A balanced whale optimization algorithm for constrained engineering design problems

In this study, two novel effective strategies composed of Lévy flight and chaotic local search are synchronously introduced into the whale optimization algorithm (WOA) to guide the swarm and further promote the harmony between the inclusive exploratory and neighborhood-informed capacities of the conventional technique and investigate the core searching capabilities of WOA in dealing with optimization tasks. However, the conventional WOA may simply be stuck at local optima or the global best may not be obtained successfully when tackling more complex optimization landscapes, including the multimodal and high dimensional scenarios. To substantiate the efficacy of the enhanced method, it is compared to a set of well-regarded variants of particle swarm optimization and differential evolution. The used benchmark problems are composed of unimodal, multimodal, and fixed-dimensions multimodal functions. Additionally, the proposed balanced method is applied to realize three practical, well-known mathematical models such as tension/compression spring, welded beam, pressure vessel design, three-bar truss design, and I-beam design problems. The experimental results and analysis reveal that the proposed algorithm can outperform other competitors in terms of the convergence speed and the quality of solutions. Promisingly, the proposed method can be treated as an effective and efficient auxiliary tool for more complex optimization models and scenarios.     

New maximum entropy-based algorithm for structural design optimization

A new algorithm based on nonlinear transformation is proposed to improve the classical maximum entropy method and solve practical problems of reliability analysis. There are three steps in the new algorithm. Firstly, the performance function of reliability analysis is normalized, dividing by its value when each input is the mean value of the corresponding random variable. Then the nonlinear transformation of such normalized performance function is completed by using a monotonic nonlinear function with an adjustable parameter. Finally, the predictions of probability density function and/or the failure probability in reliability analysis are achieved by looking the result of transformation as a new form of performance function in the classical procedure of maximum entropy method in which the statistic moments are given through the univariate dimension reduction method. In the proposed method, the uncontrollable error of integration on the infinite interval is removed by transforming it into a bounded one. Three typical nonlinear transformation functions are studied and compared in the numerical examples. Comparing with results from Monte Carlo simulation, it is found that a proper choice of the adjustable parameter can lead to a better result of the prediction of failure probability. It is confirmed in the examples that result from the proposed method with the arctangent transformation function is better than the other transformation functions. The error of prediction of failure probability is controllable if the adjustable parameter is chosen in a given interval, but the suggested value of the adjustable parameter can only be given empirically.     

A new multi-objective discrete robust optimization algorithm for engineering design

This paper proposes a novel multi-objective discrete robust optimization (MODRO) algorithm for design of engineering structures involving uncertainties. In the present MODRO procedure, grey relational analysis (GRA), coupled with principal component analysis (PCA), was used as a multicriteria decision making model for converting multiple conflicting objectives into one unified cost function. The optimization process was iterated using the successive Taguchi approach to avoid the limitation that the conventional Taguchi method fails to deal with a large number of design variables and design levels. The proposed method was first verified by a mathematical benchmark example and a ten-bar truss design problem; and then it was applied to a more sophisticated design case of full scale vehicle structure for crashworthiness criteria. The results showed that the algorithm is able to achieve an optimal design in a fairly efficient manner attributable to its integration with the multicriteria decision making model. Note that the optimal design can be directly used in practical applications without further design selection. In addition, it was found that the optimum is close to the corresponding Pareto frontier generated from the other approaches, such as the non-dominated sorting genetic algorithm II (NSGA-II), but can be more robust as a result of introduction of the Taguchi method. Due to its independence on metamodeling techniques, the proposed algorithm could be fairly promising for engineering design problems of high dimensionality.     

Minimax approach to structural optimization problems

The problem of determining the shape of an elastic body which is optimal for a whole class of loads is formulated. Its general solution scheme, based on a minimax approach, is indicated. The optimization problem of an elastic beam is studied for both the simply supported case and the cantilevered case, and some specific properties of the optimal shape are determined.The author is grateful to Professor A. I. Lurie, Professor F. L. Chernousko, and Professor G. S. Shapiro for useful discussions of the results.     

