Distributionally robust parameter identification of a time-delay dynamical system with stochastic measurements

In this paper, we consider a parameter identification problem involving a time-delay dynamical system, in which the measured data are stochastic variable. However, the probability distribution of this stochastic variable is not available and the only information we have is its first moment. This problem is formulated as a distributionally robust parameter identification problem governed by a time-delay dynamical system. Using duality theory of linear optimization in a probability space, the distributionally robust parameter identification problem, which is a bi-level optimization problem, is transformed into a single-level optimization problem with a semi-infinite constraint. By applying problem transformation and smoothing techniques, the semi-infinite constraint is approximated by a smooth constraint and the convergence of the smooth approximation method is established. Then, the gradients of the cost and constraint functions with respect to time-delay and parameters are derived. On this basis, a gradient-based optimization method for solving the transformed problem is developed. Finally, we present an example, arising in practical fermentation process, to illustrate the applicability of the proposed method.     

Second-order contingent derivatives of set-valued mappings with application to set-valued optimization

In this paper, a family of parameterized set-valued optimization problems, whose constraint set depends on a parameter, are considered. Some calculus rules are obtained for calculating the second-order contingent derivatives of the composition and sum of two set-valued mappings. Then, by using these calculus rules, some results concerning second-order sensitivity analysis are established, and an explicit expression for the second-order contingent derivative of the (weak) perturbation mapping in the set-valued optimization problems is obtained.     

Minimax estimation for singular linear multivariate models with mixed uncertainty

The problem of minimax estimation is examined for the linear multivariate statistically indeterminate observation model with mixed uncertainty. The a priori information on the distributions of model parameters is formulated in terms of second-order moment characteristics. It is shown that in the regular case the minimax estimate is defined explicitly via the solution of the dual optimization problem. For singular models, the method of dual optimization is developed by means of using the Tikhonov regularization techniques. Several particular cases which are widely used in practice are also considered.     

互补约束优化问题的乘子序列部分罚函数算法

利用互补问题的Lagrange函数,
将互补约束优化问题(MPCC)转化为含参数的约束优化问题.
给出Lagrange乘子的简单修正公式,
并给出求解互补约束优化问题的部分罚函数法. 无须假设二阶必要条件成立,
只要算法产生的迭代点列的极限点满足互补约束优化问题的线性独立约束规范(MPCC-LICQ),
且极限点是MPCC的可行点, 则算法收敛到原问题的M-稳定点. 另外,
在上水平严格互补(ULSC)成立的条件下, 算法收敛到原问题的B-稳定点.     

Optimization of shallow arches against instability using sensitivity derivatives

In this paper, the problem of optimization of shallow frame structures, which involves coupling of axial and bending responses, is discussed. A shallow arch of given shape and given weight is optimized such that its limit point load is maximized. The crosssectional area, A(x), and the moment of inertia, I(x), of the arch obey the relationship I(x) = [A(x)]ⁿ, where N = 1, 2, 3 and is a specified constant. Analysis of the arch for its limit point calculation involves a geometric nonlinear analysis which is performed using a co-rotational formulation.

The optimization is carried out using a second-order projected Langragian algorithm, and the sensitivity derivatives of the critical load parameter with respect to the areas of the finite elements of the arch are calculated using implicit differentiation. Results are presented for an arch of a specified rise to span under two different loadings, and the limitations of the approach for the intermediate rise arches are addressed.     

Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization

This study considers the robust identification of the parameters describing a Sugeno type fuzzy inference system with uncertain data. The objective is to minimize the worst-case residual error using a numerically efficient algorithm. The Sugeno type fuzzy systems are linear in consequent parameters but nonlinear in antecedent parameters. The robust consequent parameters identification problem can be formulated as second-order cone programming problem. The optimal solution of this second-order cone problem can be interpreted as solution of a Tikhonov regularization problem with a special choice of regularization parameter which is optimal for robustness (Ghaoui and Lebret (1997). SAIM Journal of Matrix Analysis and Applications 18, 1035–1064). The final regularized nonlinear optimization problem allowing simultaneous identification of antecedent and consequent parameters is solved iteratively using a generalized Gauss–Newton like method. To illustrate the approach, several simulation studies on numerical examples including the modelling of a spectral data function (one-dimensional benchmark example) is provided. The proposed robust fuzzy identification scheme has been applied to approximate the physical fitness of patients with a fuzzy expert system. The identified fuzzy expert system is shown to be capable of capturing the decisions (experiences) of a medical expert.     

