Technical note: a systematic procedure based on CALIBRA and the Nelder & Mead algorithm for fine-tuning metaheuristics

The problem of setting the parameter values of a metaheuristic algorithm that optimise its performance is complex and time-consuming. Although the performance of a metaheuristic can be very sensitive to the parameter values, it is usual in the literature that the selection of the value parameters is not enough justified. There are in the literature two procedures that facilitate the task of fine-tuning: CALIBRA and the Nelder & Mead (N&M) algorithm. We propose a hands-off systematic procedure for fine-tuning metaheuristics that takes the advantages of CALIBRA and the N&M algorithm.     

Construction of D-Optimal Experimental Designs for Nonparametric Regression Models

Under study is the problem of a D-optimal experimental design for the problem of nonparametric kernel smoothing. Modification is proposed for the process of calculating the Fisher information matrix. D-optimal designs are constructed for one and several target points for the problems of nonparametric kernel smoothing using a uniform kernel, the Gauss and Epanechnikov kernels. Comparison is performed between Fedorov’s algorithm and direct optimization methods (such as the Nelder–Mead method and the method of differential evolution). The features of the application of the optimality criterion for the experimental design of the problems with several target points were specified for the cases of various kernels and bandwidths.     

Optimal control of chaotic systems with input saturation using an input-state linearization scheme

Chaos is undesirable in many engineering applications since it causes a serious degradation of the system performance and restricts the system’s operating range. Therefore, the problem of controlling chaos has attracted intense interest in recent years. This paper proposes an approach for optimizing the control of chaotic systems with input saturation using an input-state linearization scheme. In the proposed approach, the optimal system gains are identified using the Nelder–Mead simplex algorithm. This algorithm does not require the derivatives of the cost function (or the performance index) to be optimized, and is therefore particularly applicable to problems with undifferentiable elements or discontinuities. Two numerical simulations are performed to demonstrate the feasibility and effectiveness of the proposed method.     

Robust optimal control for a batch nonlinear enzyme-catalytic switched time-delayed process with noisy output measurements

In this paper, we consider a nonlinear switched time-delayed (NSTD) system with an unknown time-varying function describing the batch culture. The output measurements are noisy. According to the actual fermentation process, this time-varying function appears in the form of a piecewise-linear function with unknown kinetic parameters and switching times. The quantitative definition of biological robustness is given to overcome the difficulty of accurately measuring intracellular material concentrations. Our main goal is to estimate these unknown quantities by using noisy output measurements and biological robustness. This estimation problem is formulated as a robust optimal control problem (ROCP) governed by the NSTD system subject to continuous state inequality constraints. The ROCP is approximated as a sequence of nonlinear programming subproblems by using some techniques. Due to the highly complex nature of these subproblems, we propose a hybrid parallel algorithm, based on Nelder–Mead method, simulated annealing and the gradients of the constraint functions, for solving these subproblems. The paper concludes with simulation results.     

An extension of the simplex method to constrained nonlinear optimization

The simplex algorithm of Nelder and Mead is extended to handle nonlinear optimization problems with constraints. To prevent the simplex from collapsing into a subspace near the constraints, a delayed reflection is introduced for those points moving into the infeasible region. Numerical experience indicates that the proposed algorithm yields good results in the presence of both inequality and equality constraints, even when the constraint region is narrow. We note that it may be possible to modify and improve the algorithm by trying out variants.     

On an unreliable-server retrial queue with customer feedback and impatience

Analysis of an unreliable-server retrial queue with customer's feedback and impatience is presented. Truncated classical and constant retrial policies are taken into account. This system is analyzed as a process of quasi-birth-and-death (QBD). The quasi-progression algorithm is applied to compute the rate matrix of QBD model. A recursive solver algorithm for computing the stationary probabilities is also developed. To make the investigated system viable economically, a cost function is developed to decide the optimum values of servers, mean service rate and mean repair rate. Quasi-Newton method, pattern search method and Nelder–Mead simplex direct search method are employed to implement the optimization tasks. Under optimum operating conditions, numerical results are provided for a comparison of retrial policies. We also give a potential application to illustrate the system's applicability.     

Cure Cycle Design for Thermosetting-Matrix Composites Fabrication under Uncertainty

Design of the optimal cure temperature cycle is imperative for low-cost of manufacturing thermosetting-matrix composites. Uncertainties exist in several material and process parameters, which lead to variability in the process performance and product quality. This paper addresses the problem of determining the optimal cure temperature cycles under uncertainty. A stochastic model is developed, in which the parameter uncertainties are represented as probability density functions, and deterministic numerical process simulations based on the governing process physics are used to determine the distributions quantifying the output parameter variability. A combined Nelder–Mead Simplex method and the simulated annealing algorithm is used in conjunction with the stochastic model to obtain time-optimal cure cycles, subject to constraints on parameters influencing the product quality. Results are presented to illustrate the effects of a degree of parameter uncertainty, constraint values, and material kinetics on the optimal cycles. The studies are used to identify a critical degree of uncertainty in practice above which a rigorous analysis and design under uncertainty is warranted; below this critical value, a deterministic optimal cure cycle may be used with reasonable confidence.     

