Composite Nonsmooth Optimization Using Approximate Generalized Gradient Vectors

A new approximation method is presented for directly minimizing a composite nonsmooth function that is locally Lipschitzian. This method approximates only the generalized gradient vector, enabling us to use directly well-developed smooth optimization algorithms for solving composite nonsmooth optimization problems. This generalized gradient vector is approximated on each design variable coordinate by using only the active components of the subgradient vectors; then, its usability is validated numerically by the Pareto optimum concept. In order to show the performance of the proposed method, we solve four academic composite nonsmooth optimization problems and two dynamic response optimization problems with multicriteria. Specifically, the optimization results of the two dynamic response optimization problems are compared with those obtained by three typical multicriteria optimization strategies such as the weighting method, distance method, and min–max method, which introduces an artificial design variable in order to replace the max-value cost function with additional inequality constraints. The comparisons show that the proposed approximation method gives more accurate and efficient results than the other methods.     

New Reflection Generator for Simulated Annealing in Mixed-Integer/Continuous Global Optimization

To reduce the well-known jamming problem in global optimization algorithms, we propose a new generator for the simulated annealing algorithm based on the idea of reflection. Furthermore, we give conditions under which the sequence of points generated by this simulated annealing algorithm converges in probability to the global optimum for mixed-integer/continuous global optimization problems. Finally, we present numerical results on some artificial test problems as well as on a composite structural design problem.     

Fast algorithms for designing variable FIR notch filters

The paper presents fast algorithms for designing finite impulse response (FIR) notch filters. The aim is to design a digital FIR notch filter so that the magnitude of the filter has a deep notch at a specified frequency, and as the notch frequency changes, the filter coefficients should be able to track the notch fast in real time. The filter design problem is first converted into a convex optimization problem in the autocorrelation domain. The frequency response of the autocorrelation of the filter impulse response is compared with the desired filter response and the integral square error is minimized with respect to the unknown autocorrelation coefficients. Spectral factorization is used to calculate the coefficients of the filter. In the optimization process, the computational advantage is obtained by exploiting the structure of the Hessian matrix which consists of a Toeplitz plus a Hankel matrix. Two methods have been used for solving the Toeplitz‐plus‐Hankel system of equations. In the first method, the computational time is reduced by using Block–Levinson's recursion for solving the Toeplitz‐plus‐Hankel system of matrices. In the second method, the conjugate gradient method with different preconditioners is used to solve the system. Comparative studies demonstrate the computational advantages of the latter. Both these algorithms have been used to obtain the autocorrelation coefficients of notch filters with different orders. The original filter coefficients are found by spectral factorization and each of these filters have been tested for filtering synthetic as well as real‐life signals. Copyright © 2007 John Wiley & Sons, Ltd.     

The analysis of expected fitness and success ratio of two heuristic optimizations on two bimodal MaxSAT problems

Heuristic algorithms, especially hill-climbing algorithms, are prone to being trapped by local optimization. Many researchers have focused on analyzing convergence and runtime of some heuristic algorithms on SAT-solving problems. However, there is rare work on analyzing success ratio (the ratio of the number of runs that find the global maxima of optimization versus the total runs) and expected fitness at each generation. Expected fitness at each generation could lead us to better understand the expected fitness of a heuristic algorithm could find on the problem to be solve at a certain generation. Success ratio help us understand with what a probability a heuristic algorithm could find the global optimization of the problem to be solved. So, this paper analyzed expected fitness and success ratio of two hill-climbing algorithms on two bimodal MaxSAT problems by using Markov chain. The theoretical and experimental results showed that though hill-climbing algorithms (both hill-climbing RandomWalk and Local (1+1)EA) converged faster, they could not always find global maxima of bimodal SAT-solving problems. The success ratios of hill-climbing algorithms usually have their own limits. The limit of success ratio is dependent on the SAT-solving problem itself and the expected distribution of initial bit string, and independent on whether hill-climbing RandomWalk or Local (1+1)EA is used.     

On the analysis of a dynamic evolutionary algorithm

Evolutionary algorithms are applied as problem-independent optimization algorithms. They are quite efficient in many situations. However, it is difficult to analyze even the behavior of simple variants of evolutionary algorithms like the (1+1) EA on rather simple functions. Nevertheless, only the analysis of the expected run time and the success probability within a given number of steps can guide the choice of the free parameters of the algorithms. Here static (1+1) EAs with a fixed mutation probability are compared with dynamic (1+1) EAs with a simple schedule for the variation of the mutation probability. The dynamic variant is first analyzed for functions typically chosen as example-functions for evolutionary algorithms. Afterwards, it is shown that it can be essential to choose the suitable variant of the (1+1) EA. More precisely, functions are presented where each static (1+1) EA has exponential expected run time while the dynamic variant has polynomial expected run time. For other functions it is shown that the dynamic (1+1) EA has exponential expected run time while a static (1+1) EA with a good choice of the mutation probability has polynomial run time with overwhelming probability.     

