Probabilistic nonconvex constrained optimization with fixed number of function evaluations

A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.     

Probabilistic learning on manifolds (PLoM) with partition

The probabilistic learning on manifolds (PLoM) introduced in 2016 has solved difficult supervised problems for the “small data” limit where the number N of points in the training set is small. Many extensions have since been proposed, making it possible to deal with increasingly complex cases. However, the performance limit has been observed and explained for applications for which N is very small and for which the dimension of the diffusion-map basis is close to N. For these cases, we propose a novel extension based on the introduction of a partition in independent random vectors. We take advantage of this development to present improvements of the PLoM such as a simplified algorithm for constructing the diffusion-map basis and a new mathematical result for quantifying the concentration of the probability measure in terms of a probability upper bound. The analysis of the efficiency of this extension is presented through two applications.     

A pegging algorithm for separable continuous nonlinear knapsack problems with box constraints

This article proposes an efficient pegging algorithm for solving separable continuous nonlinear knapsack problems with box constraints. A well-known pegging algorithm for solving this problem is the Bitran–Hax algorithm, a preferred choice for large-scale problems. However, at each iteration, it must calculate an optimal dual variable and update all free primal variables, which is time consuming. The proposed algorithm checks the box constraints implicitly using the bounds on the Lagrange multiplier without explicitly calculating primal variables at each iteration as well as updating the dual solution in a more efficient manner. Results of computational experiments have shown that the proposed algorithm consistently outperforms the Bitran–Hax in all baseline testing and two real-time application models. The proposed algorithm shows significant potential for many other mathematical models in real-world applications with straightforward extensions.     

含摩擦柱铰链平面多体系统动力学的建模和数值方法

以含摩擦柱铰链平面多体系统为研究对象,建立其动力学方程并给出相应的数值计算方法。首先,建立了含摩擦转动柱铰链的力学模型。在此基础上,应用第一类Lagrange 方程给出了该类系统的动力学方程,将Lagrange乘子与柱铰链的法向约束力建立了对应关系,并给出了柱铰链摩擦力的广义力。由于摩擦力的存在,使得该方程是关于Lagrange 乘子的分段连续的非线性代数方程组,该文对此采用混合算法:对于连续段(物体相对转动的角速度不为零时),采用拟牛顿算法和龙格-库塔法求解方程;在不连续点(物体相对转动的角速度为零时),通过粒子群算法(PSO)、试算法和龙格-库塔法求解方程,克服了方程在不连续处Lagrange 乘子(法向约束力)的初值不易选取的困难。最后,通过算例说明了该算法的有效性和可行性。     

A localized version of the method of Lagrange multipliers and its applications

This paper describes a novel version of the method of Lagrange multipliers for an improved modeling of multi-point constraints that emanate from contact-impact problems, partitioned structural analysis using parallel computers, and structural inverse problems. It is shown that the classical method of Lagrange multipliers can lead to a non-unique set of constraint conditions for the modeling of interfaces involving more than two or multi-point substructural interface nodes. The proposed version of the method of Lagrange multipliers leads not only to unique construction of constraints but also encounters no singularity in modeling an arbitrary number of multi-point constraints. An important utilization of the present method is in the regularized modeling of interfaces whose rigidities are radically different from one to another. The present approach is demonstrated via several examples for its simplicity in modeling constraints, ease of implementation and computational advantages. Received November 1998     

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.     

Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers

The use of Cartesian meshes independent of the geometry has some advantages over the traditional meshes used in the finite element method. The main advantage is that their use together with an appropriate hierarchical data structure reduces the computational cost of the finite element analysis. This improvement is based on the substitution of the traditional mesh generation process by an optimized procedure for intersecting the Cartesian mesh with the boundary of the domain and the use efficient solvers based on the hierarchical data structure. One major difficulty associated to the use of Cartesian grids is the fact that the mesh nodes do not, in general, lie over the boundary of the domain, increasing the difficulty to impose Dirichlet boundary conditions. In this paper, Dirichlet boundary conditions are imposed by means of the Lagrange multipliers technique. A new functional has been added to the initial formulation of the problem that has the effect of stabilizing the problem. The technique here presented allows for a simple definition of the Lagrange multipliers field that even allow us to directly condense the degrees of freedom of the Lagrange multipliers at element level. Copyright © 2014 John Wiley & Sons, Ltd.     

