On Noisy Extensions of Nonholonomic Constraints

We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange–d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. Next, for an arbitrary nonholonomic system, we consider two different ways of including stochasticity in the constraints. We show that when the noise preserves the linearity of the constraints, then energy is preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. We study in detail a class of Lagrangian mechanical systems on semidirect products of Lie groups, with “rolling ball type” constraints. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.     

A cutting plane method for solving harvest scheduling models with area restrictions

We describe a cutting plane algorithm for an integer programming problem that arises in forest harvest scheduling. Spatial harvest scheduling models optimize the binary decisions of cutting or not cutting forest management units in different time period subject to logistical, economic and environmental restrictions. One of the most common constraints requires that the contiguous size of harvest openings (i.e., clear-cuts) cannot exceed an area threshold in any given time period or over a set of periods called green-up. These so-called adjacency or green-up constraints make the harvest scheduling problem combinatorial in nature and very hard to solve. Our proposed cutting plane algorithm starts with a model without area restrictions and adds constraints only if a violation occurs during optimization. Since violations are less likely if the threshold area is large, the number of constraints is kept to a minimum. The utility of the approach is illustrated by an application, where the landowner needs to assess the cost of forest certification that involves clear-cut size restrictions stricter than what is required by law. We run empirical tests and find that the new method performs best when existing models fail: when the number of units is high or the allowable clear-cut size is large relative to average unit size. Since this scenario is the norm rather than the exception in forestry, we suggest that timber industries would greatly benefit from the method. In conclusion, we describe a series of potential applications beyond forestry.     

A NURBS-based Multi-Material Interpolation (N-MMI) for isogeometric topology optimization of structures

In this paper, the main intention is to propose a new Multi-material Isogeometric Topology Optimization (M-ITO) method for the optimization of multiple materials distribution, where an improved Multi-Material Interpolation model is developed using Non-Uniform Rational B-splines (NURBS), namely the “NURBS-based Multi-Material Interpolation (N-MMI)”. In the N-MMI model, three key components are involved: (1) multiple Fields of Design Variables (DVFs): NURBS basis functions with control design variables are applied to construct DVFs with the sufficient smoothness and continuity; (2) multiple Fields of Topology Variables (TVFs): each TVF is expressed by a combination of all DVFs to present the layout of a distinct material in the design domain; (3) Multi-material interpolation: the material property at each point is equal to the summation of all TVFs interpolated with constitutive elastic properties. DVFs and TVFs are in the decoupled expression and optimized in a serial evolving mechanism. This feature can ensure the constraint functions are separate and linear with respect to TVFs, which can be beneficial to lower the complexity of numerical computations and eliminate numerical troubles in the multi-material optimization. Two kinds of multi-material topology optimization problems are discussed, i.e., one with multiple volume constraints and the other with the total mass constraint. Finally, several numerical examples in 2D and 3D are provided to demonstrate the effectiveness of the M-ITO method.     

Self-adaptive velocity particle swarm optimization for solving constrained optimization problems

Particle swarm optimization (PSO) is originally developed as an unconstrained optimization technique, therefore lacks an explicit mechanism for handling constraints. When solving constrained optimization problems (COPs) with PSO, the existing research mainly focuses on how to handle constraints, and the impact of constraints on the inherent search mechanism of PSO has been scarcely explored. Motivated by this fact, in this paper we mainly investigate how to utilize the impact of constraints (or the knowledge about the feasible region) to improve the optimization ability of the particles. Based on these investigations, we present a modified PSO, called self-adaptive velocity particle swarm optimization (SAVPSO), for solving COPs. To handle constraints, in SAVPSO we adopt our recently proposed dynamic-objective constraint-handling method (DOCHM), which is essentially a constituent part of the inherent search mechanism of the integrated SAVPSO, i.e., DOCHM + SAVPSO. The performance of the integrated SAVPSO is tested on a well-known benchmark suite and the experimental results show that appropriately utilizing the knowledge about the feasible region can substantially improve the performance of the underlying algorithm in solving COPs.     

Using artificial reaction force to design compliant mechanism with multiple equality displacement constraints

Monolithic compliant mechanisms are elastic workpieces which transmit force and displacement from an input position to an output position. Continuum topology optimization is suitable to generate the optimized topology, shape and size of such compliant mechanisms. The optimization strategy for a single input single output compliant mechanism under volume constraint is known to be best implemented using an optimality criteria or similar mathematical programming method. In this standard form, the method appears unsuitable for the design of compliant mechanisms which are subject to multiple outputs and multiple constraints. Therefore an optimization model that is subject to multiple design constraints is required. With regard to the design problem of compliant mechanisms subject to multiple equality displacement constraints and an area constraint, we here present a unified sensitivity analysis procedure based on artificial reaction forces, in which the key idea is built upon the Lagrange multiplier method. Because the resultant sensitivity expression obtained by this procedure already compromises the effects of all the equality displacement constraints, a simple optimization method, such as the optimality criteria method, can then be used to implement an area constraint. Mesh adaptation and anisotropic filtering method are used to obtain clearly defined monolithic compliant mechanisms without obvious hinges. Numerical examples in 2D and 3D based on linear small deformation analysis are presented to illustrate the success of the method.     

