regression problems, will be generalized to nonparametric LTS neural networks for nonlinear regression problems. Emphasis is put particularly on the robustness against outliers. This provides alternative learning machines when faced with general nonlinear learning problems. Simple weight updating rules based on gradient descent and iteratively reweighted least squares (IRLS) algorithms will be provided. The important parameter of trimming percentage for the data at hand can be determined by cross validation. Some simulated and real-world data will be used to illustrate the use of LTS neural networks. We will compare the robustness against outliers for usual neural networks with least squares criterion and the proposed LTS neural networks. Simulation results show that the LTS neural networks proposed in this paper have good robustness against outliers.

Pontryagin maximum principle for optimal sampled-data control problems

In this short communication, we first recall a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. This result was recently obtained in [L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales, arXiv:1501.07361, 2015]. Then we discuss the maximization condition for optimal sampled-data controls that can be seen as an average of the weak maximization condition stated in the classical Pontryagin maximum principle for optimal (permanent) controls. Finally, applying this theorem, we solve a linear-quadratic example based on the classical parking problem.

An Implicit Smooth Conjugate Projection Gradient Algorithm for Optimization with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with equilibrium constraints. At first, by using a generalized complementarity function, the discussed problem is transformed into a family of general nonlinear optimization problems containing additional variable μ. Furthermore, combining the idea of penalty function, an auxiliary problem with inequality constraints is presented. And then, by providing explicit searching direction, we establish a new conjugate projection gradient method for optimization with nonlinear complementarity constraints. Under some suitable conditions, the proposed method is proved to possess global and superlinear convergence rate.

Trigonometric Fourier-series solutions of the point reactor kinetics equations

In this paper, a new method based on the Fourier series is introduced to obtain approximate solutions to the systems of the point kinetics equations. These systems are stiff involving equations with slowly and rapidly varying components. They are solved numerically using Fourier series expansion over a partition of the total time interval. Approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high order derivatives of the dependent variables at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. The obtained algebraic system is similar to the Vandermonde system. Evaluation of the inverse of the Vandermonde matrix is required to determine the coefficients of the Fourier series. Because the obtained Vandermonde matrix has a special structure, due to the properties of the sine and cosine functions, a new formula is introduced to determine its inverse using standard computations. The new method solves the general linear and non-linear kinetics problems with six groups of delayed neutrons. The validity of the algorithm is tested with five different types of reactivities including step reactivity insertion, ramp input, oscillatory reactivity changes, a reactivity as a function of the neutron density and finally temperature feedback reactivity. Comparisons are made with analytical and conventional numerical methods used to solve the point kinetics equations. The results confirm the theoretical analysis and indicate the validity of the method for all applications considered.

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numerical results for several typical test problems are reported.

