Numerical comparison between two cost functions in contact shape optimization

A finite element approach to shape optimization in a 2D frictionless contact problem for two different cost functions is presented in this work. The goal is to find an appropriate shape for the contact boundary, performing an almost constant contact-stress distribution. The whole formulation, including the mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by a direct modification of the geometry throughB-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data. Augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization) are used to obtain numerical results.     

A Generalized Logistic Link Function for Cumulative Link Models in Ordinal Regression

Ordinal regression is a kind of regression analysis used for predicting an ordered response variable. In these problems, the patterns are labelled by a set of ranks with an ordering among the different categories. The most common type of ordinal regression model is the cumulative link model. The cumulative link model relates an unobserved continuous latent variable with a monotone link function. Logit and probit functions are examples of link functions used in cumulative link models. In this paper, a novel generalized link function based on a generalization of the logistic distribution is proposed. The generalized link function proposed is able to reproduce other different link functions by changing two real parameters: \(\alpha \) and \(\lambda \). The generalized link function has been included in a cumulative link model where the latent function is determined by a standard neural network in order to test the performance of the proposal. For this model, a reformulation of the tunable thresholds and distribution parameters was applied to convert the constrained optimization problem into an unconstrained optimization problem. Experimental results demonstrate that our proposed approach can achieve competitive generalization performance.     

A shape preserving interpolant with tension controls

The monotonically and convexly constrained (MONCON) weighted v-spline interpolant is presented, which is the C¹ piecewise cubic solution to a constrained optimization problem. In addition to preserving the local monotonicity and local convexity of the data, the method has tension parameters which can be used to modify the shape of the interpolating function. Local monotonicity constraints are given that don't necessarily force the derivatives to be zero where the data changes from increasing to decreasing.     

Optimum design of prestressed concrete beams using constrained differential evolution algorithm

This paper is concerned with the cost minimization of prestressed concrete beams using a special differential evolution-based technique. The optimum design is posed as single-objective optimization problem in presence of constraints formulated in accordance with the current European building code. The design variables include geometrical dimensions that define the shape of the cross section and the amount of prestressing steel. A special (μ?+?λ)-constrained differential evolution method is performed in order to solve the optimization problem. Its search mechanism depends on several mutation strategies whereas an archiving-based adaptive tradeoff model is in charge of selecting a specific constraint-handling technique. Finally, numerical examples are included to illustrate the application of the presented approach.     

Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic

A fuzzy finite element model updating (FFEMU) method is presented in this study for the damage detection problem. The uncertainty caused by the measurement noise in modal parameters is described by fuzzy numbers. Inverse analysis is formulated as a constrained optimization problem at each α-cut level. Membership functions of each updating parameter which correspond to reduction in bending stiffness of the finite elements is determined by minimizing an objective function using a hybrid version of genetic algorithms (GA) and particle swarm optimization method (PSO) which is very efficient in terms of accuracy and robustness. Practical evaluation of the approximate bounds of the interval modal parameters in FFEMU iterations is addressed. A probabilistic analysis is performed using Monte Carlo simulation (MCS) and the results are compared with presented FFEMU method. It is apparent from numerical simulations that the proposed method is well capable in finding the membership functions of the updating parameters within reasonable accuracy. It is also shown that the results obtained by FFEMU are in good agreement with the MCS results while FFEMU is not as computationally expensive as the MCS method. Nevertheless, the proposed FFEMU do not required derivatives of the objective function like existing methods except in the deterministic case.     

On convex transformability and the solution of nonconvex problems via the dual of Falk

In structural optimization, most successful sequential approximate optimization (SAO) algorithms solve a sequence of strictly convex subproblems using the dual of Falk. Previously, we have shown that, under certain conditions, a nonconvex nonlinear (sub)problem may also be solved using the Falk dual. In particular, we have demonstrated this for two nonconvex examples of approximate subproblems that arise in popular and important structural optimization problems. The first is used in the SAO solution of the weight minimization problem, while the topology optimization problem that results from volumetric penalization gives rise to the other. In both cases, the nonconvex subproblems arise naturally in the consideration of the physical problems, so it seems counter productive to discard them in favor of using standard, but less well-suited, strictly convex approximations. Though we have not required that strictly convex transformations exist for these problems in order that they may be solved via a dual approach, we have noted that both of these examples can indeed be transformed into strictly convex forms. In this paper we present both the nonconvex weight minimization problem and the nonconvex topology optimization problem with volumetric penalization as instructive numerical examples to help motivate the use of nonconvex approximations as subproblems in SAO. We then explore the link between convex transformability and the salient criteria which make nonconvex problems amenable to solution via the Falk dual, and we assess the effect of the transformation on the dual problem. However, we consider only a restricted class of problems, namely separable problems that are at least C ¹ continuous, and a restricted class of transformations: those in which the functions that represent the mapping are each continuous, monotonic and univariate.     

