Proximal Methods in View of Interior-Point Strategies

This paper deals with regularized penalty-barrier methods for convex programming problems. In the spirit of an iterative proximal regularization approach, an interior-point method is constructed, in which at each step a strongly convex function has to be minimized and the prox-term can be scaled by a variable scaling factor. The convergence of the method is studied for an axiomatically given class of barrier functions. According to the results, a wide class of barrier functions (in particular, logarithmic and exponential functions) can be applied to design special algorithms. For the method with a logarithmic barrier, the rate of convergence is investigated and assumptions that ensure linear convergence are given.     

Free material optimization via mathematical programming

This paper deals with a central question of structural optimization which is formulated as the problem of finding the stiffest structure which can be made when both the distribution of material as well as the material itself can be freely varied. We consider a general multi-load formulation and include the possibility of unilateral contact. The emphasis of the presentation is on numerical procedures for this type of problem, and we show that the problems after discretization can be rewritten as mathematical programming problems of special form. We propose iterative optimization algorithms based on penalty-barrier methods and interior-point methods and show a broad range of numerical examples that demonstrates the efficiency of our approach. Supported by the project 03ZO7BAY of BMBF (Germany) and the GIF-contract 10455-214.06/95.     

A primal-dual trust region algorithm for nonlinear optimization

This paper concerns general (nonconvex) nonlinear optimization when first and second derivatives of the objective and constraint functions are available. The proposed method is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penalty-barrier function that involves both primal and dual variables. Each subproblem is solved using a second-derivative Newton-type method that employs a combined trust region and line search strategy to ensure global convergence. It is shown that the trust-region step can be computed by factorizing a sequence of systems with diagonally-modified primal-dual structure, where the inertia of these systems can be determined without recourse to a special factorization method. This has the benefit that off-the-shelf linear system software can be used at all times, allowing the straightforward extension to large-scale problems. Numerical results are given for problems in the COPS test collection.Mathematics Subject Classification (2000): 49M37, 65F05, 65K05, 90C30This paper is dedicated to Roger Fletcher on the occasion of his 65th birthday     

Three-term conjugate approach for structural reliability analysis

In this paper, a nonlinear conjugate structural first-order reliability method is proposed using three-term conjugate discrete map-based sensitivity analysis to enhance convergence properties as stable results and efficient computational burden of nonlinear reliability problems. The concept of finite-step length strategy is incorporated into this method to enhance the stability of the iterative formula for highly nonlinear limit state function, while three-term conjugate search direction combining with a finite-step size is utilized to enhance the efficiency of the sensitivity vector in the proposed iterative reliability formula. The proposed three-term discrete conjugate search direction is developed based on the sufficient descent condition to provide the stable results, theoretically. The efficiency and robustness of the proposed three-term conjugate formula are investigated through several nonlinear/ complex structural examples and are compared with several modified existing iterative formulas. Comparative results illustrate that the three-term conjugate-based finite step length formula provides superior efficiency and robustness than other studied methods.     

A Globally Convergent Lagrangian Barrier Algorithm for Optimization with General Inequality Constraints and Simple Bounds

We consider the global and local convergence properties of a class of Lagrangian barrier methods for solving nonlinear programming problems. In such methods, simple bound constraints may be treated separately from more general constraints. The objective and general constraint functions are combined in a Lagrangian barrier function. A sequence of such functions are approximately minimized within the domain defined by the simple bounds. Global convergence of the sequence of generated iterates to a first-order stationary point for the original problem is established. Furthermore, possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero. This paper is a companion to previous work of ours on augmented Lagrangian methods.

     

A modification of the stiefel-bettis method for nonlinearly damped oscillators

We report a modification of the Stiefel-Bettis method which is of trigonometric order one and of polynomial order two for the general second order initial value problems. We also discuss the modified Stiefel-Bettis method made explicit for the undamped nonlinear oscillators. Numerical solution of problems are given to illustrate the methods.     

Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared-operator method, the modified squared-operator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrödinger equations with periodic potentials, and isolated solitons in Ginzburg–Landau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.     

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.     

Wavelet Collocation Method for Optimal Control Problems

A Haar wavelet technique is discussed as a method for discretizing the nonlinear system equations for optimal control problems. The technique is used to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. A nonlinear programming solver can then be used to solve optimal control problems that are rather general in form. Here, general Bolza optimal control problems with state and control constraints are considered. Examples of two kinds of optimal control problems, continuous and discrete, are solved. The results are compared to those obtained by using other collocation methods.     

一般单调变分不等式的一个改进的预估-校正算法

最近何炳生等提出了解大规模单调变分不等式的一种预估-校正算法,然而,这个方法在计算每一个试验点时需要一次投影运算,因而计算量较大．为了克服这个缺点,我们提出了一个解一般大规模g-单调变分不等式的新的预估-校正算法,该方法使用了一个非常有效的预估步长准则,每个步长的选取只需要计算一次投影,这将大大减少计算量．数值试验说明我们的算法比最新文献中出现的投影类方法有效．     

Numerical Experiments with an Interior-Exterior Point Method for Nonlinear Programming

The paper presents an algorithm for solving nonlinear programming problems. The algorithm is based on the combination of interior and exterior point methods. The latter is also known as the primal-dual nonlinear rescaling method. The paper shows that in certain cases when the interior point method (IPM) fails to achieve the solution with the high level of accuracy, the use of the exterior point method (EPM) can remedy this situation. The result is demonstrated by solving problems from COPS and CUTE problem sets using nonlinear programming solver LOQO that is modified to include the exterior point method subroutine.     

