Generation of efficient frontiers in multi-objective optimization problems by generalized data envelopment analysis

In many practical problems such as engineering design problems, criteria functions cannot be given explicitly in terms of design variables. Under this circumstance, values of criteria functions for given values of design variables are usually obtained by some analyses such as structural analysis, thermodynamical analysis or fluid mechanical analysis. These analyses require considerably much computation time. Therefore, it is not unrealistic to apply existing interactive optimization methods to those problems. On the other hand, there have been many trials using genetic algorithms (GA) for generating efficient frontiers in multi-objective optimization problems. This approach is effective in problems with two or three objective functions. However, these methods cannot usually provide a good approximation to the exact efficient frontiers within a small number of generations in spite of our time limitation. The present paper proposes a method combining generalized data envelopment analysis (GDEA) and GA for generating efficient frontiers in multi-objective optimization problems. GDEA removes dominated design alternatives faster than methods based on only GA. The proposed method can yield desirable efficient frontiers even in non-convex problems as well as convex problems. The effectiveness of the proposed method will be shown through several numerical examples.     

基于排队模型的电动汽车充电桩数量优化设计

通过对电动汽车充电行为的分析,采用顾客带有不耐烦和止步行为的有限容量M/M/C/N排队模型对充电场所充电服务系统进行建模.通过求解模型的平衡方程,获得系统稳态下的队长分布及其它多项性能指标.以系统性能指标为基础,从经济效益与社会效益的角度提出了充电服务系统优化设计的目标函数,并通过数字例子对优化模型进行了说明.分析了模型参数对系统充电桩最优配置数的影响,从中可看出,系统容量、顾客的不耐烦率及顾客的止步概率等参数对充电桩数量的优化设计都会产生不可忽视的影响.     

Controlled Nonlinear Stochastic Delay Equations: Part II: Approximations and Pipe-Flow Representations

This is the second part of a work dealing with key issues that have not been addressed in the modeling and numerical optimization of nonlinear stochastic delay systems. We consider new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. Part I was concerned with issues concerning the class of admissible controls and their approximations, since the classical definitions are inadequate for our models. This part is concerned with transportation equation representations and their approximations. Such representations of nonlinear stochastic delay models have been crucial in the development of numerical algorithms with much reduced memory and computational requirements. The representations for the new models are not obvious and are developed. They also provide a template for the adaptation of the Markov chain approximation numerical methods.     

Controlled Nonlinear Stochastic Delay Equations: Part I: Modeling and Approximations

This two-part paper deals with ??foundational?? issues that have not been previously considered in the modeling and numerical optimization of nonlinear stochastic delay systems. There are new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. There are two basic and interconnected themes for these models. The first, dealt with in this part, concerns the definition of admissible control. The classical definition of an admissible control as a nonanticipative relaxed control is inadequate for these models and needs to be extended. This is needed for the convergence proofs of numerical approximations for optimal controls as well as to have a well-defined model. It is shown that the new classes of admissible controls do not enlarge the range of the value functions, is closed (together with the associated paths) under weak convergence, and is approximatable by ordinary controls. The second theme, dealt with in Part II, concerns transportation equation representations, and their role in the development of numerical algorithms with much reduced memory and computational requirements.     

Elliptic cone optimization and primal–dual path-following algorithms

In elliptic cone optimization problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of the so-called elliptic cones. We present some general classes of optimization problems that can be cast as elliptic cone programmes such as second-order cone programmes and circular cone programmes. We also describe some real-world applications of this class of optimization problems. We study and analyse the Jordan algebraic structure of the elliptic cones. Then, we present a glimpse of the duality theory associated with elliptic cone optimization. A primal–dual path-following interior-point algorithm is derived for elliptic cone optimization problems. We prove the polynomial convergence of the proposed algorithms by showing that the logarithmic barrier is a strongly self-concordant barrier. The numerical examples show the path-following algorithms are efficient.     

An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels

This paper proposes a new design optimization framework for suspension systems considering the kinematic characteristics, such as the camber angle, caster angle, kingpin inclination angle, and toe angle in the presence of uncertainties. The coordinates of rear inner hardpoints of upper control arm and lower control arm of double wishbone suspension are considered as the design variables, as well as the uncertain parameters. In this way, the actual values of the design variables will vary surrounding their nominal values. The variations result in uncertainties that are described as interval variables with lower and upper bounds. The kinematic model of the suspension is developed in software ADAMS. A high-order response surface model using the zeros of Chebyshev polynomials as sampling points is established, termed as Chebyshev metamodel, to approximate the kinematic model. The Chebyshev meta-model is expected to provide higher approximation accuracy. Interval uncertain optimization problems usually involve a nested computationally expensive double-loop optimization process, in which the inner loop optimization is to calculate the bounds of the interval design functions, while the outer loop is to search the optimum for the deterministic optimization problem. To reduce the computational cost, the interval arithmetic is introduced in the inner loop to improve computational efficiency without compromising numerical accuracy. The numerical results show the effectiveness of the proposed design method.     

