Optimal Smoothing for Convex Polytopes

It is proved that, given a convex polytope P in Rⁿ, togetherwith a collection of compact convex subsets in the interiorof each facet of P, there exists a smooth convex body arbitrarilyclose to P that coincides with each facet precisely along theprescribed sets, and has positive curvature elsewhere. 2000Mathematics Subject Classification 53A07, 52B11, 53C45.     

Optimal Smoothing of Demand for Industrial Gas

The problem of smoothing the demand for industrial gas when there are large tariff incentives for doing so is considered. Two cases are discussed:

1. a)
   where the only control variables are the maximum hourly and daily flow rates;
    
2. b)
   where, in addition, gas storage tanks can be installed to help smooth the demand.
    

In case (a) an analytic solution to the problem of establishing the optimal hourly and daily maximum settings is derived. In case (b) the influence of the storage capacity on the demand distribution is analysed, and a method of solution is proposed. This can then be used, together with the solution for case (a), to establish the optimal number of storage tanks to install.These ideas are applied to the gas usage in a steel factory.     

Optimal Actuator Placement for Dynamical Systems

Vibrations occur in many industrial areas and produce undesirable side effects. To avoid respectively suppress these effects actuators can be attached to the structure. Actuators can transfer forces into the structure, which can be used to control the system behavior of the structure in order to achieve, for example, vibration absorption, noise reduction and/or exact positioning. The appropriate positioning of actuators is of significant importance for the controllability of the structure. We present a method for determining the optimal actuator placement. We consider applications, where the dynamics of a system can be described by a finite dimensional, controllable and observable state-space model. The concept of controllability by Kálmán forms the basis for the definition of controllability measures. In this context, the controllability gramian plays an important role. In our approach actuators are placed in such a way that the selected controllability measure becomes maximal. This leads to an optimization problem with binary and continuous variables and linear matrix inequalities. The numerical results show the optimal actuator placement for a two dimensional flexible clamped plate. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Robotic Component Placement

An increasingly common feature of manufacturing companies is the use of robotic component placement machines in the assembly of electrical goods. These machines place electronic components at predefined positions on printed circuit boards. A robotic arm, which carries one component at a time, collects components from feeders and places them, under computer control, onto the printed circuit board.Using examples provided by a major New Zealand manufacturer, we investigated the problem of assigning component reels to feeder positions and then constructing efficient placement sequences for such a robotic placement machine. An integer programming formulation is developed for this problem that, unlike previous solution techniques in the literature, can produce solutions which provably minimize total placement time for all company supplied sample problems. This solution method generates solutions approximately 10% better than those generated manually by the manufacturer's engineers.     

一种求解不等式约束优化问题的光滑化算法

利用光滑函数建立了不等式约束优化问题KT条件的一个扰动方程组,提出了一个新的内点型算法.该算法在有限步终止时当前迭代点即为优化问题的一个精确稳定点.在一定条件下算法具有全局收敛性,数值试验表明该算法是有效的.     

Optimal Covering of Plane Domains by Circles Via Hyperbolic Smoothing

We consider the problem of optimally covering plane domains by a given number of circles. The mathematical modeling of this problem leads to a min–max–min formulation which, in addition to its intrinsic multi-level nature, has the significant characteristic of being non-differentiable. In order to overcome these difficulties, we have developed a smoothing strategy using a special class C^∞ smoothing function. The final solution is obtained by solving a sequence of differentiable subproblems which gradually approach the original problem. The use of this technique, called Hyperbolic Smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essential of the method is presented. For the purpose of illustrating both the actual working and the potentialities of the method, a set of computational results is presented.     

The Optimal Choice of the Smoothing Constant in an Inventory Control System

In controlling stock it is necessary to make some estimate of future demand for each stock item. Exponential smoothing is a convenient and automatic means of making such an estimate and the optimal choice of the smoothing constant alpha will depend upon the order of smoothing employed, the relative magnitudes of the random and systematic variations in demand which may occur, the replenishment lead time and also the manner in which the rule used for generating replenishment orders is formulated. The dependence both of optimal alpha and of the necessary safety stock upon these factors has been investigated by means of a model which is believed to be of wide application in industry, and a method has been developed for setting up decision rules for both single and multi-stage systems.     

