Feasibility and Dominance Rules in the Electromagnetism-Like Algorithm for Constrained Global Optimization

In many engineering optimization problems, the objective and the constraints which come from complex analytical models are often black-box functions with extensive computational effort. In this case, it is necessary for optimization process to use sampling data to fit surrogate models so as to reduce the number of objective and constraint evaluations as soon as possible. In addition, it is sometimes difficult for the constrained optimization problems based on surrogate models to find a feasible point, which is the premise of further searching for a global optimal feasible solution. For this purpose, a new Kriging-based Constrained Global Optimization (KCGO) algorithm is proposed. Unlike previous Kriging-based methods, this algorithm can dispose black-box constrained optimization problem even if all initial sampling points are infeasible. There are two pivotal phases in KCGO algorithm. The main task of the first phase is to find a feasible point when there is no feasible data in the initial sample. And the aim of the second phase is to obtain a better feasible point under the circumstances of fewer expensive function evaluations. Several numerical problems and three design problems are tested to illustrate the feasibility, stability and effectiveness of the proposed method.

Computationally expensive constrained optimization problems are challenging owing to their high complexity and computational cost. To solve these problems efficiently, a kriging-assisted bi-objective constrained global optimization (BOCGO) algorithm is developed, where three phases with three bi-objective subproblems are performed. In phase I, the constraints are searched locally and globally to find the feasible region. Once a feasible region has been located, the two terms of the constrained expected improvement function are utilized to exploit and explore the feasible region in phase II. As the kriging models are accurate enough in the concerned region, a local search is processed to improve the optimal solution in phase III. The capability of the BOCGO algorithm is demonstrated by comparison with two classical and two state-of-the-art algorithms on 20 problems and an engineering simulation problem. The results show that the BOCGO algorithm performs better in more than three-fifths of problems, illustrating its effectiveness and robustness.

In this paper, a convex filled function method for solving constrained global optimization problems is proposed. A convex filled function is proposed by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a convex filled function method for obtaining a global minimizer or an approximate global minimizer of the constrained global optimization problem is presented. Some numerical results demonstrate the efficiency of this global optimization method for solving constrained global optimization problems.

This article presents a numerical study of two augmented Lagrangian algorithms to solve continuous constrained global optimization problems. The algorithms approximately solve a sequence of bound constrained subproblems whose objective function penalizes equality and inequality constraints violation and depends on the Lagrange multiplier vectors and a penalty parameter. Each subproblem is solved by a population-based method that uses an electromagnetism-like (EM) mechanism to move points towards optimality. Three local search procedures are tested to enhance the EM algorithm. Benchmark problems are solved in a performance evaluation of the proposed augmented Lagrangian methodologies. A comparison with other techniques presented in the literature is also reported.

In order to improve the solution accuracy of the lightning attachment algorithm in dealing with constrained optimization problem, an improved constrained lightning attachment algorithm is proposed in this paper. Firstly, the parameter setting of the constraint processing technology was improved, and the constraint violation degree of all individuals in the population was added to further distinguish between feasible individuals and infeasible individuals; Secondly, several stages in the lightning attachment procedure optimization algorithm are improved, including the performance enhancement stage, the uplink leader stage, the downlink leader stage and so on. The improved constrained lightning attachment procedure optimization algorithm proposed in this paper is compared with the other three constrained optimization algorithms on the standard test function g01-g13. The results show that the algorithm proposed in this paper has certain advantages in solving precision.

The estimation of anisotropic parameters is a critical step in the work flow for seismic anisotropic imaging. Walkaway(WVSP) or 3DVSP provides a valuable way to measure seismic anisotropy, as WVSP/3DVSP geometry is generally characterized by wide seismic ray angle coverage and well-determined vertical velocities and first break (FB) times. Most of the presently available anisotropic inversion methods are valid only for flat-layered or lateral homogenous velocity models. Anisotropic parameters in the presence of laterally inhomogeneous geological structures such as salt diapirs and faults often need to be estimated for accurate seismic imaging. Therefore, a new methodology was developed to estimate anisotropic parameters in lateral inhomogeneous velocity models. First, an anisotropic Eikonal solver (referred to as a fast marching scheme) is employed to efficiently and accurately calculate FB times in 2D or 3D grid-based inhomogeneous anisotropic velocity models. Then, a constrained global optimization algorithm known as simulated annealing is used to efficiently search and invert the anisotropic parameters based on the least square errors between calculated and observed FB times. Both numerical and field WVSP data examples are used to demonstrate the feasibility of this methodology in estimating anisotropic parameters in lateral inhomogeneous anisotropic models from WVSP or 3DVSP surveys.

A recently proposed swarm inspired optimization algorithm based on the navigation approach of Moths in space entitle as Moth-Flame Optimization (MFO) algorithm, is used for solve equality and inequality constrained optimization, real challenging layout problems. The navigating strategy of moths in universe entitles transverse orientation, a well active mechanism for travel so far distance in the straight direction. In fact, artificial lights trick moths, so they follow a deadly spiral path. MFO algorithm gives the competitive results with both continuous and discrete control variables. Real Challenging Constrained Optimization is a way of optimising an objective function in presence of constraints on some control variables. MFO have an ability to solve both constraints that may be either hard constrained or soft constrained. A statical representation of MFO algorithm expresses the best objective function value with reference to accuracy and standard deviation over recently proposed and most popular optimization algorithms. Fourteen constrained benchmark function of real engineering problems have been calculated and gained solutions were compared with the solution obtained by various recognized algorithms. The results obtained through MFO algorithm represent better solutions in the field of engineering design problems among many recently developed algorithms.