Modified barrier functions (theory and methods)

The nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and/or the objective function into an equivalent problem, the classical Lagrangian which has important properties on the primal and the dual spaces.The application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions (MBF's) and MBF Methods (MBFM's). Being classical Lagrangians (CL's) for an equivalent problem, the MBF's combine the best properties of the CL's and classical barrier functions (CBF's) but at the same time are free of their most essential deficiencies.Due to the excellent MBF properties, new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed.The MBFM have up to a superlinear rate of convergence and are to the classical barrier functions (CBF's) method as the Multipliers Method for Augmented Lagrangians is to the Classical Penalty Function Method. Based on the dual theory associated with MBF, the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed. The application of the MBF to linear (LP) and quadratic (QP) programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the CBF methods.The numerical realization of the MBFM leads to the Newton Modified Barrier Method (NMBM). The excellent MBF properties allow us to discover that for any nondegenerate constrained optimization problem, there exists a hot start, from which the NMBM has a better rate of convergence, a better complexity bound, and is more stable than the interior point methods, which are based on the classical barrier functions.     

求解一般约束优化问题的修正BFGS信赖域算法

借鉴无约束优化问题的BFGS信赖域算法,建立了非线性一般约束优化问题的BFGS信赖域算法,并证明了算法的全局收敛性．数值实验表明,算法是有效的．     

参外激励下强非线性广义Van der Pol方程的亚谐共振解（英文）

In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.     

Nonlinear conjugate gradient methods with structured secant condition for nonlinear least squares problems

In this paper, we deal with conjugate gradient methods for solving nonlinear least squares problems. Several Newton-like methods have been studied for solving nonlinear least squares problems, which include the Gauss-Newton method, the Levenberg-Marquardt method and the structured quasi-Newton methods. On the other hand, conjugate gradient methods are appealing for general large-scale nonlinear optimization problems. By combining the structured secant condition and the idea of Dai and Liao (2001) [20], the present paper proposes conjugate gradient methods that make use of the structure of the Hessian of the objective function of nonlinear least squares problems. The proposed methods are shown to be globally convergent under some assumptions. Finally, some numerical results are given.     

Variational iteration method for solving non-linear partial differential equations

In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.     

Feature extraction algorithms for constrained global optimization II. Batch process scheduling application

The feature extraction algorithms developed in part I of this series are applied to solve nonconvex mixed integer nonlinear programming problems which arise in the optimal scheduling of multipurpose chemical plants. A general formulation of the multipurpose plant scheduling problem is developed which considers the allocation of plant equipment and secondary, limited resources to recipe operations so as to satisfy given production requirements while minimizing cost. Results obtained with a test example involving 135 binary and 922 continuous variables show that the successive refinement strategy is effective in identifying dominant regions of the solution space. Furthermore it is shown that multiple moment based characterization methods are superior to the interval analysis method reported in the literature. Trials using a second, larger nonlinear test problem involving 356 binary and 2402 continuous variables demonstrate that the focused successive refinement strategy is more efficient than a constant resolution strategy which employs genetic algorithm constructions. Although the conventional genetic algorithm can be significantly improved by introducing a heuristic mutation strategy which increases the likelihood of constant feasibility, the successive refinement strategy remains dominant. These studies demonstrate that the feature extraction strategy employing successive refinements and relatively low order moment based region characterization methods, offers an effective approach to solving an important class of large scale MINLP problems with multiple local optima.     

Self-adaptive operator splitting methods for monotone variational inequalities

Summary. Solving a variational inequality problem VI(Ω,F) is equivalent to finding a solution of a system of nonsmooth equations (a hard problem). The Peaceman-Rachford and /or Douglas-Rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems, because they solve the original problem via solving a series of systems of nonlinear smooth equations (a series of easy problems). Although the solution of VI(Ω,F) is invariant under multiplying F by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is a constant in the original operator splitting methods. In general, it is difficult to choose a proper parameter β for individual problems. In this paper, we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates. Exact and inexact forms of the modified method with self-adaptive variable parameter are suggested and proved to be convergent under mild assumptions. Finally, preliminary numerical tests show that the self-adaptive adjustment rule is proper and necessary in practice.     

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

The effective and efficient numerical solution of Riemann‐Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann‐Hilbert problems, the resulting numerical methods have been shown, in practice, to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann‐Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painlevé II equation and the modified Korteweg–de Vries equation to explicitly demonstrate the practical validity of the theory. © 2014 Wiley Periodicals, Inc.     

A new two-parameter family of nonlinear conjugate gradient methods

Conjugate gradient methods are an important class of methods for unconstrained optimization, especially for large-scale problems. Recently, they have been much studied. In this paper, we propose a new two-parameter family of conjugate gradient methods for unconstrained optimization. The two-parameter family of methods not only includes the already existing three practical nonlinear conjugate gradient methods, but has other family of conjugate gradient methods as subfamily. The two-parameter family of methods with the Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the two-parameter family of methods. The numerical results show that this method is efficient for the given test problems. In addition, the methods related to this family are uniformly discussed.