An Adaptive Mesh Generation Scheme for the Finite Element Method

재난 상황에서 방재시스템의 구조물이 제대로 역할을 수행하려면 구조의 자동화된 상태 파악 기능이 필수적이다. 이 기능은 실시간 연산면에서 매우 효율적이어야 한다. 유한요소법은 구조해석법으로 가장 많이 사용되는 방법으로 자리 잡고 있으며, 자동화된 구조물의 유한요소 해석에서의 요소망은 적응적 형성 기법으로 생산하여야 한다. 적응적 요소망 기능이 없으면, 같은 요소망으로 유한요소 해석을 수행하는데, 이 경우에는 불필요한 부분의 많은 요소에 의한 연산 비효율성이 있을 수 있고, 일반적으로 초기와 최종 요소망 모양만 검토하기 때문에 해석 중 생길 수 있는 과도하게 변형된 요소에 의한 오차 증가도 초래할 수 있다. 다소 복잡한 동적 및 비선형 해석에서 이러한 문제가 자주 발생한다. 본 연구는 시간영역 동적해석에서 전 단계 해석 결과를 사용하여 계산된 대표 변형률 값을 오차 평가에 사용하고, 요소 세분화는 절점 이동인 r-법과 요소 분할인 h-법의 효율적인 조합을 분산 매개변수(dispersion parameter)를 활용하여 계산하고, 모양계수(shape factor) 개념으로 과대하게 변형된 요소 모양을 방지하는 동적인 적응적 유한 요소망 형성 전략을 제시한다. 적용한 예제를 통하여 정확성과 연산 효율성을 검증하고 나아가 방법의 간단함이 지진 하중, 풍하중 및 비선형 해석이 필요한 방재 구조시스템의 복잡한 해석에도 효율적으로 사용될 수 있는 가능성을 보여 준다. Ideal structures in hazard mitigation systems must have automated state identification capabilities in order to function acceptably in case of a severe hazard. This capability must be efficient in terms of real time computations. The finite element method has become the most widely used method of structural analysis and in an automated finite element structural analysis, an adaptive mesh generation scheme has become an essential component. Without an adaptive scheme, the same mesh is used throughout the analysis and this causes computational inefficiency due to fine mesh where they are not needed and inaccuracy due to overly distorted elements formed during the analysis that may proceed undetected as only the initial and the final element shapes are generally checked in practice. This is especially true for dynamic and nonlinear problems in structural analyses. Thus, the finite element mesh for these types of analyses must be dynamically adaptive and computationally efficient. In this paper, an efficient adaptive finite element mesh generation scheme for dynamic analyses of planar problems is described. Representative strain values are used for error estimates and refinements of meshes use combinations of the h-method (node movement) and the r-method (element division) based on a dispersion parameter. A coefficient that depends on the shape of element is used to correct overly distorted elements. A case study shows the validity and computational efficiency of the scheme. The study also demonstrates the potential applicability of the scheme's effective use in complex structural problems such as those under severe environmental hazards such as seismic loads, erratic wind loads, and general nonlinear problems.

Non-similar solutions for mixed convection along a wedge embedded in a porous medium saturated by a non-Newtonian nanofluid

Purpose – The purpose of this paper is to present a boundary layer analysis for the mixed convection past a vertical wedge in a porous medium saturated with a power law type non-Newtonian nanofluid. Numerical results for friction factor, surface heat transfer rate and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter Nr, Brownian motion parameter Nb, thermophoresis parameter Nt, Lewis number Le and the power law exponent n. The dependency of the friction factor, surface heat transfer rate (Nusselt number) and mass transfer rate on these parameters has been discussed. Design/methodology/approach – This general non-linear problem cannot be solved in closed form and, therefore, a numerical solution is necessary to describe the physics of the problem. An implicit, tri-diagonal finite-difference method has proven to be adequate and sufficiently accurate for the solution of this kind of problems. Therefore, it is adopted in the present study. Variable step sizes were used. The convergence criterion employed in this study is based on the difference between the current and the previous iterations. When this difference reached 10−5 for all the points in the η directions, the solution was assumed to be converged, and the iteration process was terminated. Findings – The results indicate that as the buoyancy ratio parameter (Nr) and thermophoresis parameter (Nt) increase, the friction factor increases whereas the heat transfer rate (Nusselt number) and mass transfer rate (Sherwood number) decrease. As the Brownian motion parameter (Nb) increases, the friction factor and surface mass transfer rates increase whereas the surface heat transfer rate decreases. As Le increases, mass transfer rates increase. As the power law exponent n increases, the heat and mass transfer rates increase. Research limitations/implications – The analysis is valid for natural convection dominated regime. The combined forced and natural convection dominated regimes will be reported in a future work. Practical implications – The approach used is useful in optimizing the porous media heat transfer problems in geothermal energy recovery, crude oil extraction, ground water pollution, thermal energy storage and flow through filtering media. Originality/value – The results of the study may be of some interest to the researchers of the field of porous media heat transfer. Porous foam and microchannel heat sinks used for electronic cooling are optimized utilizing the porous medium. The utilization of nanofluids for cooling of microchannel heat sinks requires understanding of fundamentals of nanofluid convection in porous media.

