A real-coded biogeography-based optimization with mutation

Biogeography-based optimization (BBO) is a new biogeography inspired algorithm for global optimization. There are some open research questions that need to be addressed for BBO. In this paper, we extend the original BBO and present a real-coded BBO approach, referred to as RCBBO, for the global optimization problems in the continuous domain. Furthermore, in order to improve the diversity of the population and enhance the exploration ability of RCBBO, the mutation operator is integrated into RCBBO. Experiments have been conducted on 23 benchmark problems of a wide range of dimensions and diverse complexities. The results indicate the good performance of the proposed RCBBO method. Moreover, experimental results also show that the mutation operator can improve the performance of RCBBO effectively.     

Novel biogeography-based optimization algorithm with hybrid migration and global-best Gaussian mutation

The Biogeography-Based Optimization algorithm and its variants have been used widely for optimization problems. To get better performance, a novel Biogeography-Based Optimization algorithm with Hybrid migration and global-best Gaussian mutation is proposed in this paper. Firstly, a linearly dynamic random heuristic crossover strategy and an exponentially dynamic random differential mutation one are presented to form a hybrid migration operator, and the former is used to get stronger local search ability and the latter strengthen the global search ability. Secondly, a new global-best Gaussian mutation operator is put forward to balance exploration and exploitation better. Finally, a random opposition learning strategy is merged to avoid getting stuck in local optima. The experiments on the classical benchmark functions and the complexity functions from CEC-2013 and CEC-2017 test sets, and the Wilcoxon, Bonferroni-Holm and Friedman statistical tests are used to evaluate our algorithm. The results show that our algorithm obtains better performance and faster running speed compared with quite a few state-of-the-art competitive algorithms. In addition, experimental results on Minimum Spanning Tree and K-means clustering optimization show that our algorithm can cope with these two problems better than the comparison algorithms.     

A semismooth Newton method for topology optimization

This paper deals with elliptic optimal control problems for which the control function is constrained to assume values in {0, 1}. Based on an appropriate formulation of the optimality system, a semismooth Newton method is proposed for the solution. Convergence results are proved, and some numerical tests illustrate the efficiency of the method.     

On a biologically inspired topology optimization method

This work concerns the development of a biologically inspired methodology for the study of topology optimization in engineering and natural systems. The methodology is based on L systems and its turtle interpretation for the genotype–phenotype modeling of the topology development. The topology is analyzed using the finite element method, and optimized using an evolutionary algorithm with the genetic encoding of the L system and its turtle interpretation, as well as, body shape and physical characteristics. The test cases considered in this work clearly show the suitability of the proposed method for the study of engineering and natural complex systems.     

An effective krill herd algorithm with migration operator in biogeography-based optimization

Krill herd (KH) is a novel search heuristic method. To improve its performance, a biogeography-based krill herd (BBKH) algorithm is presented for solving complex optimization tasks. The improvement involves introducing a new krill migration (KM) operator when the krill updating to deal with optimization problems more efficiently. The KM operator emphasizes the exploitation and lets the krill cluster around the best solutions at the later run phase of the search. The effects of these enhancements are tested by various well-defined benchmark functions. Based on the experimental results, this novel BBKH approach performs better than the basic KH and other optimization algorithms.     

Convergence of a random optimization method for constrained optimization problems

Matyas' random optimization method (Ref. 1) is applied to the constrained nonlinear minimization problem, and its convergence properties are studied. It is shown that the global minimum can be found with probability one, even if the performance function is multimodal (has several local minima) and even if its differentiability is not ensured.The author would like to thank Professors Y. Sawaragi (Kyoto University), T. Soeda (Tokushima University), and T. Shoman (Tokushima University) for their kind advice.     

Adaptive mesh method for topology optimization of fluid flow

In order to solve the topology optimization problems of fluid flow and obtain higher resolution of the interface with a minimum of additional expense, an automatic local adaptive mesh refinement method is proposed. The optimization problem is solved by a simple but robust optimality criteria (OC) algorithm. A material distribution information based adaptive mesh method is adopted during the optimization process. The optimization procedure is provided and verified with several benchmark examples.     

Stochastic optimization using a trust-region method and random models

Mathematical Programming - In this paper, we propose and analyze a trust-region model-based algorithm for solving unconstrained stochastic optimization problems. Our framework utilizes random...     

Expected number of steps of a random optimization method

Some explanatory remarks are presented, relevant to the objections by Pang (Ref. 1) to Dorea's paper (Ref. 2).     

Expected number of steps of a random optimization method

In this paper, we give an estimate of the expected number of steps of Matya's random optimization method applied to the constrained nonlinear minimization problem. It is also shown that, in a sense, this random optimization method can be optimized by the uniform distribution, in which case the exact value of the expected number of steps is computed.     

Zeroth-order optimization with orthogonal random directions

Mathematical Programming - We propose and analyze a randomized zeroth-order optimization method based on approximating the exact gradient by finite differences computed in a set of orthogonal...     