An inverse convergence approach for arguments of Jacobian elliptic functions

Computing the value of the Jacobian elliptic functions, given the argument u and the parameter m, is a problem, whose solution can be found either tabulated in tables of elliptic functions [1] or by use of existing software, such as Mathematica, etc. The inverse problem, finding the argument, given the Jacobian elliptic function and the parameter m, is a problem whose solution is found only in tables of elliptic functions. Standard polynomial inverse interpolation procedures fail, due to ill conditioning of the system of the unknowns. In this paper, we describe a numerical procedure based on the convergence of the unknowns of the solution, by the use of arithmetical method, as an alternative way of solving the problem. The method gives very good results with no significant error, in the computation of the argument of the Jacobian elliptic function given the Jacobian elliptic function and the parameter. This new procedure is important in problems involving cavities or inclusions of ellipsoidal shape encountered in the mechanical design of bearings, filters, and composite materials. They are also important in the modeling of porosity of bones. This porosity may lead to osteoporosis, a disease which affects bone mineral density in humans with bad consequences. Also these procedures are of importance in problems encountered in the physics discipline such as in the analysis of the dependence of the maximum tunneling current on external magnetic field for large area Josephson junctions with overlap boundary conditions.     

一阶多智能体受扰系统的自抗扰分布式优化算法

研究一类具有未知外部干扰的一阶多智能体系统的分布式优化问题.在分布式优化任务中,每个智能体只被容许利用自己的局部目标函数和邻居的状态信息,设计一个分布式优化算法,使全局目标函数取得最小值,其中全局目标函数是所有局部目标函数之和.针对该问题,首先提出由扩张状态观测器和优化算法组成的自抗扰分布式优化算法.其次,在Lyapu...     

A polynomial algorithm for a lot-sizing problem with backlogging,outsourcing and limited inventory

This paper addresses a real-life production planning problem arising in a manufacturer of luxury goods. This problem can be modeled as a single item dynamic lot-sizing model with backlogging, outsourcing and inventory capacity. Setup cost is included in the production cost function, and the production level at each period is unbounded. The holding, backlogging and outsourcing cost functions are assumed to be linear. The backlogging level at each period is also limited. The goal is to satisfy all demands in the planning horizon at minimal total cost. We show that this problem can be solved in O(T⁴ log T) time where T is the number of periods in the planning horizon.     

The steady-state region of attraction under linear feedback control: A numerical approach

In this paper, a numerical approach to operability analysis is developed. Based on the concept of the steady-state region of attraction, which indicates the initial operating conditions that can be driven to a given setpoint, this analysis helps design engineers assess whether a process can be effectively controlled by linear control systems. Using steady-state operating data that are generated with process simulators (such as ASPEN PLUS^®), the steady-state region of attraction are calculated by solving the proposed constrained nonlinear optimization problem. The objective function for the optimization problem is developed using the ellipsoid theory while the constraint functions are formulated from the complement set of the initially defined constraint set and the boundary of the available input space (AIS). Furthermore, as the solution of a nonlinear optimization problem generally depends on the initial starting point, a method of generating a set of ellipsoids in the hyperspace to start the calculation is also developed. The proposed constrained nonlinear optimization problem can be solved using numerical tools such as MATLAB^®. This analysis can be performed based on process flowsheets with process simulators and thus is useful in early stages of process design. The use of the proposed numerical framework is illustrated by a case study of a methyl acetate reactive distillation column.     