The tangent function through a functional equation

This paper describes a demonstration experiment which is designed to assist students to develop, quickly, an understanding of the Nelder‐Mead simplex method.     

Hybridizing local search algorithms for global optimization

In this paper, we combine two types of local search algorithms for global optimization of continuous functions. In the literature, most of the hybrid algorithms are produced by combination of a global optimization algorithm with a local search algorithm and the local search is used to improve the solution quality, not to explore the search space to find independently the global optimum. The focus of this research is on some simple and efficient hybrid algorithms by combining the Nelder–Mead simplex (NM) variants and the bidirectional random optimization (BRO) methods for optimization of continuous functions. The NM explores the whole search space to find some promising areas and then the BRO local search is entered to exploit optimal solution as accurately as possible. Also a new strategy for shrinkage stage borrowed from differential evolution (DE) is incorporated in the NM variants. To examine the efficiency of proposed algorithms, those are evaluated by 25 benchmark functions designed for the special session on real-parameter optimization of CEC2005. A comparison study between the hybrid algorithms and some DE algorithms and non-parametric analysis of obtained results demonstrate that the proposed algorithms outperform most of other algorithms and their difference in most cases is statistically considerable. In a later part of the comparative experiments, a comparison of the proposed algorithms with some other evolutionary algorithms reported in the CEC2005 confirms a better performance of our proposed algorithms.     

Mesh-based Nelder–Mead algorithm for inequality constrained optimization

Despite the lack of theoretical and practical convergence support, the Nelder–Mead (NM) algorithm is widely used to solve unconstrained optimization problems. It is a derivative-free algorithm, that attempts iteratively to replace the worst point of a simplex by a better one. The present paper proposes a way to extend the NM algorithm to inequality constrained optimization. This is done through a search step of the mesh adaptive direct search (Mads) algorithm, inspired by the NM algorithm. The proposed algorithm does not suffer from the NM lack of convergence, but instead inherits from the totality of the Mads convergence analysis. Numerical experiments show an important improvement in the quality of the solutions produced using this search step.     

Tabu Search directed by direct search methods for nonlinear global optimization

In recent years, there has been a great deal of interest in metaheuristics in the optimization community. Tabu Search (TS) represents a popular class of metaheuristics. However, compared with other metaheuristics like genetic algorithm and simulated annealing, contributions of TS that deals with continuous problems are still very limited. In this paper, we introduce a continuous TS called Directed Tabu Search (DTS) method. In the DTS method, direct-search-based strategies are used to direct a tabu search. These strategies are based on the well-known Nelder–Mead method and a new pattern search procedure called adaptive pattern search. Moreover, we introduce a new tabu list conception with anti-cycling rules called Tabu Regions and Semi-Tabu Regions. In addition, Diversification and Intensification Search schemes are employed. Numerical results show that the proposed method is promising and produces high quality solutions.     

Shuffled complex evolution approach for effective and efficient global minimization

The degree of difficulty in solving a global optimization problem is in general dependent on the dimensionality of the problem and certain characteristics of the objective function. This paper discusses five of these characteristics and presents a strategy for function optimization called the shuffled complex evolution (SCE) method, which promises to be robust, effective, and efficient for a broad class of problems. The SCE method is based on a synthesis of four concepts that have proved successful for global optimization: (a) combination of probabilistic and deterministic approaches; (b) clustering; (c) systematic evolution of a complex of points spanning the space, in the direction of global improvement; and (d) competitive evolution. Two algorithms based on the SCE method are presented. These algorithms are tested by running 100 randomly initiated trials on eight test problems of differing difficulty. The performance of the two algorithms is compared to that of the controlled random search CRS2 method presented by Price (1983, 1987) and to a multistart algorithm based on the simplex method presented by Nelder and Mead (1965).This work was partially funded by the National Science Foundation, Grant ECE-86-10584, and by the National Weather Service, Grant NA85AA-H-HY088.     

基于IMO-SLFDA的大规模分布式光伏并网的智能配电网相位平衡研究

分布式光伏的大量接入智能配电网后,可能导致三相电流的失衡,进而破坏配电系统的安全稳定性。对此,本文构建了以电流不平衡和电能损失最小化为目标的含分布式光伏的配电网优化的多目标模型,旨在解决大规模分布式光伏发电并网后配电网相位平衡的问题;然后,用随机单纯形法对混合蛙跳算法进行优化,改进了蛙跳算法求解优化问题时极易陷入局部最优以及计算效率较低的缺点,并和决策算法相结合,提出适用本文算例的改进的多目标混合蛙跳决策算法,确保能以极快的搜索速度和较高的计算精度得到最优解;最后,以IEEE-123节点三相不平衡测试系统为例,通过控制变量的相关操作实现配电系统的三相平衡。对比分析基础案例和优化算例的差异,验证了本文所提算法的先进性和实用性。     

Multi-direction-based Nelder–Mead method

The original Nelder–Mead (NM) method tends to be used to optimize low-dimensional functions. This article provides a modified NM that has the capability of large-scale optimization. The modification of NM is characterized by (a) working with a population of points, (b) mining multiple search directions through two strategies – point-grouping and variable-centroid multi-direction (VCMD), thus giving rise to VCMD plus grouping (VCMD_g) and (c) introducing random coefficients into NM and performing mutation on best-points, producing a random NM (NM_r). The combination of NM_r and VCMD_g, NM_r-VCMD_g, is just the modified NM in this work.     

