不等式约束优化问题的Hestenes-Powell增广拉格朗日函数的精确性质

增广拉格朗日函数法是用无约束极小化技术求解约束优化问题的一类重要方法.本文对不等式约束优化问题的Hestenes-Powell增广拉格朗日函数(简记为HP-ALF)的精确性质作了详尽讨论.在适当的假设下,建立了原不等式约束优化问题的极小点和HP-ALF在原问题变量空间或者原问题变量空间与乘子变量空间的积空间上的无约束极小点之间的相互对应关系;获得了关于HP-ALF的精确性的许多新结果.本文给出的性质说明HP-ALF是一个连续可微的精确乘子罚函数,且用经典的乘子法可求得不等式约束优化问题的最优解和对应的拉格朗日乘子值.     

Active constrained truncated Newton method for simple-bound optical tomography

In the past, nonlinear unconstrained optimization of the optical imaging problem has focused on Newton-Raphson techniques. Besides requiring expensive computation of the Jacobian, the unconstrained minimization with Tikhonov regularization can pose significant storage problems for large-scale reconstructions, involving a large number of unknowns necessary for realization of optical imaging. We formulate the inverse optical imaging problem as both simple-bound constrained and unconstrained minimization problems in order to illustrate the reduction in computational time and storage associated with constrained image reconstructions. The forward simulator of excitation and generated fluorescence, consisting of the Galerkin finite-element formulation, is used in an inverse algorithm to find the spatial distribution of absorption and lifetime that minimizes the difference between predicted and synthetic frequency-domain measurements. The inverse approach employs the truncated Newton method with trust region and a modification of automatic reverse differentiation to speed the computation of the optimization problem. The reconstruction results confirm that the physically based, constrained minimization with efficient optimization schemes may offer a more logical approach to the large-scale optical imaging problem than unconstrained minimization with regularization.     

Energy-based r-adaptivity: a solution strategy and applications to fracture mechanics

This paper deals with energy based r-adaptivity in finite hyperelastostatics. The focus lies on the development of a numerical solution strategy. Although the concept of improving the accuracy of a finite element solution by minimizing the discrete potential energy with respect to the material node point positions is well-known, the numerical implementation of the underlying minimization problem is difficult. In this paper, energy based r-adaptivity is defined as a minimization problem with inequality constraints. The constraints are introduced to restrict the maximum distortion of the finite element mesh. As a solution strategy for the constrained problem, we use a classical barrier method. Beside the theoretical aspects and the implementation, a numerical experiment is presented. We illustrate the performance of the proposed r-adaptivity in the case of a cracked specimen.     

Using multicriterion optimization for strength design of composite laminates

Using multicriterion optimization methods for weight minimization of failure strength controlled composite structures is described. A composite lay-up design problem for minimizing the number of layers under strength constraints with respect to multiple loading conditions is formulated. The constrained problem is transferred into a sequence of unconstrained problems and solved with an interactive descent method. A typical design cycle that comprises the finite element mesh generator and solver as well as the laminate analysis and optimization modules is illustrated with a numerical example.     

An Evolutionary Lagrange Method For Mixed-Integer Constrained Optimization Problems

This study proposes a method for solving mixed-integer constrained optimization problems using an evolutionary Lagrange method. In this approach, an augmented Lagrange function is used to transform the mixed-integer constrained optimization problem into an unconstrained min—max problem with decision-variable minimization and Lagrange-multiplier maximization. The mixed-integer hybrid differential evolution (MIHDE) is introduced into the evolutionary min—max algorithm to accomplish the implementation of the evolutionary Lagrange method. MIHDE provides a mixed coding to denote genetic representations of teal and integer variables, and a rounding operation is used to guide the genetic evolution of integer variables. To fulfill global convergence, self-adaptation for penalty parameters is involved in the evolutionary min—max algorithm so that small penalty parameters can be used, not affecting the final search results. Some numerical experiments are tested to evacuate the performance of the proposed method. Numerical experiments demonstrate that the proposed method converges to better solutions than the conventional penalty function method     

Application of the MFS to inverse obstacle scattering problems

In this paper, the method of fundamental solutions (MFS) is used to detect the shape, size and location of a scatterer embedded in a host acoustic homogeneous medium from scant measurements of the scattered acoustic pressure in the vicinity of the obstacle. A nonlinear constrained minimization regularized MFS technique is proposed for the numerical solution of the inverse problem in question. The stability of the technique is investigated by inverting measurements contaminated by random noise. The results of several numerical experiments are presented.     

Stable numerical integration of dynamical systems subject to equality state-space constraints

Numerical simulation of constrained dynamical systems is known to exhibit stability problems even when the unconstrained system can be simulated in a stable manner. We show that not the constraints themselves, but the transformation of the continuous set of equations to a discrete set of equations is the true source of the stability problem. A new theory is presented that allows for stable numerical integration of constrained dynamical systems. The derived numerical methods are robust with respect to errors in the initial conditions and stable with respect to errors made during the integration process. As a consequence, perturbations in the initial conditions are allowed. The new theory is extended to the case of constrained mechanical systems. Some numerical results obtained when implementing the numerical method here developed are shown.     