Constrained optimization of nacelle shapes in Euler flow using semianalytical sensitivity analysis

The problem of aerodynamic shape optimization in Euler flow is addressed. B-splines are used for parametrization of the shape. Cheap gradient calculation is obtained via sensitivity analysis and the solution of an adjoint equation; pseudo secondorder spatial accuracy is achieved by means of a semianalytical formulation. As a validation of the approach, several inverse and constrained optimization test problems are presented with emphasis on civil engine nacelle design. The handling of nondifferentiable quantities (such as maxima) in cost functions is allowed for via the use of the Kreisselmeier-Steinhauser function.     

Simultaneous Optimization for Concave Costs: Single Sink Aggregation or Single Source Buy-at-Bulk

We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information that needs to be transmitted is f(j). One natural and important (though not necessarily comprehensive) class of functions is those which are concave, non-decreasing, and satisfy f(0) = 0. Our goal is to find a tree which is a good approximation simultaneously to the optimum trees for all such functions. This problem is motivated by aggregation in sensor networks, as well as by buy-at-bulk network design. We present a randomized tree construction algorithm that guarantees E[max_f C_f/C^*(f)] ≤ 1 + log k, where C_f is a random variable denoting the cost of the tree for function f and C^*(f) is the cost of the optimum tree for function f. To the best of our knowledge, this is the first result regarding simultaneous optimization for concave costs. We also show how to derandomize this result to obtain a deterministic algorithm that guarantees max_f C_f/C^*(f) = O(log k). Both these results are much stronger than merely obtaining a guarantee on max_f E[C_f/C^*(f)]. A guarantee on max_f E[C_f/C^*(f)] can be obtained using existing techniques, but this does not capture simultaneous optimization since no one tree is guaranteed to be a good approximation for all f simultaneously. While our analysis is quite involved, the algorithm itself is very simple and may well find practical use. We also hope that our techniques will prove useful for other problems where one needs simultaneous optimization for concave costs.     

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.     

Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method

Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems’ objective functions have no derivatives. According to former research studies, some search directions are obtained using the quadratic hypothesis of objective functions. Based on derivatives, quadratic function assumptions, and directional derivatives, the computational formulas of numerical first-order partial derivatives, second-order partial derivatives, and numerical second-order mixed partial derivatives were constructed. Based on the coordinate transformation relation, a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction. A numerical algorithm was proposed, taking the second order approximation direction as an example. A large stepsize numerical algorithm based on coordinate transformation was proposed. Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function. The numerical second order approximation direction with the numerical mixed partial derivatives showed good results. Its calculated amount is 0.2843% of that of without second-order mixed partial derivative. In the process of rotating the local coordinate system 360°, because the objective function is more complex than the quadratic function, if the numerical direction derivative is used instead of the analytic partial derivative, the optimization direction varies with a range of 103.05°. Because theoretical error is in the numerical negative gradient direction, the calculation with the coordinate transformation is 94.71% less than the calculation without coordinate transformation. If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation, the sawtooth phenomenon occurs. When each numerical mixed partial derivative takes more than one point, the optimization results cannot be improved. The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained, but does not require derivability and does not take into account truncation error and rounding error. Thus, the application scopes of many optimization methods are extended.      

On a class of branching problems in broadcasting and distribution

We introduce the following network optimization problem: given a directed graph with a cost function on the arcs, demands at the nodes, and a single source s, find the minimum cost connected subgraph from s such that its total demand is no less than lower bound D. We describe applications of this problem to disaster relief and media broadcasting, and show that it generalizes several well-known models including the knapsack problem, the partially ordered knapsack problem, the minimum branching problem, and certain scheduling problems. We prove that our problem is strongly NP-complete and give an integer programming formulation. We also provide five heuristic approaches, illustrate them with a numerical example, and provide a computational study on both small and large sized, randomly generated problems. The heuristics run efficiently on the tested problems and provide solutions that, on average, are fairly close to optimal.     

Vector Image Segmentation by Piecewise Continuous Approximation

In this article we present an approach to the segmentation problem by a piecewise approximation of the given image with continuous functions. Unlike the common approach of Mumford and Shah in our formulation of the problem the number of segments is a parameter, which can be estimated. The problem can be stated as: Compute the optimal segmentation with a fixed number of segments, then reduce the number of segments until the segmentation result fulfills a given suitability. This merging algorithm results in a multi-objective optimization, which is not only resolved by a linear combination of the contradicting error functions. To constrain the problem we use a finite dimensional vector space of functions in our approximation and we restrict the shape of the segments. Our approach results in a multi-objective optimization: On the one hand the number of segments is to be minimized, on the other hand the approximation error should also be kept minimal. The approach is sound theoretically and practically: We show that for L ²-images a Pareto-optimal solution exists and can be computed for the discretization of the image efficiently.     