Global optimization by random perturbation of the gradient method with a fixed parameter

The paper deals with the global minimization of a differentiable cost function mapping a ball of a finite dimensional Euclidean space into an interval of real numbers. It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbation avoids convergence to local minima. The stated results suggest an algorithm for the numerical approximation of global minima: experiments are performed for the problem of fitting a sum of exponentials to discrete data and to a nonlinear system involving about 5000 variables. The effect of the random perturbation is examined by comparison with the purely deterministic gradient method.     

Novel reliability-based optimization method for thermal structure with hybrid random,interval and fuzzy parameters

In this paper, novel reliability-based optimization model and method are proposed for thermal structure design with random, interval and fuzzy uncertainties in material properties, external loads and boundary conditions. Random variables are used to quantify the probabilistic uncertainty with sufficient sample data; whereas, interval variables and fuzzy variables are adopted to model the non-probabilistic uncertainty associated with objective limited information and subjective expert opinions, respectively. Using the interval ranking strategy, the level-cut limit state function is precisely quantified to represent the safety state. The eventual safety possibility is derived based on multiple integral, where the cut levels of different fuzzy variables are considered to be independent. Then a hybrid reliability-based optimization model is established with considerable computational cost caused by three-layer nested loop. To improve the computational efficiency, a subinterval vertex method is presented to replace the inner-loop and middle-loop. Comparing numerical results with traditional reliability model, a mono-objective example and a multi-objective example are provided to demonstrate the feasibility of proposed method for hybrid reliability analysis and optimization in practical engineering.     

CARTopt: a random search method for nonsmooth unconstrained optimization

A random search algorithm for unconstrained local nonsmooth optimization is described. The algorithm forms a partition on $\mathbb{R}^{n}$ using classification and regression trees (CART) from statistical pattern recognition. The CART partition defines desirable subsets where the objective function f is relatively low, based on previous sampling, from which further samples are drawn directly. Alternating between partition and sampling phases provides an effective method for nonsmooth optimization. The sequence of iterates {z _k } is shown to converge to an essential local minimizer of f with probability one under mild conditions. Numerical results are presented to show that the method is effective and competitive in practice.     

Interval optimization based line sampling method for fuzzy and random reliability analysis

For structural system with fuzzy variables as well as random variables, a novel algorithm for obtaining membership function of fuzzy reliability is presented on interval optimization based Line Sampling (LS) method. In the presented algorithm, the value domain of the fuzzy variables under the given membership level is firstly obtained according to their membership functions. Then, in the value domain of the fuzzy variables, bounds of reliability of the structure are obtained by the nesting analysis of the interval optimization, which is performed by modern heuristic methods, and reliability analysis, which is achieved by the LS method in the reduced space of the random variables. In this way the uncertainties of the input variables are propagated to the safety measurement of the structure, and the membership function of the fuzzy reliability is obtained. The presented algorithm not only inherits the advantage of the direct Monte Carlo method in propagating and distinguishing the fuzzy and random uncertainties, but also can improve the computational efficiency tremendously in case of acceptable precision. Several examples are used to illustrate the advantages of the presented algorithm.     

Robust optimization of consumption with random endowment

We consider the problem of optimal consumption for an investor who is risk and uncertainty averse. We model these preferences of the investor with the help of a convex risk-measure. Apart from consumption the agent has the possibility to invest initial capital and random endowment in a market where stock-prices are semimartingales. We formulate this as a maximin problem that will be solved by duality methods.     

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.     

An isogeometric boundary element approach for topology optimization using the level set method

Among the various types of structural optimization, topology has been occupying a prominent place over the last decades. It is considered the most versatile because it allows structural geometry to be determined taking into account only loading and fixing constraints. This technique is extremely useful in the design phase, which requires increasingly complex computational modeling. Modern geometric modeling techniques are increasingly focused on the use of NURBS basis functions. Consequently, it seems natural that topology optimization techniques also use this basis in order to improve computational performance. In this paper, we propose a way to integrate the isogeometric boundary techniques to topology optimization through the level set function. The proposed coupling occurs by describing the normal velocity field from the level set equation as a function of the normal shape sensitivity. This process is not well behaved in general, so some regularization technique needs to be specified. Limiting to plane linear elasticity cases, the numerical investigations proposed in this study indicate that this type of coupling allows to obtain results congruent with the current literature. Moreover, the additional computational costs are small compared to classical techniques, which makes their advantage for optimization purposes evident, particularly for boundary element method practitioners.     

Improved logarithmic linearizing method for optimization problems with free-sign pure discrete signomial terms

Free-sign pure discrete signomial (FPDS) terms are vital to and are frequently observed in many nonlinear programming problems, such as geometric programming, generalized geometric programming, and mixed-integer non-linear programming problems. In this study, all variables in the FPDS term are discrete variables. Any improvement to techniques for linearizing FPDS term contributes significantly to the solving of nonlinear programming problems; therefore, relative techniques have continually been developed. This study develops an improved exact method to linearize a FPDS term into a set of linear programs with minimal logarithmic numbers of zero-one variables and constraints. This method is tighter than current methods. Various numerical experiments demonstrate that the proposed method is significantly more efficient than current methods, especially when the problem scale is large.