Solving non-linear, non-smooth and non-convex optimal power flow problems using chaotic invasive weed optimization algorithms based on chaos

Invasive Weed Optimization (IWO) algorithm is a simple but powerful algorithm which is capable of solving general multi-dimensional, linear and nonlinear optimization problems with appreciable efficiency. Recently IWO algorithm is being used in several engineering design owing to its superior performance in comparison with many other existing algorithms. This paper presents a Chaotic IWO (CIWO) algorithms based on chaos, and investigates its performance for optimal settings of Optimal Power Flow (OPF) control variables of OPF problem with non-smooth and non-convex generator fuel cost curves (non-smooth and non-convex OPF). The performance of CIWO algorithms are studied and evaluated on the standard IEEE 30-bus test system with different objective functions. The experimental results suggest that IWO algorithm holds immense promise to appear as an efficient and powerful algorithm for optimization in the power system.

Deautoconvolution: A new decomposition approach versus TIGRA and local regularization

Abstract Solving an autoconvolution equation is a nonlinear ill-posed inverse problem. Besides standard methods for general nonlinear problems several customized methods for deautoconvolution are available. Recently, a new decomposition approach for solving ill-posed quadratic equations, e.g. autoconvolutions, has been proposed. In this article we compare the new approach to the TIGRA method of Ramlau and to the local regularization method of Dai and Lamm. Numerical tests show that the new method yields better approximations to the unknown true solution than existing methods in comparable computation time.

The incremental subgradient methods on distributed estimations in-network

A unified framework on distributed estimations in-network using incremental subgradient (IS) methods is introduced. The IS methods meet the requirement of the asynchronously processing, which are efficient solvers developed recently focusing on separable non-differentiable convex optimization problems. The main contribution of this paper is to formalize distributed estimators as equivalent separable convex optimization problems, where general skills and several specific cases on signal estimations are presented. Analytical and simulation results show that the IS framework can solve general nonlinear estimation problems in-network, and achieve comparable performances as the centralized estimators.

A new cubic B-spline method for approximating the solution of a class of nonlinear second-order boundary value problem with two dependent variables

In this paper, we will apply cubic B-splines on a uniform mesh to explore the numerical solutions and numerical derivatives of a class of nonlinear second-order boundary value problems with two dependent variables. Our new method is based on the cubic spline interpolation. The analytical solutions and any-order derivatives can be well approximated with 4th order accuracy. Furthermore, our new method is also able to solve general nonlinear 4th-order two-point boundary value problems. Numerical results show that our method is very practical and effective. a0(x)u 00 +a1(x)u 0 +a2(x)u +a3(x)v 00 +a4(x)v 0 +a5(x)v +G1(x;u;v) = f1(x); b0(x)v 00 +b1(x)v 0 +b2(x)v +b3(x)u 00 +b4(x)u 0 +b5(x)u +G2(x;u;v) = f2(x);

A Filter Method with Unified Step Computation for Nonlinear Optimization

We present a filter line search method for solving general nonlinear and nonconvex optimization problems. The method is of the filter variety but uses a robust (always feasible) subproblem based on an exact penalty function to compute a search direction. This contrasts traditional filter methods that use a (separate) restoration phase designed to reduce infeasibility until a feasible subproblem is obtained. Therefore, an advantage of our approach is that every trial step is computed from subproblems that value reducing both the constraint violation and the objective function. Moreover, our step computation involves subproblems that are computationally tractable and utilize second derivative information when it is available. The formulation of each subproblem and the choice of weighting parameter is crucial for obtaining an efficient, robust, and practical method. Our strategy is based on steering methods designed for exact penalty functions but is fortified with a trial step convexification scheme that en...

Iterative Methods for Solving General Nonlinear Quasi-Variational Inequalities

This paper presents some iterative methods for solving the general nonlinear quasi-variational inequality problem and consider the convergence criteria of the iterative schemes under certa in conditions. Some special cases are also discussed.

Smoothing Methods for Nonlinear Complementarity Problems

In this paper, we present some new smoothing techniques to solve general nonlinear complementarity problems. Under a weaker condition than monotonicity as on the original problems, we prove convergence of our methods. We also present an error estimate under a general monotonicity condition. Some numerical tests confirm the efficiency of the proposed methods.