Maximum stiffness design of laminated composite plates via a constrained global optimization approach

The optimal lamination arrangements of laminated composite plates with maximum stiffness subject to side constraints are investigated via a constrained multi-start global optimization approach. In the optimal design process, the deformation analysis of laminated composite plates is accomplished by utilizing a shear deformable laminated composite finite element and the optimal design problem, which has been converted into an unconstrained minimization problem via the general augmented Lagrangian method, is solved by utilizing the proposed unconstrained multi-start global optimization technique to determine the optimal fiber angles and layer group thicknesses of the laminated composite plates for attaining maximum stiffness and simultaneously satisfying the imposed side constraints. The feasibility of the proposed constrained multi-start global optimization algorithm is validated by means of a simple but representative example and its applications are demonstrated by means of a number of examples on the maximum stiffness design of symmetrically laminated composite plates. The effects of length-to-thickness ratio, aspect ratio, and number of layer groups upon the optimum fiber angles and layer group thicknesses of the plates are investigated.     

Three-dimensional unconstrained and constrained image-reconstruction techniques applied to fluorescence, frequency-domain photon migration

The development of near-infrared (NIR) optical imaging for biomedical optical imaging is hampered by the computational intensiveness of large-scale three-dimensional (3-D) image reconstruction and the potential lack of endogenous contrast for detection of relevant tissue features. In this contribution the inverse optical imaging problem is formulated in three dimensions in a noncompressive geometry as a simple-bound constrained minimization problem in order to recover the interior fluorescence properties of exogenous contrast agent from frequency-domain photon migration measurements at the boundary. The solution of the forward optical diffusion problem for the frustum shape containing fluorescence inclusions of 10:1 contrast is accomplished by use of the Galerkin finite-element formulation. The inverse approach employs the truncated Newton method with trust region and a modification of automatic reverse differentiation to speed the computation of the optimization problem. The image-reconstruction results confirm that the constrained minimization may offer a more logical approach for the 3-D optical imaging problem than unconstrained optimization.     

OPTIMAL DESIGN OF TORISPHERICAL PRESSURE VESSEL END CLOSURES

The problem of minimizing the maximum shearing stresses occurring in torispherical heads of two and three radii is presented. A system model is formulated from the design variables describing the head shapes and this model is optimized by using a penalty function procedure combined with a variable metric unconstrained minimization technique. Various examples of the algorithm are illustrated and the procedure is generally applicable as a design tool.     

The Estimation of Parameters in Nonlinear,Implicit Models

The estimation of parameters in nonlinear algebraic models is considered for a general class of problems. Included are problems where both the independent and dependent variables are subject to error and problems where the model is algebraically implicit. The maximum likelihood principle leads to an equality constrained minimization problem. An algorithm to achieve this minimization is obtained through the use of Lagrange multipliers.     

Advanced modeling techniques for micromagnetic systems

We present a review of micromagnetic and magnetotransport modeling methods which go beyond the standard model. We first give a brief overview of the standard micromagnetic model, which for (i) the steady-state (equilibrium) solution is based on the minimization of the free energy functional, and for (ii) the dynamical solution, relies on the numerical solution of the Landau-Lifshitz-Gilbert (LLG) equation. We present three complements to the standard model, i.e., (i) magnetotransport calculations based on ohmic conduction in the presence of the anisotropic magnetoresistance (AMR) effect, (ii) magnetotransport calculations based on spin-dependent tunneling in the presence of single charge tunneling (Coulomb blockade) effect, and (iii) stochastic micromagnetics, which incorporates the effects of thermal fluctuations via a white-noise thermal field in the LLG equation. All three complements are of practical importance: (i) magnetotransport model either in the ohmic or tunneling transport regimes, enables the conversion of the micromagnetic results to the measurable quantity of magnetoresistance ratio, while (ii) stochastic modeling is essential as the dimensions of the micromagnetic system reduces to the deep submicron regime and approaches the superparamagnetic limit.     

Generalized co-ordinate partitioning in static equilibrium analysis of large-scale mechanical systems

This paper presents a computer-based method for automatic formulation and efficient numerical solution of static equilibrium equations for nonlinear constrained mechanical systems with conservative forces. Nonlinear holonomic constraint equations and a potential energy function are written in terms of a maximal set of Cartesian generalized co-ordinates. A stable static equilibrium configuration is found by minimizing the potential energy of the system, subject to the kinematic constraint equations, i.e. constrained optimization. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent and independent co-ordinates and construct an influence coefficient matrix that relates variations in dependent and independent co-ordinates. This information is employed to convert the constrained optimization problem to an unconstrained optimization problem. A simple example is presented to illustrate the method. An algorithm that may be used in analysis of large-scale systems is presented.     