A Trajectory-Based Method for Constrained Nonlinear Optimization Problems

A trajectory-based method for solving constrained nonlinear optimization problems is proposed. The method is an extension of a trajectory-based method for unconstrained optimization. The optimization problem is transformed into a system of second-order differential equations with the aid of the augmented Lagrangian. Several novel contributions are made, including a new penalty parameter updating strategy, an adaptive step size routine for numerical integration and a scaling mechanism. A new criterion is suggested for the adjustment of the penalty parameter. Global convergence properties of the method are established.     

Stochastic independent modal-space control of distributed-parameter systems

A method capable in theory of estimating and controlling all the modes of a distributed-parameter system is presented. A linear distributed estimator using a distributed Kalman function is defined. It is shown that a particular choice of the Kalman function, in conjunction with the independent modal-space control method, leads to an infinite set of independent second-order modal-space Kalman filters cascaded with spatial modal filters. Independent modal-space Kalman filters experience no computational difficulties, regardless of the order of the system, and closed-form solutions for the modal Kalman gain matrices can be obtained with relative ease, making real-time implementation feasible. It is also shown that the independent modal-space Kalman filters are theoretically free of observation spillover.This work was supported in part by NSF Grant No. PFR-80-20623.     

Stochastic second-order cone programming: Applications models

Second-order cone programs are a class of convex optimization problems. We refer to them as deterministic second-order cone programs (DSCOPs) since data defining them are deterministic. In DSOCPs we minimize a linear objective function over the intersection of an affine set and a product of second-order (Lorentz) cones. Stochastic programs have been studied since 1950s as a tool for handling uncertainty in data defining classes of optimization problems such as linear and quadratic programs. Stochastic second-order cone programs (SSOCPs) with recourse is a class of optimization problems that defined to handle uncertainty in data defining DSOCPs. In this paper we describe four application models leading to SSOCPs.     

On coupled transversal and axial motions of two beams with a joint

In this paper we develop and analyze a mathematical model for combined axial and transverse motions of two Euler-Bernoulli beams coupled through a joint composed of two rigid bodies. The motivation for this problem comes from the need to accurately model damping and joints for the next generation of inflatable/rigidizable space structures. We assume Kelvin-Voigt damping in the two beams whose motions are coupled through a joint which includes an internal moment. The resulting equations of motion consist of four, second-order in time, partial differential equations, four second-order ordinary differential equations, and certain compatibility boundary conditions. The system is re-cast as an abstract second-order differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove the system is well posed, and that with positive damping parameters the resulting semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator is characterized.     

Strength vs. toughness optimization of microstructured composites

Aim of this paper is to present a new fractal approach linking the macroscopic mechanical properties of micro- and nano-structured materials with the main parameters: composition, grain size and structural dimension, as well as contiguity and mean free path. Assuming the key role played by the interfaces, the proposed fractal energy approach unifies the influences of all the above parameters, through the introduction of a fractal structural parameter (FSP), which represents an extension of the Gurland’s structural parameter. This modeling approach is assessed through an extensive comparison with experimental data on poly crystalline diamond (PCD) and WC–Co alloys. The results clearly show that the theoretical fractal predictions are in a fairly good agreement with the experiments on both hardness and toughness. This new synthetic parameter is thus proposed to investigate, design and optimize new micro- and nano-grained materials. Eventually, FSP-based optimization maps are developed, that allow to design new materials with high hardness and toughness.     

指数分布冷贮备系统产品的统计分析——转换开关指数型的情形

给出了全样本场合下指数分布冷贮备系统产品寿命分布中参数θ≠λ时的矩估计和极大似然估计,通过Monte-Carlo给出了参数矩估计的精度,考察了1000次满足条件时所需要的模拟次数,随着样本量的增大,矩估计存在的比率逐渐增大,而极大似然估计的结果与样本有关.同时给出了参数θ=λ时的矩估计、极大似然估计和逆矩估计,通过Monte-Carlo模拟考察了参数点估计精度,认为矩估计比较优.文章还给出了求参数区间估计的两种方法——精确方法和近似方法,通过Monte-Carlo模拟认为精确方法精度较高.     