Smoothing Method for Minimax Problems

In this paper, we propose a smoothing method for minimax problem. The method is based on the exponential penalty function of Kort and Bertsekas for constrained optimization. Under suitable condition, the method is globally convergent. Preliminary numerical experiments indicate the promising of the algorithm.     

Smoothing Functions and Smoothing Newton Method for Complementarity and Variational Inequality Problems

This paper provides for the first time some computable smoothing functions for variational inequality problems with general constraints. This paper proposes also a new version of the smoothing Newton method and establishes its global and superlinear (quadratic) convergence under conditions weaker than those previously used in the literature. These are achieved by introducing a general definition for smoothing functions, which include almost all the existing smoothing functions as special cases.     

Lagrangian Smoothing Heuristics for Max-Cut

This paper presents a smoothing heuristic for an NP-hard combinatorial problem. Starting with a convex Lagrangian relaxation, a pathfollowing method is applied to obtain good solutions while gradually transforming the relaxed problem into the original problem formulated with an exact penalty function. Starting points are drawn using different sampling techniques that use randomization and eigenvectors. The dual point that defines the convex relaxation is computed via eigenvalue optimization using subgradient techniques. The proposed method turns out to be competitive with the most recent ones. The idea presented here is generic and can be generalized to all box-constrained problems where convex Lagrangian relaxation can be applied. Furthermore, to the best of our knowledge, this is the first time that a Lagrangian heuristic is combined with pathfollowing techniques. The work was supported by the German Research Foundation (DFG) under grant No 421/2-1.     

Optimal Sensor Placement Method Considering the Importance of Structural Performance Degradation for the Allowable Loadings for Damage Identification

Considering the importance of damage for the structural performance and for decreasing the identification error, this paper proposes an optimal sensor placement method based on a weighted standard deviation norm (WSDN) index. The standard deviation of the identified damage parameters is solved using the series expansion theory and probabilistic method to quantify the effect of a measurement error on damage identification. The damage estimation weight (DEW) index, which can reflect the importance of each element in the structural capabilities, is established based on a performance-damage curve. A significant DEW for a specified element indicates that the element is important for the structure and that its identification error should be small. The WSDN index is obtained from the Hadamard product of the standard deviations (SDs) and DEWs. Thus, the identification error of the entire structure is measured using the weighting coefficient. The optimal sensor placement (OSP) procedure is performed by minimizing the WSDN index. The proposed method can clearly decrease the uncertainties of the identification results for the important elements. Other OSP criteria, including the condition number, information entropy, and standard deviation norm, which aim to decrease the identification error, are discussed in this paper for comparison with the proposed method. Two numerical examples and an experiment, which pertain to the deformation performance, buckling features, and dynamic characteristics, are discussed to verify the advantages of the proposed method.     

Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity

Using Bourgain spaces and the generator of dilation P=3t ? _t +x ? _x , which almost commutes with the linear Korteweg-de Vries operator, we show that a solution of the initial value problem associated for the coupled system of equations of Korteweg-de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has an analyticity in time and a smoothing effect up to real analyticity if the initial data only have a single point singularity at x=0.     

An Optimal Deterministic Algorithm for Computing the Diameter of a Three-Dimensional Point Set

We describe a deterministic algorithm for computing the diameter of a finite set of points in R ³ , that is, the maximum distance between any pair of points in the set. The algorithm runs in optimal time O(nlog n) for a set of n points. The first optimal, but randomized, algorithm for this problem was proposed more than 10 years ago by Clarkson and Shor [11] in their ground-breaking paper on geometric applications of random sampling. Our algorithm is relatively simple except for a procedure by Matoušek [25] for the efficient deterministic construction of epsilon-nets. This work improves previous deterministic algorithms by Ramos [31] and Bespamyatnikh [7], both with running time O(nlog ² n) . The diameter algorithm appears to be the last one in Clarkson and Shor's paper that up to now had no deterministic counterpart with a matching running time. Received May 10, 2000, and in revised form November 3, 2000. Online publication June 22, 2001.     