Subset simulation for multi-objective optimization

Subset simulation is an efficient Monte Carlo technique originally developed for structural reliability problems, and further modified to solve single-objective optimization problems based on the idea that an extreme event (optimization problem) can be considered as a rare event (reliability problem). In this paper subset simulation is extended to solve multi-objective optimization problems by taking advantages of Markov Chain Monte Carlo and a simple evolutionary strategy. In the optimization process, a non-dominated sorting algorithm is introduced to judge the priority of each sample and handle the constraints. To improve the diversification of samples, a reordering strategy is proposed. A Pareto set can be generated after limited iterations by combining the two sorting algorithms together. Eight numerical multi-objective optimization benchmark problems are solved to demonstrate the efficiency and robustness of the proposed algorithm. A parametric study on the sample size in a simulation level and the proportion of seed samples is performed to investigate the performance of the proposed algorithm. Comparisons are made with three existing algorithms. Finally, the proposed algorithm is applied to the conceptual design optimization of a civil jet.     

A practical fractional numerical optimization method for designing economically and environmentally friendly super-tall buildings

Due to the complexity of super tall buildings, many well-known optimization algorithms are not well applicable. Using structural lateral system of super tall buildings as engineering background, the paper developed a practical fractional numerical optimization method (FNOM), which applies fractional strategy and quasi-constant assumption, to reduce material cost and embodied carbon cost by searching the optimal structural dimensions. Firstly, two kinds of relationships among optimization variables (structural dimensions), driven design constraints (the interstory drift and the natural period) and optimization objective (cost including material cost and embodied carbon cost) are mathematically modelled. Genetic algorithm (GA) is then introduced to search the optimal structural dimensions based on the quasi-constant assumption of virtual work and internal work of the inactive components. Thirdly, fractional strategy is applied to create assemblies composed of different structural component sets, and the assemblies are then to be optimized in proper sequences. Fourthly, FNOM is implemented as a user-friendly software called C-FNO to practically support the preliminary design of super-tall buildings. Finally, a 700 m high super-tall building is employed to illustrate FNOM by using C-FNO, and the results show that only three design constraints of the interstory drift, the natural period and the stress ratio need to be solved during each optimization step. Belt truss, mega column, outrigger truss and shear wall of the super tall building should be optimized in sequence to save more cost. A great amount of cost can be still saved for the super tall building with the normal traditional design.     

Nonlinear optimization and support vector machines

Support Vector Machine (SVM) is one of the most important class of machine learning models and algorithms, and has been successfully applied in various fields. Nonlinear optimization plays a crucial role in SVM methodology, both in defining the machine learning models and in designing convergent and efficient algorithms for large-scale training problems. In this paper we present the convex programming problems underlying SVM focusing on supervised binary classification. We analyze the most important and used optimization methods for SVM training problems, and we discuss how the properties of these problems can be incorporated in designing useful algorithms.     

Lamination Design Optimization for Continuous Fiber Reinforced Composites of Variable Thicknesses北大核心CSCD

由于具备高的比强度、比刚度,利用连续纤维增强复合材料代替传统金属材料以实现结构轻量化正受到设计者们的广泛关注。然而,结构的复杂性给复合材料的铺层设计与优化带来了很大的挑战。针对航空用复合材料铺层设计约束多的问题,通过逐步构建设计变量准确表达结构的铺层信息。基于经典遗传算法框架,结合各设计变量特点,定义了铺层优化算法中的遗传算子,通过引入“修复”策略保证了每一代解都能满足设计约束,分布在可行域区间内。最后利用精英保留策略提高了算法的局部寻优能力,可以降低复杂复合材料结构铺层设计的计算成本。通过解决经典benchmark问题并与已有优化结果的比较,验证了前述铺层优化算法的全局、局部寻优能力,为工程实际中的复合材料铺层设计优化提供了理论支撑。     