Enriched self-adjusted performance measure approach for reliability-based design optimization of complex engineering problems

For reliability-based design optimization (RBDO) of practical structural/mechanical problems under highly nonlinear constraints, it is an important characteristic of the performance measure approach (PMA) to show robustness and high convergence rate. In this study, self-adjusted mean value is used in the PMA iterative formula to improve the robustness and efficiency of the RBDO-based PMA for nonlinear engineering problems based on dynamic search direction. A novel merit function is applied to adjust the modified search direction in the enriched self-adjusted mean value (ESMV) method, which can control the instability and value of the step size for highly nonlinear probabilistic constraints in RBDO problems. The convergence performance of the enriched self-adjusted PMA is illustrated using four nonlinear engineering problems. In particular, a complex engineering example of aircraft stiffened panel is used to compare the RBDO results of different reliability methods. The results demonstrate that the proposed self-adjusted steepest descent search direction can improve the computational efficiency and robustness of the PMA compared to existing modified reliability methods for nonlinear RBDO problems.     

Nonmonotone algorithm for minimax optimization problems

Many real life problems can be stated as a minimax optimization problem, such as the problems in economics, finance, management, engineering and other fields. In this paper, we present an algorithm with nonmonotone strategy and second-order correction technique for minimax optimization problems. Using this scheme, the new algorithm can overcome the difficulties of the Maratos effect occurred in the nonsmooth optimization, and the global and superlinear convergence of the algorithm can be achieved accordingly. Numerical experiments indicate some advantages of this scheme.     

Improved optimization methods for image registration problems

Numerical Algorithms - In this paper, we propose new multilevel optimization methods for minimizing continuously differentiable functions obtained by discretizing models for image registration...     

Approximate solutions to idealized structural dynamic optimization problems

This paper presents an application of two solution methods to the least-weight optimization of a simple sandwich beam with a frequency constraint. The first method, an adaptation of a numericalshooting technique used in optimal control, is found to give good results if unknown initial conditions can be reasonably approximated. The second method, a perturbation method used widely in theoretical mechanics and aerodynamics, yields approximate analytical expressions. These expressions can be used in turn to estimate starting values for the numerical technique. Converged numerical results are presented for a least-weight cantilever beam with fixed fundamental frequency and for a beam on simple supports with fixed fundamental frequency.     

Outer approximation algorithm for nondifferentiable optimization problems

It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ⁿ can be expressed as minimizing max{g(x, y)|y X}, whereg is a support function forf[f(x) g(x, y), for ally X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.This research was supported by the Science and Engineering Research Council, UK, and by the National Science Foundation under Grant No. ECS-79-13148.     

An algorithm for composite nonsmooth optimization problems

Nonsmooth optimization problems are divided into two categories. The first is composite nonsmooth problems where the generalized gradient can be approximated by information available at the current point. The second is basic nonsmooth problems where the generalized gradient must be approximated using information calculated at previous iterates.Methods for minimizing composite nonsmooth problems where the nonsmooth function is made up from a finite number of smooth functions, and in particular max functions, are considered. A descent method which uses an active set strategy, a nonsmooth line search, and a quasi-Newton approximation to the reduced Hessian of a Lagrangian function is presented. The Theoretical properties of the method are discussed and favorable numerical experience on a wide range of test problems is reported.This work was carried out at the University of Dundee from 1976–1979 and at the University of Kentucky at Lexington from 1979–1980. The provision of facilities in both universities is gratefully acknowledged, as well as the support of NSF Grant No. ECS-79-23272 for the latter period. The first author also wishes to acknowledge financial support from a George Murray Scholarship from the University of Adelaide and a University of Dundee Research Scholarship for the former period.     

Outer approximation algorithm for nondifferentiable optimization problems

Omissions from the list of references of Ref. 1 are corrected.     

Evolutionary structural optimization for problems with stiffness constraints

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.     