基于动态参照点的损失厌恶投资组合优化模型

在连续时间下,考虑损失厌恶投资者参照点的动态调整特征,构建基于动态参照点的损失厌恶投资组合模型,使用鞅方法对模型进行求解,得到最优风险资产权重的解析表达式。并计算损失厌恶投资者在参照点动态调整条件下的预期最优期末财富。进一步应用数值算例,分析投资者的参照点动态调整幅度和损失厌恶水平对模型最优风险资产权重和预期最优期末财富的影响。     

Mesh Spacing Estimates and Efficiency Considerations for Moving Mesh Systems

Adaptive numerical methods for solving partial differential equations (PDEs) that control the movement of grid points are called moving mesh methods. In this paper, these methods are examined in the case where a separate PDE, that depends on a monitor function, controls the behavior of the mesh. This results in a system of PDEs: one controlling the mesh and another solving the physical problem that is of interest. For a class of monitor functions resembling the arc length monitor, a trade off between computational efficiency in solving the moving mesh system and the accuracy level of the solution to the physical PDE is demonstrated. This accuracy is measured in the density of mesh points in the desired portion of the domain where the function has steep gradient. The balance of computational efficiency versus accuracy is illustrated numerically with both the arc length monitor and a monitor that minimizes certain interpolation errors. Physical solutions with steep gradients in small portions of their domain are considered for both the analysis and the computations.     

Optimal Placement of a Deposit between Markets: Riemann-Finsler Geometrical Approach

By using the Riemann-Finsler geometry, we study the existence and location of the optimal points for a general cost function involving Finsler distances. Our minimization problem provides a model for the placement of a deposit within a domain with several markets such that the total transportation cost is minimal. Several concrete examples are studied either by precise mathematical tools or by evolutionary (computer assisted) techniques. The research of Alexandru Kristály was supported by the Grant PN II, ID_527/2007 and CNCSIS Project AT 8/70. ágoston Róth was supported by the Research Center of Sapientia Project 1122.     

Smoothing algorithms for nonlinear finite-dimensional systems

Systems are considered where the state evolves either as a diffusion process or as a finitestate Markov process, and the measurement process consists either of a nonlinear function of the state with additive white noise or as a counting process with intensity dependent on the state. Fixed interval smooting is considered, and the first main result obtained expresses a smoothing probability or a probability density symmetrically in terms of forward filtered, reverse-time filterd and unfiltered quantities; an associated result replaces the unfiltered and reverse-time filtered qauantities by a likelihood function. Then stochastic differential equationsare obtained for the evolution of the reverse-time filtered probability or probability density and the reverse-time likelihood function. Lastly, a partial differential equation is obtained linking smoothed and forward filterd probabilities or probability densities; in all instances considered, this equation is not driven by any measurement process. The different approaches are also linked to known techniques applicable in the linear-Gaussian case.     

Predictor-Corrector Smoothing Methods for Monotone LCP

In this paper, we analyze the global and local convergence properties of two predictor-corrector smoothing methods, which are based on the framework of the method in [1], for monotone linear complementarity problems (LCPs). The difference between the algorithm in [1] and our algorithms is that the neighborhood of smoothing central path in our paper is different to that in [1]. In addition, the difference between Algorithm 2.1 and the algorithm in [1] exists in the calculation of the predictor step. Comparing with the results in [1],the global and local convergence of the two methods can be obtained under very mild conditions. The global convergence of the two methods do not need the boundness of the inverse of the Jacobian. The superlinear convergence of Algorithm 2.1‘ is obtained under the assumption of nonsingularity of generalized Jacobian of Φ(x,y) at the limit point and Algorithm 2.1 obtains superlinear convergence under the assumption of strict complementarity at the solution. The efficiency of the two methods is tested by numerical experiments.