Multiple robust H∞ controller design using the nonsmooth optimization method

This paper proposes a systematic technique to design multiple robust H_∞ controllers. The proposed technique achieves a desired robust performance objective, which is impossible to achieve with a single robust controller, by dividing the uncertainty set into several subsets and by designing a robust controller to each subset. To achieve this goal with a small number of divisions of the uncertainty set, an optimization problem is formulated. Since the cost function of this optimization problem is not a smooth function, a numerical nonsmooth optimization algorithm is proposed to solve this problem. This method avoids the use of Lyapunov variables, and therefore it leads to a moderate size optimization problem. A numerical example shows that the proposed multiple robust control method can improve the closed‐loop performance when a single robust controller cannot achieve satisfactory performance. Copyright © 2009 John Wiley & Sons, Ltd.     

Some problems in distributed computational geometry

In a planar geometric network vertices are located in the plane, and edges are straight line segments connecting pairs of vertices, such that no two of them intersect. In this paper we study distributed computing in asynchronous, failure-free planar geometric networks, where each vertex is associated to a processor, and each edge to a bidirectional message communication link. Processors are aware of their locations in the plane.We consider fundamental computational geometry problems from the distributed computing point of view, such as finding the convex hull of a geometric network and identification of the external face. We also study the classic distributed computing problem of leader election, to understand the impact that geometric information has on the message complexity of solving it.We obtain an O(nlog²n) message complexity algorithm to find the convex hull, and an O(nlogn) message complexity algorithm to identify the external face of a geometric network of n processors. We present a matching lower bound for the external face problem. We prove that the message complexity of leader election in a geometric ring is Ω(nlogn), hence geometric information does not help in reducing the message complexity of this problem.     

Functional gradient as a tool for semi-analytical optimization for structural buckling

A functional gradient for the buckling load (P) with respect to the structural shape function (S) is introduced for shape optimization. The gradient (P_,S) is derived analytically by functional differentiation of the governing equation with respect to S, where both P and the deflection (W) are considered as functionals of S. It is found that P_,S is a functional of W and is not dependent explicitly on S. The P_,S is combined with a simple gradient projection optimization method and a finite element program. The analytical gradient expression is independent of the FE model but requires its input. At each optimization step the buckling problem is solved numerically and the gradient is calculated. Then, a new improved design is computed. In the first part of the study 1D beam problems are solved. Results are compared with exact solutions and found to be more accurate than common numerical optimizations. An example of a clamped beam with two simple supports is studied. The optimal area distribution of a general unimodal 1D beam is found to be composed of sections with the same shape as for the basic simply supported case. The proposed method is generalized to 2D plates. The optimization is combined with a commercial FE package. For the case of simple square plates the optimization produces smooth thickness distribution showing higher buckling load improvement than common numerical optimization. The method was then applied to square plates with a central circular hole, where non-uniform FE meshes were used. The optimal shapes exhibited singularities resembling commonly used stiffeners.     

Stochastic utility-based flow control algorithm for services with time-varying rate requirements

In this paper, we study a utility based flow control problem for a communication network. In most previous works on utility based flow control, the utility function of each user, which represents its satisfaction to the allocated data rate, is assumed to be fixed. This implies that the degree of the rate requirement of each user is assumed to be fixed over the entire duration of its session. However, in communication networks, many services are variable rate services, i.e., the degree of their rate requirement varies over time, which cannot be modeled with traditional static utility functions. To resolve this issue and appropriately model services with variable rate requirements, we propose a stochastic utility function that varies stochastically according to the variation of the degree of the rate requirement of a service. We formulate a flow control problem as a stochastic optimization problem with stochastic utility functions that aims at maximizing the average network utility while satisfying the constraint on link capacity and QoS requirement. By solving the stochastic optimization problem, we develop a distributed flow control algorithm that converges to the optimal rate allocation.