LMI优化的一种原对偶中心路径算法

系统和控制理论中许多重要的问题,都可转化为具有线性目标函数、线性矩阵不等式约束的LMI优化问题,从而使其在数值上易于求解.本文给出一种求解LMI优化问题的原对偶中心路径算法,该算法利用牛顿方法求解中心路径方程得到牛顿系统,并将该牛顿系统对称化以避免得到非对称化的搜索方向.文章详细分析了算法的计算复杂性.     

An adjoint-based pressure correction scheme

Incompressible flows are solenoidal, i. e. the divergence of the flow field is zero. This is an algebraic constraint on the solution in time. The pressure has to be determined, so that the constraint is fulfilled. To calculate the pressure often a Poisson equation is derived, which is then solved by an iterative method. Instead of this the constraint is here formulated as an optimization problem. The objective functional is taken as the square of the norm of the divergence. A gradient based optimization is performed to calculate the pressure in every time step of the simulation. By this an alternative iterative scheme is derived. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized convexity in non-regular programming problems with inequality-type constraints

Convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush-Kuhn-Tucker-invexity (hereafter KKT-invexity), that is both necessary and sufficient to obtain Karush-Kuhn-Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by Izmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work, we generalize Martin's result to non-regular problems by defining an analogous concept, the 2-KKT-invexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by Izmailov.     

Dynamic Global Constraints in Backtracking Based Environments

Global constraints provide strong filtering algorithms to reduce the search space when solving large combinatorial problems. In this paper we propose to make the global constraints dynamic, i.e., to allow extending the set of constrained variables during search. We describe a generic dynamisation technique for an arbitrary monotonic global constraint and we compare it with the semantic-based dynamisation for the alldifferent constraint. At the end we sketch a dynamisation technique for non-monotonic global constraints. A comparison with existing methods to model dynamic problems is given as well.     

A combined constraint handling framework: an empirical study

This paper presents a new combined constraint handling framework (CCHF) for solving constrained optimization problems (COPs). The framework combines promising aspects of different constraint handling techniques (CHTs) in different situations with consideration of problem characteristics. In order to realize the framework, the features of two popular used CHTs (i.e., Deb’s feasibility-based rule and multi-objective optimization technique) are firstly studied based on their relationship with penalty function method. And then, a general relationship between problem characteristics and CHTs in different situations (i.e., infeasible situation, semi-feasible situation, and feasible situation) is empirically obtained. Finally, CCHF is proposed based on the corresponding relationship. Also, for the first time, this paper demonstrates that multi-objective optimization technique essentially can be expressed in the form of penalty function method. As CCHF combines promising aspects of different CHTs, it shows good performance on the 22 well-known benchmark test functions. In general, it is comparable to the other four differential evolution-based approaches and five dynamic or ensemble state-of-the-art approaches for constrained optimization.     

A dynamic programming approach to the optimization of elastic trusses

The optimal design of elastic trusses is discussed from a dynamic programming point of view. Emphasis is placed on minimum volume design of statically determinate trusses with displacement and stress constraints in the discrete case, i.e., when the cross-sectional areas of the bars are available from a discrete set of values. This, a design constraint usually very difficult to handle with standard nonlinear programming algorithms, is naturally incorporated in the present formulation. In addition, the functional equation approach is shown to furnish a direct solution to the problem of determining a design, among all possible ones satisfying certain volume and displacement constraints, for which the maximum stress is a minimum. A successive approximation approach is briefly indicated as an extension of the method to solve statically indeterminate trusses. Finally, several numerical examples are presented and the main features of the methods are briefly exposed.     

具有动应力、位移和稳定性约束的离散变量桁架结构布局优化设计算法

给出了在动应力、动位移和动稳定约束下离散变量结构布局优化设计问题的数学模型,用“拟静力”算法,将具有动应力约束、动位移约束和动稳定约束的离散变量结构布局优化设计问题化为静应力、静位移和静稳定约束的优化问题,然后利用两级优化算法求解该模型．优化过程由两级组成,拓扑级优化和形状级优化．在每一级,都使用了综合算法,并且在搜索过程中都根据两类设计变量的相对差商值进行搜索．对包含稳定约束和不包含稳定约束的优化结果做了比较,结果显示稳定性约束对优化结果产生较大的影响．     

A Cutting Plane Algorithm for Linear Reverse Convex Programs

In this paper, global optimization of linear programs with an additional reverse convex constraint is considered. This type of problem arises in many applications such as engineering design, communications networks, and many management decision support systems with budget constraints and economies-of-scale. The main difficulty with this type of problem is the presence of the complicated reverse convex constraint, which destroys the convexity and possibly the connectivity of the feasible region, putting the problem in a class of difficult and mathematically intractable problems. We present a cutting plane method within the scope of a branch-and-bound scheme that efficiently partitions the polytope associated with the linear constraints and systematically fathoms these portions through the use of the bounds. An upper bound and a lower bound for the optimal value is found and improved at each iteration. The algorithm terminates when all the generated subdivisions have been fathomed.     