This paper describes a multi‐start with clustering strategy for use on constrained optimization problems. It is based on the characteristics of non‐linear constrained global optimization problems and extends a strategy previously tested on unconstrained problems. Earlier studies of multi‐start with clustering found in the literature have focused on unconstrained problems with little attention to non‐linear constrained problems. In this study, variations of multi‐start with clustering are considered including a simulated annealing or random search procedure for sampling the design domain and a quadratic programming (QP) sub‐problem used in cluster formation. The strategies are evaluated by solving 18 non‐linear mathematical problems and six engineering design problems. Numerical results show that the solution of a one‐step QP sub‐problem helps predict possible regions of attraction of local minima and can enhance robustness and effectiveness in identifying local minima without sacrificing efficiency. In comparison to other multi‐start techniques found in the literature, the strategies of this study can be attractive in terms of the number of local searches performed, the number of minima found, whether the global minimum is located, and the number of the function evaluations required. Copyright © 2002 John Wiley & Sons, Ltd.

The present work proposes a multi-objective improved teaching-learning based optimization (MO-ITLBO) algorithm for unconstrained and constrained multi-objective function optimization. The MO-ITLBO algorithm is the improved version of basic teaching-learning based optimization (TLBO) algorithm adapted for multi-objective problems. The basic TLBO algorithm is improved to enhance its exploration and exploitation capacities by introducing the concept of number of teachers, adaptive teaching factor, tutorial training and self-motivated learning. The MO-ITLBO algorithm uses a grid-based approach to adaptively assess the non-dominated solutions (i.e. Pareto front) maintained in an external archive. The performance of the MO-ITLBO algorithm is assessed by implementing it on unconstrained and constrained test problems proposed for the Congress on Evolutionary Computation 2009 (CEC 2009) competition. The performance assessment is done by using the inverted generational distance (IGD) measure. The IGD measures obtained by using the MO-ITLBO algorithm are compared with the IGD measures of the other state-of-the-art algorithms available in the literature. Finally, Lexicographic ordering is used to assess the overall performance of competitive algorithms. Results have shown that the proposed MO-ITLBO algorithm has obtained the 1st rank in the optimization of unconstrained test functions and the 3rd rank in the optimization of constrained test functions.

Trust-region based adaptive radial basis function algorithm for global optimization of expensive constrained black-box problems

An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should be portable, easy to learn, use, and maintain, and sufficiently fast for some production work. The motivation, design principles, uses, and capabilities for this environment are outlined. The environment includes an interval data type, a symbolic form of automatic differentiation to obtain an internal representation for functions, a special technique to allow conditional branches with operator overloading and interval computations, and generic routines to give interval and noninterval function and derivative information. Some of these generic routines use a special version of the backward mode of automatic differentiation. The package also includes dynamic data structures for exhaustive search algorithms.

Multiobjective optimization with ϵ-constrained method for solving real-parameter constrained optimization problems

This paper focuses on solving the problem of composite constrained convex optimization with a sum of smooth convex functions and non-smooth regularization terms (ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm) subject to locally general constraints. Motivated by the modern large-scale information processing problems in machine learning (the samples of a training dataset are randomly decentralized across multiple computing nodes), each of the smooth objective functions is further considered as the average of several constituent functions. To address the problem in a decentralized fashion, we propose a novel computation-efficient decentralized stochastic gradient algorithm, which leverages the variance reduction technique and the decentralized stochastic gradient projection method with constant step-size. Theoretical analysis indicates that if the constant step-size is less than an explicitly estimated upper bound, the proposed algorithm can find the exact optimal solution in expectation when each constituent function (smooth) is strongly convex. Concerning the existing decentralized schemes, the proposed algorithm not only is suitable for solving the general constrained optimization problems but also possesses low computation cost in terms of the total number of local gradient evaluations. Furthermore, the proposed algorithm via differential privacy strategy can effectively mask the privacy of each constituent function, which is more practical in applications involving sensitive messages, such as military affairs or medical treatment. Finally, numerical evidence is provided to demonstrate the appealing performance of the proposed algorithm.

A surrogate-assisted evolution strategy for constrained multi-objective optimization

In this paper, a modified version of spider monkey optimization (SMO) algorithm for solving constrained optimization problems has been proposed. To the best of author's knowledge, this is the first attempt to develop a version of SMO which can solve constrained continuous optimization problems by using the Deb's technique for handling constraints. The proposed algorithm is named constrained spider monkey optimization (CSMO) algorithm. The performance of CSMO is investigated on the well-defined constrained optimization problems of CEC2006 and CEC2010 benchmark sets. The results of the proposed algorithm are compared with constrained versions of particle swarm optimization, artificial bee colony and differential evolution. Outcome of the experiment and the discussion of results demonstrate that CSMO handles the global optimization task very well for constrained optimization problems and shows better performance in comparison with compared algorithms. Such an outcome will be an encouragement for the research community to further explore the potential of SMO in solving benchmarks as well as real-world problems, which are often constrained in nature.