P2q hierarchical decomposition algorithm for quantile optimization: application to irrigation strategies design

Decision theory dealing with uncertainty is usually considering criteria such as expected, minimum or maximum values. In economic areas, the quantile criterion is commonly used and provides significant advantages. This paper gives interest to the quantile optimization in decision making for designing irrigation strategies. We developed P2q, a hierarchical decomposition algorithm which belongs to the branching methods family. It consists in repeating the creation, evaluation and selection of smaller promising regions. Opposite to common approaches, the main criterion of interest is the α-quantile where α is related to the decision maker risk acceptance. Results of an eight parameters optimization problem are presented. Quantile optimization provided optimal irrigation strategies that differed from thus reached with expected value optimization, responding more accurately to the decision maker preferences.     

Note: An algorithm to solve polyhedral convex set optimization problems

We provide a counterexample to the remark in Löhne and Schrage [An algorithm to solve polyhedral convex set optimization problems, Optimization 62 (2013), pp. 131-141] that every solution of a polyhedral convex set optimization problem is a pre-solution. A correct statement is that every solution of a polyhedral convex set optimization problem obtained by the algorithm SetOpt is a pre-solution. We also show that every finite infimizer and hence every solution of a polyhedral convex set optimization problem contains a pre-solution.     

Improved particle swarm algorithm for hydrological parameter optimization

In this paper, a new method named MSSE-PSO (master-slave swarms shuffling evolution algorithm based on particle swarm optimization) is proposed. Firstly, a population of points is sampled randomly from the feasible space, and then partitioned into several sub-swarms (one master swarm and other slave swarms). Each slave swarm independently executes PSO or its variants, including the update of particles’ position and velocity. For the master swarm, the particles enhance themselves based on the social knowledge of master swarm and that of slave swarms. At periodic stage in the evolution, the master swarm and the whole slave swarms are forced to mix, and points are then reassigned to several sub-swarms to ensure the share of information. The process is repeated until a user-defined stopping criterion is reached. The tests of numerical simulation and the case study on hydrological model show that MSSE-PSO remarkably improves the accuracy of calibration, reduces the time of computation and enhances the performance of stability. Therefore, it is an effective and efficient global optimization method.     

Nonmonotone trust region algorithm for unconstrained optimization problems

In this paper, a nonmonotone trust region algorithm for unconstrained optimization problems is presented. In the algorithm, a kind of nonmonotone technique, which is evidently different from Grippo, Lampariello and Lucidi’s approach, is used. Under mild conditions, global and local convergence results of the algorithm are established. Preliminary numerical results show that the new algorithm is efficient.     

Globally convergent algorithm for nonlinearly constrained optimization problems

A new globally convergent algorithm for minimizing an objective function subject to equality and inequality constraints is presented. The algorithm determines a search direction by first solving a linear program and using the information gained thereby to define a quadratic approximation, with a guaranteed solution, to the original problem; the solution of the quadratic problem is the desired search direction. The algorithm incorporates a new method for choosing the penalty parameter. Numerical results illustrate the performance of the algorithm.The author wishes to thank Professor D. Q. Mayne and Dr. F. A. Pantoja for critically reviewing the first draft of this paper, for their suggestions, criticism, and contributions to some of the proofs. Support of the UK Science Research and Engineering Council is gratefully acknowledged.     

A modified ODE-based algorithm for unconstrained optimization problems

This paper presents a modified ODE-based algorithm for unconstrained optimization problems. It combines the idea of IMPBOT algorithm with nonmonotone and subspace techniques. The main feature of this method is that at each iteration, a lower dimensional system of linear equations is solved to obtain a trial step. Under some standard assumptions, the method is proven to be globally convergent. Numerical results show the efficiency of this proposed method in practical computation.     

A saddle-point theorem with application to structural optimization

The relaxation for optimal complicance design is independent of whether the underlying elastic problem is formulated in terms of displacements or strains. For the purposes of numerical experimentation and computation, it may be advantageous to formulate optimal design problems in terms of displacements as is done in Ref. 1. The relaxed problem delivered by the displacement-based formulation is of min-min-max type. Because of this, efficient numerical implementation is hampered by the order of the last two min-max operations. We show here that the last two min-max operations may be exchanged, facilitating an efficient numerical algorithm. We remark that the rigorous results given here corroborate the numerical methods and experiments given in Ref. 1.This work was supported by NSF Grant DMS-92-05158.