Optimal Design of Hole Reinforcements for Composite Structures

In many technical applications like aerospace and automotive structures, holes in thin-walled composite components are necessary for some reason. It easily happens that the presence of a hole results in a detrimental stress concentration in the vicinity of the hole with a strength degradation and premature failure of the structure, respectively. In order to avoid the aforementioned overloading and to achieve a sufficient strength, in practice, a local reinforcement is employed. In the present study, reinforcements by elliptic doublers,as well as doublers adapted to reinforcement requirements in a layerwise manner, are considered. The increasing demands of a low weight and high strength for modern structures lead to the problem of an optimal reinforcement design. For this purpose, an appropriate optimization model is set up, a structural model is developed to describe the mechanical behavior (displacements, stresses, etc.) of such structures, and the techniques of mathematical structural optimization are used to find an optimal design in a systematic manner. In this study, the finite-element method is applied to the structural analysis. Eventually, an appropriate mathematical optimization algorithm is used to approach the desired design optimum in an iterative way. The implemented procedure works with a good reliability and efficiency and yields optimal reinforcement designs which are very useful for direct engineering applications.     

Sobre la necesidad de controlar el error de discretización de elementos finitos en optimización de forma estructural con algoritmos evolutivos

This work analyzes the influence of the discretization error contained in the Finite Element (FE) analyses of each design configuration proposed by the structural shape optimization algorithms over the behavior of the algorithm. The paper clearly shows that if FE analyses are not accurate enough, the final solution provided by the optimization algorithm will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper proposes the combination of two strategies to reduce the computational cost related to the use of mesh adaptivity in evolutionary optimization algorithms: (a) the use of the algorithm described by Bugeda et al. [1] which reduces the computational cost associated to the adaptive FE analysis of each geometrical configuration and, (b) the successive increase of the required accuracy of the FE analyses in order to obtain a considerable reduction of the computational cost in the early stages of the optimization process.     

Integrated design and optimization of a complex nonlinear hybrid electric powertrain including simulations in time domain

This paper deals with the design and optimization of hybrid electric powertrains. Therefore basic relations of the behavior of hybrid electric powertrain systems and the controller design are introduced. Based on models of typical hybrid electric system components principal optimization approaches with respect to performance parameters like efficiency, availability, lifetime, etc. are shown. Hereby an optimization algorithm based on a global optimization technique is applied. Using the example of a fuel cell based hybrid electric powertrain system the approaches are introduced and compared to each using time-domain simulations integrated in optimization algorithms. The results show that both approaches are appropriate to design the system as well as the controllers. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust parameter design for multi-response problems

Most real world search and optimization problems naturally involve multiple responses. In this paper we investigate a multiple response problem within desirability function framework and try to determine values of input variables that achieve a target value for each response through three meta-heuristic algorithms such as genetic algorithm (GA), simulated annealing (SA) and tabu search (TS). Each algorithm has some parameters that need to be accurately calibrated to ensure the best performance. For this purpose, a robust calibration is applied to the parameters by means of Taguchi method. The computational results of these three algorithms are compared against each others. The superior performance of SA over TS and TS over GA is inferred from the obtained results in various situations.     

On the analysis of a simple evolutionary algorithm on quadratic pseudo-boolean functions

Evolutionary algorithms are randomized search heuristics, which are often used as function optimizers. In this paper the well-known (1+1) Evolutionary Algorithm ((1+1) EA) and its multistart variants are studied. Several results on the expected runtime of the (1+1) EA on linear or unimodal functions have already been presented by other authors. This paper is focused on quadratic pseudo-boolean functions, i.e., polynomials of degree 2, a class of functions containing NP-hard optimization problems. Subclasses of the class of all quadratic functions are identified where the (1+1) EA is efficient, for other subclasses the (1+1) EA has exponential expected runtime, but a large enough success probability within polynomial time such that a multistart variant of the (1+1) EA is efficient. Finally, a particular quadratic function is identified where the EA and its multistart variants fail in polynomial time with overwhelming probability.     

The analysis of evolutionary algorithms on sorting and shortest paths problems

The analysis of evolutionary algorithms is up to now limited to special classes of functions and fitness landscapes. E.g., it is not possible to characterize the set of TSP instances (or another NP-hard combinatorial optimization problem) which are solved by a generic evolutionary algorithm (EA) in an expected time bounded by some given polynomial. As a first step from artificial functions to typical problems from combinatorial optimization, we analyze simple EAs on well-known problems, namely sorting and shortest paths. Although it cannot be expected that EAs outperform the well-known problem specific algorithms on these simple problems, it is interesting to analyze how EAs work on these problems. The following results are obtained:– Sorting is the maximization of sortedness which is measured by one of several well-known measures of presortedness. The different measures of presortedness lead to fitness functions of quite different difficulty for EAs.– Shortest paths problems are hard for all types of EA, if they are considered as single-objective optimization problems, whereas they are easy as multi-objective optimization problems.     