An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems

In this paper, we present two families of second-order and third-order explicit methods for numerical integration of initial-value problems of ordinary differential equations. Firstly, a family of second-order methods with two free parameters is derived by considering a suitable rational approximation to the theoretical solution of the problem at some grid points. Imposing that the principal term of the local truncation error of this family vanishes, we obtain an expression for one of the parameters in terms of the other. With this approach, a new one-parameter family of third-order methods is obtained. By selecting any 3(2) pair of second and third order methods, they can be implemented as an embedded type method, thus leading to a variable step-size formulation. We have considered one 3(2) pair of second and third order methods and made a comparison of numerical results with several ode solvers which are currently used in practice. The comparison of numerical results shows that the embedded 3(2) pair outperforms the methods considered for comparison.     

Parameter estimation of chaotic system with time-delay: A differential evolution approach

It is an important issue to estimate parameters of chaotic system in nonlinear science. In this paper, parameter estimation problem of chaotic system with time-delay is considered. Parameters and time-delay are estimated together by treating time-delay as an additional parameter. The parameter estimation problem is converted to an multi-dimensional optimization problem. A differential evolution (DE) algorithm, which possess a powerful searching capability for finding the solutions for a given optimization problem, is applied to solve this optimization problem. Two illustrative examples are given to verify the effectiveness of the proposed method.     

Estimating Functions for Nonlinear Time Series Models

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.     

Adaptive disturbance transmission decoupling in active vehicle suspensions

With active vehicle suspension, one can tailor a vehicles response to load and inertial without affecting the vehicle response to road disturbances. This decoupling is achieved in [1] and [2] using a filtered combination of measured signals. These filters require exact knowledge of certain vehicle parameters including vehicle mass to achieve the desired decoupling. Here we propose a parameter adaptive version of these filters which does not require knowledge of vehicle parameters. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polyhedral approximation of ellipsoidal uncertainty sets via extended formulations: a computational case study

Robust optimization is an important technique to immunize optimization problems against data uncertainty. In the case of a linear program and an ellipsoidal uncertainty set, the robust counterpart turns into a second-order cone program. In this work, we investigate the efficiency of linearizing the second-order cone constraints of the latter. This is done using the optimal linear outer-approximation approach by Ben-Tal and Nemirovski (Math Oper Res 26:193–205, 2001) from which we derive an optimal inner approximation of the second-order cone. We examine the performance of this approach on various benchmark sets including portfolio optimization instances as well as (robustified versions of) the MIPLIB and the SNDlib.     

A new refinement indicator for adaptive parameterization: Application to the estimation of the diffusion coefficient in an elliptic problem

We propose a new refinement indicator (NRI) for adaptive parameterization to determine the diffusion coefficient in an elliptic equation in two-dimensional space. The diffusion coefficient is assumed to be a piecewise constant space function. The unknowns are both the parameter values and the zonation. Refinement indicators are used to localize parameter discontinuities in order to construct iteratively the zonation (parameterization). The refinement indicator is obtained usually by using the first-order effect on the objective function of removing degrees of freedom for a current set of parameters. In this work, in order to reduce the computation costs, we propose a new refinement indicator based on the second-order effect on the objective function. This new refinement indicator depends on the objective function, and its first and second derivatives with respect to the parameter constraints. Numerical experiments show the high efficiency of the new refinement indicator compared to the standard one.     

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper we compute the asymptotic distribution for these estimates in the case where the underlying noise sequence has infinite fourth moment but finite second moment. In this case, the sample covariances on which the innovations algorithm are based are known to be asymptotically stable. The asymptotic results developed here are useful to determine which model parameters are significant. In the process, we also compute the asymptotic distributions of least squares estimates of parameters in an autoregressive model.     

Sensitivity analysis of optimization problems in Hilbert space with application to optimal control

A family of optimization problems in a Hilbert space depending on a vector parameter is considered. It is assumed that the problems have locally isolated local solutions. Both these solutions and the associated Lagrange multipliers are assumed to be locally Lipschitz continuous functions of the parameter. Moreover, the assumption of the type of strong second-order sufficient condition is satisfied.It is shown that the solutions are directionally differentiable functions of the parameter and the directional derivative is characterized. A second-order expansion of the optimal-value function is obtained. The abstract results are applied to state and control constrained optimal control problems for systems described by nonlinear ordinary differential equations with the control appearing linearly.