Optimal Design of Plane Frames under Uncertainty: A Comparison of Two Approaches

Using the first collapse theorem, the necessary and sufficient survival conditions of an elasto-plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the design of plane frames is considered, where the applied load is supposed to be stochastic. In the first approach the recourse problem will be formulated in the standard form of stochastic linear programming (SLP). In order to apply efficient numerical solution procedures (LP-solvers), approximate recourse problems based on discretization (RPD) and the expected value problem (EVP) are introduced. In the second approach – based on the yield condition – a quadratic cost function will be introduced. After the formulation of the stochastic optimization problem, the expected cost based optimization problem (ECBOP) and the minimum expected cost problem (MECP) are formulated as representatives of appropriate substitute problems. Subsequently, comparative numerical results using these methods are presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Performance analysis of distributed solution approaches in simulation-based optimization

Applying computationally expensive simulations in design or process optimization results in long-running solution processes even when using a state-of-the-art distributed algorithm and hardware. Within these simulation-based optimization problems the optimizer has to treat the simulation systems as black-boxes. The distributed solution of this kind of optimization problem demands efficient utilization of resources (i.e. processors) and evaluation of the solution quality. Analyzing the parallel performance is therefore an important task in the development of adequate distributed approaches taking into account the numerical algorithm, its implementation, and the used hardware architecture. In this paper, simulation-based optimization problems are characterized and a distributed solution algorithm is presented. Different performance analysis techniques (e.g. scalability analysis, computational complexity) are discussed and a new approach integrating parallel performance and solution quality is developed. This approach combines a priori and a posteriori techniques and can be applied in early stages of the solution process. The feasibility of the approach is demonstrated by applying it to three different classes of simulation-based optimization problems from groundwater management.     

A differentiated service scheme to optimize website revenues

We consider a website host server with web quality of service (QoS) capabilities to offer differentiated services. A quantitative modelling framework is set up to analyse the economic benefits of differentiated services and to build optimization models for managing the website host's connection bandwidth to the Internet (which is assumed to be the bottleneck factor determining the QoS). Three models are formulated corresponding to three operational scenarios to provide differentiated services. The first is for the marketing manager to classify visit requests as premium or basic when the information technology (IT) manager has already reserved bandwidths for the two classes, and the second is for the IT manager to allocate the total available bandwidth to each class when the marketing manager has already designated which visit requests are premium and which are basic. The third is for the joint optimization of request classification and bandwidth allocation when centralized coordination is possible. Analytic results are obtained for a special case that corresponds to very impatient customers requesting large amounts of data. Qualitative insights gained and numerical results obtained strongly support the implementation of differentiated services. More interestingly, the decentralized models that use simple and rough-cut rules yield solutions almost as good as the joint optimization model.     

A black-box scatter search for optimization problems with integer variables

The goal of this work is the development of a black-box solver based on the scatter search methodology. In particular, we seek a solver capable of obtaining high quality outcomes to optimization problems for which solutions are represented as a vector of integer values. We refer to these problems as integer optimization problems. We assume that the decision variables are bounded and that there may be constraints that require that the black-box evaluator is called in order to know whether they are satisfied. Problems of this type are common in operational research areas of applications such as telecommunications, project management, engineering design and the like.Our experimental testing includes 171 instances within four classes of problems taken from the literature. The experiments compare the performance of the proposed method with both the best context-specific procedures designed for each class of problem as well as context-independent commercial software. The experiments show that the proposed solution method competes well against commercial software and that can be competitive with specialized procedures in some problem classes.     

Comparison of non-linear optimization algorithms for yield curve estimation

The yield curve is a very important financial tool used in investment and policy decisions. Its estimation from market data is essentially a non-linear optimization problem. In this paper, we compare a diversity of non-linear optimization algorithms for estimating yield curves based on actual bond market data and conclude that certain classes of algorithms are more effective due to the nature of the problem.     