Parameter Identification Based on First Order Approximation Neural Networks

Constitutive models for structural analyses contain material parameters. Usually not all of them can be determined a priori with sufficient accuracy. They must be set such that numerical results agree with available measurements as well as possible. Hence, an inverse problem must be solved. In order to keep the number of the required numerical calculations for parameter identification as small as possible, back analyses are performed iteratively. In each iteration step, a backpropagation artificial neural network (BPANN) is trained to approximate results of already performed numerical analyses. In this paper the classical zero‐order training algorithm is extended in order to obtain first‐order approximation neural networks. Based on the trained BPANN, a prognosis of optimal parameters can be obtained.     

Optimization of Mechanically Coupled Composite Pipes by Genetic Algorithms

The optimization of composite components with regard to minimum weight and maximum load bearing capacity in consideration of multiple constraints is an optimization problem of rather high complexity. Genetic Algorithms are a good choice for solving such problems. In this paper the formulation of a Genetic Algorithm for the simultaneous optimization of two thin walled, mechanically coupled composite pipes subjected to a combination of thermal and mechanical loads is presented. The optimization goal is the minimization of the total mass of the pipe arrangement taking into account multiple design constraints. It is shown that Genetic Algorithms are valueable tools for solving optimization problems with a large number of parameters. Furthermore, it is possible to find additional, perhaps practicable, close‐to‐optimal configurations as a byproduct of the optimization process. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-class support vector machine optimized by inter-cluster distance and self-adaptive deferential evolution

Support vector machine (SVM) is a popular tool for machine learning task. It has been successfully applied in many fields, but the parameter optimization for SVM is an ongoing research issue. In this paper, to tune the parameters of SVM, one form of inter-cluster distance in the feature space is calculated for all the SVM classifiers of multi-class problems. Inter-cluster distance in the feature space shows the degree the classes are separated. A larger inter-cluster distance value implies a pair of more separated classes. For each classifier, the optimal kernel parameter which results in the largest inter-cluster distance is found. Then, a new continuous search interval of kernel parameter which covers the optimal kernel parameter of each class pair is determined. Self-adaptive differential evolution algorithm is used to search the optimal parameter combination in the continuous intervals of kernel parameter and penalty parameter. At last, the proposed method is applied to several real word datasets as well as fault diagnosis for rolling element bearings. The results show that it is both effective and computationally efficient for parameter optimization of multi-class SVM.     

Optimum stacking sequence of composite laminates for maximum strength

A method is presented for maximum strength optimum design of symmetric composite laminates subjected to in-plane and transverse loadings. The finite element method based on shear deformation theory is used for the analysis of composite laminates. Ply orientation angles are chosen as design variables. The quadratic failure criterion which is meant to predict fracture, is used as an object function for optimum stacking sequence design of a laminated plate. The Broydon-Fletcher-Goldfarb-Shanno optimization technique is employed to solve the optimization problem effectively. Numerical results are given for various loading conditions, boundary conditions, and aspect ratios. The results show that the quadratic failure criterion such as Tsai-Hill theory is effective for the optimum structural design of composite laminates.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 393–404, May–June, 1995.     

Optimal Fine‐Scale Structures in Compliance Minimization for a Shear Load

We return to a classic problem of structural optimization whose solution requires microstructure. It is well‐known that perimeter penalization assures the existence of an optimal design. We are interested in the regime where the perimeter penalization is weak; i.e., in the effect of perimeter as a selection mechanism in structural optimization. To explore this topic in a simple yet challenging example, we focus on a two‐dimensional elastic shape optimization problem involving the optimal removal of material from a rectangular region loaded in shear. We consider the minimization of a weighted sum of volume, perimeter, and compliance (i.e., the work done by the load), focusing on the behavior as the weight ɛ of the perimeter term tends to 0. Our main result concerns the scaling of the optimal value with respect to ɛ. Our analysis combines an upper bound and a lower bound. The upper bound is proved by finding a near‐optimal structure, which resembles a rank‐2 laminate except that the approximate interfaces are replaced by branching constructions. The lower bound, which shows that no other microstructure can be much better, uses arguments based on the Hashin‐Shtrikman variational principle. The regime being considered here is particularly difficult to explore numerically due to the intrinsic nonconvexity of structural optimization and the spatial complexity of the optimal structures. While perimeter has been considered as a selection mechanism in other problems involving microstructure, the example considered here is novel because optimality seems to require the use of two well‐separated length scales.© 2016 Wiley Periodicals, Inc.