Adjustable robust solutions of uncertain linear programs

We consider linear programs with uncertain parameters, lying in some prescribed uncertainty set, where part of the variables must be determined before the realization of the uncertain parameters (``non-adjustable variables'), while the other part are variables that can be chosen after the realization (``adjustable variables'). We extend the Robust Optimization methodology ([1, 3-6, 9, 13, 14]) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterpart (RC), however, in most cases the ARC is computationally intractable (NP-hard). This difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data. The ensuing Affinely Adjustable Robust Counterpart (AARC) problem is then shown to be, in certain important cases, equivalent to a tractable optimization problem (typically an LP or a Semidefinite problem), and in other cases, having a tight approximation which is tractable. The AARC approach is illustrated by applying it to a multi-stage inventory management problem.Research was partially supported by the Israeli Ministry of Science grant #0200-1-98, the Israel Science Foundation founded by The Israel Academy of Sciences and Humanities, grant #683/99-10.0, and the Fund for Promotion of Research at the Technion.     

基于T-S模型的不确定时滞系统的保代价控制

基于T—S模糊模型，研究具有范数有界、时变参数不确定性和时滞的非线性系统的保代价控制问题，得到存在稳定模糊保代价控制器的充分条件，并推算出相应的线性矩阵不等式(LMI)形式。     

Representation theorems for probability functions satisfying spectrum exchangeability in inductive logic

We prove de Finetti style representation theorems covering the class of all probability functions satisfying spectrum exchangeability in polyadic inductive logic and give an application by characterizing those probability functions satisfying spectrum exchangeability which can be extended to a language with equality whilst still satisfying that property.     

Asset-selling problem with an uncertain deadline,quitting offer,and search skipping option

This paper presents a discrete-time sequential stochastic asset-selling problem with an infinite planning horizon, where the process of selling the asset may reach a deadline at any point in time with a probability. It is assumed that a quitting offer is available at every point in time and search skipping is permitted. Thus, decisions must be made as to whether or not to accept the quitting offer, to accept an appearing buyer’s offer, and to conduct a search for a buyer. The main purpose of this paper is to clarify the properties of the optimal decision rules in relation to the model’s parameters.     

Joint swing-up and stabilization of the Reaction Wheel Pendulum using Discontinuous Integral algorithm

In this paper, a third-order discontinuous integral controller is designed to jointly swing-up and robustly stabilize in finite-time the upright position of the Reaction Wheel Pendulum (RWP) system, despite some uncertainties and perturbations. To show this, a numerical estimation of the domain of attraction of the controller is used. The control algorithm produces a continuous control signal, thus reducing the usual chattering effect of the sliding-mode controllers. The theoretical results of the paper are verified by simulations and experiments in a laboratory setup.     

On Topological Derivatives for Elastic Solids with Uncertain Input Data

In this paper, a new approach to the derivation of the worst scenario and the maximum range scenario methods is proposed. The derivation is based on the topological derivative concept for the boundary-value problems of elasticity in two and three spatial dimensions. It is shown that the topological derivatives can be applied to the shape and topology optimization problems within a certain range of input data including the Lamé coefficients and the boundary tractions. In other words, the topological derivatives are stable functions and the concept of topological sensitivity is robust with respect to the imperfections caused by uncertain data. Two classes of integral shape functionals are considered, the first for the displacement field and the second for the stresses. For such classes, the form of the topological derivatives is given and, for the second class, some restrictions on the shape functionals are introduced in order to assure the existence of topological derivatives. The results on topological derivatives are used for the mathematical analysis of the worst scenario and the maximum range scenario methods. The presented results can be extended to more realistic methods for some uncertain material parameters and with the optimality criteria including the shape and topological derivatives for a broad class of shape functionals. This research is partially supported by the Brazilian Agency CNPq under Grant 472182/2007-2, FAPERJ under Grant E-26/171.099/2006 (Rio de Janeiro) and Brazilian-French Research Program CAPES/COFECUB under Grant 604/08 between LNCC in Petrópolis and IECN in Nancy, and by the Research Grant CNRS-CSAV between Institut Elie Cartan in Nancy and the Institut of Mathematics in Prague. The support is gratefully acknowledged.     

Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both uncertain

Several researchers have recently derived formulae for economic order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected annual cost. The purpose of this paper is threefold. First, this study extends the previous result to the EOQ formula, taking into account the scenario where the quantity backordered and the quantity received are both uncertain. Second, the complete squares method can readily derive global optimal expressions from a non-convex objective function in an algebraic manner. Third, the explicit identification of some analytic cases can be obtained: it is not as easy to do this using decomposition by projection. A numerical example has been solved to illustrate the solution procedure. Finally, some special cases can be deduced from the EOQ model under study, and concluding remarks are drawn.     

不确定二维多项式族的稳定性检验集合

不确定二维多项式族,包括多胞形二维多项式族,区间二维多项式族,钻石形二维多项式族,其鲁棒稳定性可由其棱边多项式的稳定性确定.如果二维多项式族的不确定参数较多,被检验棱边多项式的数目可能出现所谓组合爆炸问题.为解决这一问题,提出一检验集合,该集合根据棱边二维多项式的凸方向来构造,仅对属于该检验集合棱边二维多项式进行稳定性检验,从而大大地减少被检验的棱边二维多项式数目.给出一例来说明这种新方法的可行性.     

Triangular configuration of uncertain systems stabilizable by means of feedback controller

The uncertain linear systems with time-varying delays and limited measurable states variables are shown to have special configuration, called triangular configuration, to be stabilizable by means of state variable feedback controller. This configuration indicates the location of uncertainties among system parameters including input and output coefficients.