On Problems in the Calculus of Variations in Increasingly Elongated Domains

This paper deals with minimization problems in the calculus of variations set in a sequence of domains, the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. The authors study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions.     

Some Extensions to a Direct Optimization Method

A direct method for the global extremization of a class of integrals, introduced in Refs. 1–3, is generalized to allow for constraints in the form of differential conditions and by considering the so-called infinite-horizon case.     

On a Class of Direct Optimization Problems

Based on an earlier publication (Ref. 1), a coordinate transformation is proposed, which allows the direct global extremization of a class of integrals without the use of comparison methods such as variational or field techniques. This direct method is shown to be applicable to a class of unconstrained optimal control problems. A motivation for the proposed method as well as applications are presented.     

Direct Methods in the Parametric Moving Boundary Variational Problem

In this article, we consider the moving boundary variational problem in a parametric form. By means of the tools of the nonsmooth analysis and exact penalty functions, a new form of necessary conditions for an extremum is obtained. The new conditions make it possible to construct new (“direct”) numerical algorithms. Numerical experiments have demonstrated efficiency of the proposed algorithms and the expediency to further promotion of the approach.     

Modified Two-Point Stepsize Gradient Methods for Unconstrained Optimization

For unconstrained optimization, the two-point stepsize gradient method is preferable over the classical steepest descent method both in theory and in real computations. In this paper we interpret the choice for the stepsize in the two-point stepsize gradient method from the angle of interpolation and propose two modified two-point stepsize gradient methods. The modified methods are globally convergent under some mild assumptions on the objective function. Numerical results are reported, which suggest that improvements have been achieved.     

A New Gradient Method for Ill-Posed Problems

In this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method.     

A New Notion of Conjugacy for Isoperimetric Problems

For problems in the calculus of variations with isoperimetric side constraints, we provide in this paper a set of points whose emptiness, independently of nonsingularity assumptions, is equivalent to the nonnegativity of the second variation along admissible variations. The main objective of introducing a characterization of this condition should be, of course, to obtain a simpler way of verifying it. There are two other sets of points available in the literature, introduced by Loewen and Zheng (1994) and Zeidan (1996), for which this necessary condition implies their emptiness. However, we show that verifying membership of these sets may be more difficult than checking directly if that condition holds. Contrary to this behavior, we prove that the desired objective of characterizing that condition is achieved by means of the set introduced in this paper.     

A New Gradient Method with an Optimal Stepsize Property

The gradient method for the symmetric positive definite linear system is as follows

Steepest Descent, CG, and Iterative Regularization of Ill-Posed Problems

The state of the art iterative method for solving large linear systems is the conjugate gradient (CG) algorithm. Theoretical convergence analysis suggests that CG converges more rapidly than steepest descent. This paper argues that steepest descent may be an attractive alternative to CG when solving linear systems arising from the discretization of ill-posed problems. Specifically, it is shown that, for ill-posed problems, steepest descent has a more stable convergence behavior than CG, which may be explained by the fact that the filter factors for steepest descent behave much less erratically than those for CG. Moreover, it is shown that, with proper preconditioning, the convergence rate of steepest descent is competitive with that of CG.This revised version was published online in October 2005 with corrections to the Cover Date.     

Nonmonotone Globalization Techniques for the Barzilai-Borwein Gradient Method

In this paper we propose new globalization strategies for the Barzilai and Borwein gradient method, based on suitable relaxations of the monotonicity requirements. In particular, we define a class of algorithms that combine nonmonotone watchdog techniques with nonmonotone linesearch rules and we prove the global convergence of these schemes. Then we perform an extensive computational study, which shows the effectiveness of the proposed approach in the solution of large dimensional unconstrained optimization problems.     

核属于H函数类的多维积分方程近似解直接方法的优化

本文我们确定了核属于 H函数类的多维第二类 Fredholm积分方程类在自适直接方法意义下的最优近似解的精确阶估计 ,并给出了最优算法 .     

On the Behavior of the Gradient Norm in the Steepest Descent Method

It is well known that the norm of the gradient may be unreliable as a stopping test in unconstrained optimization, and that it often exhibits oscillations in the course of the optimization. In this paper we present results descibing the properties of the gradient norm for the steepest descent method applied to quadratic objective functions. We also make some general observations that apply to nonlinear problems, relating the gradient norm, the objective function value, and the path generated by the iterates.     

Semicontinuity and Supremal Representation in the Calculus of Variations

We study the weak* lower semicontinuity properties of functionals of the form

A projected‐steepest‐descent potential‐reduction algorithm for convex programming problems

A recent work of Shi (Numer. Linear Algebra Appl. 2002; 9 : 195–203) proposed a hybrid algorithm which combines a primal‐dual potential reduction algorithm with the use of the steepest descent direction of the potential function. The complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. In this paper, we make efforts to further reduce the computational costs. We notice that in order to obtain the steepest descent direction of the potential function, the Hessian matrix of second order partial derivatives of the objective function needs to be computed. To avoid this, we in this paper propose another hybrid algorithm which uses a projected steepest descent direction of the objective function instead of the steepest descent direction of the potential function. The complexity of the original potential reduction algorithm still remains valid but the overall computational cost is further reduced. Our numerical experiments are also reported. Copyright © 2004 John Wiley & Sons, Ltd.     

Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems

A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced. For a broad and important class of DAE systems (semiexplicit systems with algebraic variables of different index), a direct multiple shooting method is developed. The multiple shooting method is based on the discretization of the optimal control problem and its transformation into a finite-dimensional nonlinear programming problem (NLP). Special attention is turned to the mandatory calculation of consistent initial values at the multiple shooting nodes within the iterative solution process of (NLP). Two different methods are proposed. The projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution. An illustrative example completes this article.     

Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

Steepest descent preconditioning for nonlinear GMRES optimization

Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N‐GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, whereas the second employs a predefined small step. A simple global convergence proof is provided for the N‐GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N‐GMRES optimization is also motivated by relating it to standard non‐preconditioned GMRES for linear systems in the case of a standard quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N‐GMRES optimization algorithm is able to very significantly accelerate convergence of stand‐alone steepest descent optimization. Moreover, performance of steepest‐descent preconditioned N‐GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited‐memory Broyden–Fletcher–Goldfarb–Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest‐descent preconditioned N‐GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared with established techniques. In addition, it is argued that the real potential of the N‐GMRES optimization framework lies in the fact that it can make use of problem‐dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N‐GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example. Copyright © 2012 John Wiley & Sons, Ltd.     

A Theoretical and Numerical Comparison of Some Semismooth Algorithms for Complementarity Problems

In this paper we introduce a general line search scheme which easily allows us to define and analyze known and new semismooth algorithms for the solution of nonlinear complementarity problems. We enucleate the basic assumptions that a search direction to be used in the general scheme has to enjoy in order to guarantee global convergence, local superlinear/quadratic convergence or finite convergence. We examine in detail several different semismooth algorithms and compare their theoretical features and their practical behavior on a set of large-scale problems.     

Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.     

Relaxation of an area-like functional for the function $$\frac{x}{|x|}$$

We compute the relaxation