An easily implementable hierarchical heuristicfor a two-echelon spare parts distribution system

This paper addresses a two-echelon spare parts stocking and distribution system consisting of a central Distribution Center (DC) and regional facilities. Because the primary purpose for holding inventory is to provide timely repairs of customer′s equipment, we set as our objective to minimize total inventory investment subject to constraints on the delay due to parts outages. We decompose the resulting problem by level and by facility. By simplifying the expressions for the delay constraints and applying previously developed heuristics for the single-level problem [1], we are able to derive closed-form expressions for the inventory control parameters. We then develop a search algorithm (on DC fill rate) to approximate the parameters (Lagrange multipliers) in the closed-form expressions. Numerical comparisons against an analytic lower bound and, for small problems, exact solutions show the approximation to be quite accurate. We also found that it outperforms methods currently in use by the firm that motivated this work. Finally, because it yields closed-form expressions for inventory control parameters and the parameters are only updated periodically, the policy is “easily implementable” once suitable Lagrange multipliers have been computed.     

基于三场变分原理的对偶mortar有限元法

通过引入独立媒介面，将mortar有限元法由二场变分原理推广到三场变分原理。通过采用满足双正交性条件的对偶基函数离散Lagrange乘子空间，实现了Lagrange乘子的凝聚，由此提出了基于三场变分原理的对偶mortar有限元法。提出的新方法同时解决了常规mortar元的约束交叉、主从偏见及求解效率等问题。自主编制了相应的计算程序，并采用两个三维数值算例对新方法进行了验证。研究结果表明:基于三场变分原理的对偶mortar方法对界面连续性条件的求解精度高，可有效用于含约束交叉的非协调网格计算，所支持的复杂子区域划分使得有限元分析更为灵活。     

Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization

A two‐dimensional, finite deformation frictional contact formulation with Coulomb's law is presented. The approach considers multibody contact and is based on a mortar formulation. The enforcement of contact constraints is realized with dual Lagrange multipliers. These alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase. Friction kinematic variables are formulated in an objective way and enter non‐smooth complementarity functions for expressing the contact constraints. An active set strategy is derived by applying a semi‐smooth Newton method, which treats contact nonlinearities, material and geometrical nonlinearities in one single iterative scheme. By further carrying out a consistent linearization for both normal and frictional contact forces and constraints, a robust and highly efficient algorithm for linear and higher‐order (quadratic) interpolation is achieved. Efficiency of the proposed method and quality of results are demonstrated in several examples. Copyright © 2010 John Wiley & Sons, Ltd.     

A constructive approach to nonlocal variational mechanics

A novel approach to nonlocal variational problems is presented using Balakrishnan's epsilon technique. The minimum conditions are derived for the variational problem bearing restricted implicit nonlocality, and for the general problem with nonlocal constraints. The epsilon technique set up provides a simple direct method for constructing the extended Lagrangian function, and yields a constructive method for introducing the Lagrange multipliers.     

Prognostic Kalman Filter Based Bayesian Learning Model for Data Accuracy Prediction

Data is always a crucial issue of concern especially during its prediction and computation in digital revolution. This paper exactly helps in providing efficient learning mechanism for accurate predictability and reducing redundant data communication. It also discusses the Bayesian analysis that finds the conditional probability of at least two parametric based predictions for the data. The paper presents a method for improving the performance of Bayesian classification using the combination of Kalman Filter and K-means. The method is applied on a small dataset just for establishing the fact that the proposed algorithm can reduce the time for computing the clusters from data. The proposed Bayesian learning probabilistic model is used to check the statistical noise and other inaccuracies using unknown variables. This scenario is being implemented using efficient machine learning algorithm to perpetuate the Bayesian probabilistic approach. It also demonstrates the generative function for Kalman-filer based prediction model and its observations. This paper implements the algorithm using open source platform of Python and efficiently integrates all different modules to piece of code via Common Platform Enumeration (CPE) for Python.     

Processing nonlinear multipoint constraints in the finite element method

This paper reviews and evaluates two methods for processing nonlinear constraints. Both are based on the Lagrange multiplier technique. However, one of them, the constraint forces method, is often more efficient than the commonly used method of elimination. The two methods are evaluated by applying them to an example problem and comparing corresponding solution times. Guidelines are given for selecting the more efficient method for a given constraint problem.     

A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method

NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formulation proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point problem simultaneously for displacements and Lagrange multipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the contact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization provides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations.     