Upper bound limit analysis using the weak form quadrature element method

Limit analysis is a useful tool for design and safety assessment of structures in civil and geotechnical engineering. In the present study, a newly developed high order algorithm-the weak form quadrature element method is reformulated for upper bound limit analysis. The dual formulations of the kinematic theorem are employed with the nodal stresses chosen as the optimization variables. The weak form equilibrium constraint is numerically integrated by Lobatto integration and then the nodal derivatives are approximated by differential quadrature analogue. The resulting optimization problem is formulated as a standard second-order cone programming problem and solved by the optimization toolbox Mosek. This paper aims to improve the efficiency of the existing numerical limit algorithms especially for problems with singularities such as cracked structures and to overcome the well-known volumetric locking occurred for incompressible materials. Some numerical tests are given to show the accuracy and efficiency of the present method.     

A class of convexification and concavification methods for non-monotone optimization problems

A class of convexification and concavification methods are proposed for solving some classes of non-monotone optimization problems. It is shown that some classes of non-monotone optimization problems can be converted into better structured optimization problems, such as, concave minimization problems, reverse convex programming problems, and canonical D.C. programming problems by the proposed convexification and concavification methods. The equivalence between the original problem and the converted better structured optimization problem is established.     

Guaranteed Cost Observer–Based Controls for a Class of Uncertain Neutral Time-Delay Systems

In this paper, guaranteed-cost observer-based controls for a class of uncertain neutral time-delay systems are considered. The asymptotic stabilization for the uncertain neutral systems is guaranteed with an observer-based feedback control. The linear matrix inequality (LMI) approach is used to design the observer-based feedback control system. Two classes of observer-based controls are proposed and their guaranteed costs are given. The control and observer gains are given from the LMI feasible solutions. A convex optimization problem with LMIs is formulated to design the optimal guaranteed-cost observer-based controls which minimize the guaranteed cost of the system considered. A numerical example is given to illustrate the results.The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 93-2213-E-214-020     

Design of dynamic controller for neutral differential systems with delay in control input

In this paper, the design problem of dynamic output feedback controller for asymptotic stabilization of a class of neutral systems have been considered. A criterion for the existence of such controllers is derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. A parameterized characterization of the controllers is given in terms of the feasible solutions to the LMIs, which can be solved by various convex optimization algorithms. A numerical example is given to illustrate the proposed design method.     

Dynamic server allocation for unstable queueing networks with flexible servers

This paper is concerned with the dynamic assignment of servers to tasks in queueing networks where demand may exceed the capacity for service. The objective is to maximize the system throughput. We use fluid limit analysis to show that several quantities of interest, namely the maximum possible throughput, the maximum throughput for a given arrival rate, the minimum arrival rate that will yield a desired feasible throughput, and the optimal allocations of servers to classes for a given arrival rate and desired throughput, can be computed by solving linear programming problems. We develop generalized round-robin policies for assigning servers to classes for a given arrival rate and desired throughput, and show that our policies achieve the desired throughput as long as this throughput is feasible for the arrival rate. We conclude with numerical examples that illustrate the points discussed and provide insights into the system behavior when the arrival rate deviates from the one the system is designed for.     

Copositive optimization - Recent developments and applications

Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications.     

由FR共轭梯度法控制的两类优化算法的全局收敛性

１引言 考虑无约束优化问题其中ｆ：Ｒｎ→Ｒ是一阶可微函数．求解（１）的非线性共轭梯度法具有如下形式：其中ｇｋ＝　ｆ（ｘｋ）,ａｋ是通过某种线搜索获得的步长,纯量βｋ的选取使得方法（２）—（３）在ｆ（ｘ）是严格凸二次函数且采用精确线搜索时化为线性共轭梯度法［１］．比较常见的βｋ的取法有Ｆｌｅｔｃｈｅｒ－Ｒｅｅｖｅｓ（ＦＲ）公式［２］和Ｐｏｌａｋ－Ｒｉｂｉｅｒｅ－Ｐｏｌｙａｋ（ＰＲＰ）公式［３－４］等．它们分别为其中　　　取欧几里得范数．对于一般非线性函数,ＦＲ方法具有较好的理论收敛性［５－６］,而…     

Numerical aspects of X‐SIMP‐based topology optimization

The discretization of topology design problems on the basis of the finite‐element‐method results in general in large‐scale combinatorial optimization problems, which are usually relaxed by the introduction of a continuous material density function as design variable. To avoid optimal designs containing unfavourable microstructures such as the well‐known “checkerboard” patterns, the relaxed problem can be regularized by the X‐SIMP‐approach, which penalizes intermediate density values as well as high density gradients within the design domain. In this context we discuss numerical aspects of the X‐SIMP‐based regularization such as the discretization of the regularized problem, the formulation of the corresponding stiffness matrix and the numerical solution of the discretized problem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)