An Integral Method for Solving the Point Reactor Neutron Kinetics Equations with Newtonian Temperature Feedback

A numerical integral method to efficiently solve the point kinetics equations with Newtonian temperature feedback is described and investigated, which employs the better basis function (BBF) for the approximation of the neutron density in integral of one time step. The numerical evaluation is performed by the developed BBF code. The code can solve the general non-linear kinetics problems with six groups of delayed neutron. For the application purposes, the developed code and the method are tested by using a variety of problems, including ramp reactivity input with or without temperature feedback. The results are shown that the BBF method is clearly an effective and accurate numerical method for solving the point kinetics equations with Newtonian temperature feedback, and it can be used in real time power reactor forecasting in order to prevent the reactivity accidents.

Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

AbstractThe alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles. When the ADI technique is coupled with orthogonal spline collocation (OSC) for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order. In [2], we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms. A natural question that arises is: Does this method have an extension to non-rectangular regions? In this paper, we present a simple idea of how the ADI OSC technique can be extended to some such regions. Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem (TPBVP). We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.

Saliency Combined Particle Filtering for Aircraft Tracking

Vision-based aircraft tracking has been considered for emerging real-world applications, such as collision avoidance, air traffic surveillance, and target tracking for military use. However, conventional tracking methods often fail in following aircraft due to 1) variations of object shape, 2) continuously varying background, and 3) unpredictable flight motion. In this paper, we address the problems of vision-based aircraft tracking. To this ends, we propose a principled manner of improving color-based tracking algorithm by combining a biologically inspired saliency feature. More specifically, we exploit the integration of color distributions into particle filtering, which is a Monte Carlo method for general nonlinear filtering problems. To overcome the varying appearances which are usually from changing illumination and pose conditions, we update the target color model. Furthermore, we adopt a structure tensor based saliency algorithm to incorporate the saliency features into particle filter framework, which results in robustly assigning appropriate particle weights even in complex backgrounds. The rationale behind our approach is that color and saliency information are complementary, both mutually fulfilling and completing each other, especially when tracking aircraft in a harsh environment. Tests on real flight sequences reveal that the proposed system yields convincing tracking outcomes under both variations of background and sudden target motion changes.

A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows

In Xu (2013) [14], a class of parametrized flux limiters is developed for high order finite difference/volume essentially non-oscillatory (ENO) and Weighted ENO (WENO) schemes coupled with total variation diminishing (TVD) Runge–Kutta (RK) temporal integration for solving scalar hyperbolic conservation laws to achieve strict maximum principle preserving (MPP). In this paper, we continue along this line of research, but propose to apply the parametrized MPP flux limiter only to the final stage of any explicit RK method. Compared with the original work (Xu, 2013) [14], the proposed new approach has several advantages: First, the MPP property is preserved with high order accuracy without as much time step restriction; Second, the implementation of the parametrized flux limiters is significantly simplified. Analysis is performed to justify the maintenance of third order spatial/temporal accuracy when the MPP flux limiters are applied to third order finite difference schemes solving general nonlinear problems. We further apply the limiting procedure to the simulation of the incompressible flow: the numerical fluxes of a high order scheme are limited toward that of a first order MPP scheme which was discussed in Levy (2005) [3]. The MPP property is guaranteed, while designed high order of spatial and temporal accuracy for the incompressible flow computation is not affected via extensive numerical experiments. The efficiency and effectiveness of the proposed scheme are demonstrated via several test examples.

Particle filter for state estimation of jump Markov nonlinear system with application to multi-targets tracking

In this article, we first introduce the problem of state estimation of jump Markov nonlinear systems (JMNSs). Since the density evolution method for predictor equations satisfies Fokker–Planck–Kolmogorov equation (FPKE) in Bayes estimation, the FPKE in conjunction with Bayes’ conditional density update formula can provide optimal estimation for a general continuous-discrete nonlinear filtering problem. It is well known that the analytical solution of the FPKE and Bayes’ formula is extremely difficult to obtain except a few special cases. Hence, we try to design a particle filter to achieve Bayes estimation of the JMNSs. In order to test the viability of our algorithm, we apply it to multiple targets tracking in video surveillance. Before starting simulation, we introduce the ‘birth’ and ‘death’ description of targets, targets’ transitional probability model, and observation probability. The experiment results show good performance of our proposed filter for multiple targets tracking.