Tutorial on Surrogate Constraint Approaches for Optimization in Graphs

Surrogate constraint methods have been embedded in a variety of mathematical programming applications over the past thirty years, yet their potential uses and underlying principles remain incompletely understood by a large segment of the optimization community. In a number of significant domains of combinatorial optimization, researchers have produced solution strategies without recognizing that they can be derived as special instances of surrogate constraint methods. Once the connection to surrogate constraint ideas is exposed, additional ways to exploit this framework become visible, frequently offering opportunities for improvement.We provide a tutorial on surrogate constraint approaches for optimization in graphs, illustrating the key ideas by reference to independent set and graph coloring problems, including constructions for weighted independent sets which have applications to associated covering and weighted maximum clique problems. In these settings, the surrogate constraints can be generated relative to well-known packing and covering formulations that are convenient for exposing key notions. The surrogate constraint approaches yield widely used heuristics for identifying independent sets as simple special cases, and also afford previously unidentified heuristics that have greater power in these settings. Our tutorial also shows how the use of surrogate constraints can be placed within the context of vocabulary building strategies for independent set and coloring problems, providing a framework for applying surrogate constraints that can be used in other applications.At a higher level, we show how to make use of surrogate constraint information, together with specialized algorithms for solving associated sub-problems, to obtain stronger objective function bounds and improved choice rules for heuristic or exact methods. The theorems that support these developments yield further strategies for exploiting surrogate constraint relaxations, both in graph optimization and integer programming generally.     

Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.     

基于气动载荷与叶片厚度分布的叶栅气动设计方法

建立一种以载荷与叶片厚度分布为约束的叶栅气动设计计算方法,约束条件体现了气动特性与强度两方面的要求,这些约束及其它所有边界条件,都包含在相应的变分原理的驻值条件中。变分原理以周角函数为泛函的未知函数,周角函数定义在由流线坐标与流函数坐标(周向)构成的映象面上。在映象面上,求解域——叶栅通道——化作一个矩形,叶片型线映射为一条水平直线,从而避免了叶片外形未知的困难。利用有限元方法建立了计算程序,算例显示这种方法能有效地满足对叶片型线的设计要求,而且迭代计算具有良好的收敛性。     

关于铁木森柯梁动力优化的讨论

本文讨论受到多频率约束的铁木森柯梁和尤拉梁的最小重量设计问题.以对称的简支梁为例,本文揭示了铁木森柯梁的异常特征:如果在梁的中部适当地构造很高的一个薄条,相应于第一对称振形的频率可以高于反对称振型的频率;受到二组不同频率约束的铁木森柯梁很可能有同样的最小重量.这些异常特征说明,为了得到一个良态的问题,有必要将最大横断面积约束包括到问题提法中.     

An optimal choice problem for a set of checking procedures

An item having a known initial failure probability is to be controlled by some out of a finite set of possible checks. Every check costs a certain amount of money, and budget constraints must be met. A check is characterized by the probabilities of three events: (i) letting a workable item pass, (ii) overlooking a failure if one is present, (iii) introducing a failure into a workable item. We show how any subset of checks to be employed is ordered optimally, and how the optimal subsequence of checks depends on the initial failure probability and oil the budget constraint.     

求解双层规划的多目标布谷鸟算法

双层规划是一类具有主从递阶结构的优化问题,属于NP-hard范畴。本文利用KKT条件将双层规划问题转化为等价的单层约束规划问题,通过约束处理技术进一步转化为带偏好双目标无约束优化问题,提出多目标布谷鸟算法求解策略。该算法采用Pareto支配和ε-个体比较准则,充分利用种群中优秀不可行解的信息指导搜索过程;设置外部档案集存储迭代过程中的优秀个体并通过高斯扰动改善外部档案集的质量,周期性替换群体中的劣势个体,引导种群不断向可行域或最优解逼近。数值实验及其参数分析验证了算法的有效性。     

A class of generalized variable penalty methods for nonlinear programming

A class of generalized variable penalty formulations for solving nonlinear programming problems is presented. The method poses a sequence of unconstrained optimization problems with mechanisms to control the quality of the approximation for the Hessian matrix, which is expressed in terms of the constraint functions and their first derivatives. The unconstrained problems are solved using a modified Newton's algorithm. The method is particularly applicable to solution techniques where an approximate analysis step has to be used (e.g., constraint approximations, etc.), which often results in the violation of the constraints. The generalized penalty formulation contains two floating parameters, which are used to meet the penalty requirements and to control the errors in the approximation of the Hessian matrix. A third parameter is used to vary the class of standard barrier or quasibarrier functions, forming a branch of the variable penalty formulation. Several possibilities for choosing such floating parameters are discussed. The numerical effectiveness of this algorithm is demonstrated on a relatively large set of test examples.The author is thankful for the constructive suggestions of the referees.