An Artificial Intelligence Approach for Solving Stochastic Transportation Problems

Recent years witness a great deal of interest in artificial intelligence (AI) tools in the area of optimization. AI has developed a large number of tools to solve the most difficult search-and-optimization problems in computer science and operations research. Indeed, metaheuristic-based algorithms are a sub-field of AI. This study presents the use of the metaheuristic algorithm, that is, water cycle algorithm (WCA), in the transportation problem. A stochastic transportation problem is considered in which the parameters supply and demand are considered as random variables that follow the Weibull distribution. Since the parameters are stochastic, the corresponding constraints are probabilistic. They are converted into deterministic constraints using the stochastic programming approach. In this study, we propose evolutionary algorithms to handle the difficulties of the complex high-dimensional optimization problems. WCA is influenced by the water cycle process of how streams and rivers flow toward the sea (optimal solution). WCA is applied to the stochastic transportation problem, and obtained results are compared with that of the new metaheuristic optimization algorithm, namely the neural network algorithm which is inspired by the biological nervous system. It is concluded that WCA presents better results when compared with the neural network algorithm.     

Constraint relationships in linear and nonlinear finite element analyses

Finite element analysis of engineering structures commonly requires the use of inter-nodal displacement constraints. Current algorithms for this facility—elimination, Lagrange multipliers and penalty function methods—are briefly reviewed. An alternative approach is proposed which avoids some of the problems of these methods. This technique uses solution of the unconstrained stiffness equation with an extended number of right-hand sides. The resulting solutions are then combined to satisfy the constraints. This method has application in the synthesis of sub-structures with incompatible boundary variables. It is particularly efficient when the constraint affects many variables but the kinematics of the constraint can be expressed in terms of few independent variables, e.g. rigid body motion of part of the structure. In common with the Lagrange method, it produces data for use in a nonlinear analysis with constraints. The formulation of ‘rigid body’ constraints in geometric nonlinearity is detailed and demonstrated.     

On bounded approximations of periodicity for computational homogenization of Stokes flow in porous media

By separation of scales and the homogenization of a flow through porous media, a two‐scale problem arises where a Darcy‐type flow is present on the macroscale and a Stokes flow on the subscale. In this paper, the problem is given as the minimization of a potential. Additional constraints imposing periodicity in a weak sense are added using Lagrange multipliers. In particular, the upper and lower energy bounds for the corresponding strongly periodic problem are produced, quantifying the accuracy of the weakly periodic boundary conditions. A numerical example demonstrates the evaluation of energy bounds and the performance of weakly periodic boundary conditions on a representative volume element. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd     

SHAPE OPTIMIZATION OF A 2D BODY SUBJECTED TO SEVERAL LOADING CONDITIONS

An algorithm for shape optimization based on simultaneous solution of the equations and inequalities arising from Kuhn-Tucker necessary conditions is presented. Regular triangular FE assembly is proposed. Element vertices are associated with design variables directly or through spline parameters defining the boundary of the optimized body. This way, during the iteration procedure, FE assembly is automatically remeshed together with the motion of the optimized boundary. Multiple loading conditions are represented in the problem as equality conditions in the form of a set of equilibrium equations for each loading condition separately. From the necessary condition equations an additional, important relation between cost function, Lagrange multipliers associated with inequality constraints and their limit values is derived. The algorithm combines standard professional FEM programs with an optimizer proposed in the paper which is illustrated with shape optimization of several 2D bodies. The proposed approach is theoretically rigorous and relatively simple for practical applications, and allows considerations sensitivities, adjoint systems and constraints linearization to be avoided.     

Lagrange constraints for transient finite element surface contact

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.     

A bidirectional evolutionary structural optimization algorithm for mass minimization with multiple structural constraints

A bidirectional evolutionary structural optimization algorithm is presented, which employs integer linear programming to compute optimal solutions to topology optimization problems with the objective of mass minimization. The objective and constraint functions are linearized using Taylor's first-order approximation, thereby allowing the method to handle all types of constraints without using Lagrange multipliers or sensitivity thresholds. A relaxation of the constraint targets is performed such that only small changes in topology are allowed during a single update, thus ensuring the existence of feasible solutions. A variety of problems are solved, demonstrating the ability of the method to easily handle a number of structural constraints, including compliance, stress, buckling, frequency, and displacement. This is followed by an example with multiple structural constraints and, finally, the method is demonstrated on a wing-box, showing that topology optimization for mass minimization of real-world structures can be considered using the proposed methodology.