A generalized temperature-dependent,non-Newtonian Hele–Shaw flow in injection and compression moulding

A generalized Hele–Shaw flow is derived and analysed in injection and compression moulding which contains non-isothermal and non-Newtonian rheological behaviour of the molten polymers as well as non-symmetric effects. The corresponding fixed domain formulations for the flow are investigated from a mathematical point of view. Different approaches for the determination of the temperature and some viscosity models are discussed. Together with an outline of the proposed solution algorithm special finite element techniques are described for the numerical treatment of this complex non-linear moving boundary problem. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A Reduced Order Model for Control of a Single Reactor Activated Sludge Process

The IAWQ activated sludge model no 1, slightly modified by adding a nitrogen limiting growth function for heterotrophic microorganisms, has been validated on a sequencing single sludge wastewater reactor doing nitrification and denitrification. However, this model is too complex to be used for control design. The main purpose of this paper is to show how it is possible to simplify the full model from biological and mathematical considerations. Properties of observability, controllability and theoretical identifiability have been analyzed. The parameter identification has been carried out by using the simplex method of Nelder & Mead. Simulation results performed over a range of six hours (two aerobic / anoxic cycles), show that there exists a good fit between the simulated solution and the actual behavior of a lab scale pilot plant.     

Integer programming models and linearizations for the traveling car renter problem

The traveling car renter problem (CaRS) is an extension of the classical traveling salesman problem (TSP) where different cars are available for use during the salesman’s tour. In this study we present three integer programming formulations for CaRS, of which two have quadratic objective functions and the other has quadratic constraints. The first model with a quadratic objective function is grounded on the TSP interpreted as a special case of the quadratic assignment problem in which the assignment variables refer to visitation orders. The second model with a quadratic objective function is based on the Gavish and Grave’s formulation for the TSP. The model with quadratic constraints is based on the Dantzig–Fulkerson–Johnson’s formulation for the TSP. The formulations are linearized and implemented in two solvers. An experiment with 50 instances is reported.     

The Red–Blue transportation problem

This paper considers the Red–Blue Transportation Problem (Red–Blue TP), a generalization of the transportation problem where supply nodes are partitioned into two sets and so-called exclusionary constraints are imposed. We encountered a special case of this problem in a hospital context, where patients need to be assigned to rooms. We establish the problem’s complexity, and we compare two integer programming formulations. Furthermore, a maximization variant of Red–Blue TP is presented, for which we propose a constant-factor approximation algorithm. We conclude with a computational study on the performance of the integer programming formulations and the approximation algorithms, by varying the problem size, the partitioning of the supply nodes, and the density of the problem.     

An efficient grouping genetic algorithm for U-shaped assembly line balancing problems with maximizing production rate

U-type assembly line is one of the important tools that may increase companies’ production efficiency. In this study, two different modeling approaches proposed for the assembly line balancing problems have been used in modeling type-II U-line balancing problems, and the performances of these models have been compared with each other. It has been shown that using mathematical formulations to solve medium and large size problem instances is impractical since the problem is NP-hard. Therefore, a grouping genetic and simulated annealing algorithms have been developed, and a particle swarm optimization algorithm is adapted to compare with the proposed methods. A special crossover operator that always obtains feasible offspring has been suggested for the proposed grouping genetic algorithm. Furthermore, a local search procedure based on problem-specific knowledge was applied to increase the intensification of the algorithm. A set of well-known benchmark instances was solved to evaluate the effectiveness of the proposed and existing methods. Results showed that while the mathematical formulations can only be used to solve small size instances, metaheuristics can obtain high quality solutions for all size problem instances within acceptable CPU times. Moreover, grouping genetic algorithm has been found to be superior to the other methods according to the number of optimal solutions, or deviations from the lower bound values.     

Optimal Simplification of Cartographic Lines Using Shortest-path Formulations

Line simplification is a process by which unnecessary detail in cartographic data is eliminated. In the past, a variety of heuristics have been implemented to simplify lines. This paper describes practical algorithms which have been developed to provide optimal solutions. The line simplification problem is shown to have the special structure of an acyclic shortest-path problem. Three formulations are presented, which differ in the form of their objectives and/or constraints, and a solution algorithm is outlined for each formulation. Issues associated with implementation are addressed by using each algorithm to perform various degrees of simplification on a large example line. Possibilities for future research related to optimal line simplification are also discussed.