Methodologies for the optimal design of parallel manipulators

In this paper the problem of determining a manipulator design so that its workspace corresponds to a prescribed workspace is considered. Two different strategies, resulting in two different types of optimization problem are considered. The first strategy attempts to obtain a good overall approximation to the prescribed workspace and results in an unconstrained optimization problem. The second strategy entails designing a manipulator so that its workspace fully encloses the prescribed workspace and results in a constrained optimization problem. Two specific formulations of the constrained problem are proposed. The first constrained problem simply aims to fit the manipulator workspace as exactly as possible to the prescribed workspace, while still ensuring that the prescribed workspace is fully enclosed. The second constrained optimization formulation is used to design a manipulator, the workspace of which fully encloses the prescribed workspace, but which is also well‐conditioned throughout the workspace with respect to some performance measure. The particular manipulator used to illustrate and evaluate these formulations is a simple 2‐dof planar parallel manipulator, and the final formulation is also applied to a 3‐dof planar parallel manipulator. Although the manipulators studied here are simple, the objective of this study is to obtain a robust numerical methodology which can be extended to more practical and complex manipulators. Copyright © 2003 John Wiley & Sons, Ltd.     

Novel magnetic tips developed for the switching magnetization magnetic force microscopy

Using micromagnetic calculations we search for optimal magnetic properties of novel magnetic tips to be used for a Switching Magnetization Magnetic Force Microscopy (SM-MFM), a novel technique based on two-pass scanning with reversed tip magnetization. Within the technique the sum of two scans images local atomic forces and their difference maps the local magnetic forces. The tip magnetization is switched during the scanning by a small magnetic field. The technology of novel low-coercitive magnetic tips is proposed. For best performance the tips must exhibit low magnetic moment, low switching field, and single-domain state at remanence. Such tips are equipped with Permalloy objects of a precise shape that are defined on their tilted sides. We calculate switching fields of such tips by solving the micromagnetic problem to find the optimum shape and dimensions of the Permalloy objects located on the tips. Among them, hexagon was found as the best shape for the tips.     

Hysteresis properties of the two-dimensional structure of domain walls in magnetically triaxial films with a (110)-oriented surface plane

Structural transformations of a two-vortex asymmetric domain wall in a magnetically triaxial film with Goss-oriented (110) surface and cubic anisotropy have been studied by numerical micromagnetic simulation. It is established that transformations of the domain wall structure exhibit significant hysteresis that is accompanied by the related hysteresis of magnetization reversal in a field applied perpendicular to the easy magnetization axis.     

Magnetization reversal process in elliptical Permalloy nanodots

Using the Vectorial Magneto-Optic Kerr Effect (V-MOKE) and numerical simulations, we have investigated the magnetization reversal process in arrays of 15-nm-thick Permalloy nanometer-scale dots, having elliptical shape and eccentricity, varying from 1 to 2.5. V-MOKE hysteresis loops revealed that the magnetization reversal is incoherent for elements with eccentricity of 1 and 1.5, while it becomes an almost perfect coherent magnetization rotation for elements with eccentricity equal to 2 and 2.5. In the latter case, the V-MOKE loops agree well with those predicted by the Stoner–Wohlfarth model for particles with uniaxial magnetic anisotropy. We were able to reproduce the V-MOKE results with micromagnetic simulations, gaining a deeper insight into the magnetic configurations that develop during the reversal process.     

Genetic algorithm with adaptive and dynamic penalty functions for the selection of cleaner production measures: a constrained optimization problem

This paper presents a new approach of genetic algorithm (GA) to solve the constrained optimization problem. In a constrained optimization problem, feasible and infeasible regions occupy the search space. The infeasible regions consist of the solutions that violate the constraint. Oftentimes classical genetic operators generate infeasible or invalid chromosomes. This situation takes a turn for the worse when infeasible chromosomes alone occupy the whole population. To address this problem, dynamic and adaptive penalty functions are proposed for the GA search process. This is a novel strategy because it will attempt to transform the constrained problem into an unconstrained problem by penalizing the GA fitness function dynamically and adaptively. New equations describing these functions are presented and tested. The effects of the proposed functions developed have been investigated and tested using different GA parameters such as mutation and crossover. Comparisons of the performance of the proposed adaptive and dynamic penalty functions with traditional static penalty functions are presented. The result from the experiments show that the proposed functions developed are more accurate, efficient, robust and easy to implement. The algorithms developed in this research can be applied to evaluate environmental impacts from process operations.     

The role of magnetostatic interactions in the phenomenon of thermal magnetization of highly anisotropic fine-grained magnetic alloys

The phenomenon of thermal magnetization of fine-grained alloys has been studied by micromagnetic modeling in a multilayer stochastic system. It is established that thermal magnetization in this system is manifested in a certain temperature interval. The mechanism of this effect is different from that stipulated by the Zaitsev-Lileev model.