A hybrid method for solving multi-objective global optimization problems

Real optimization problems often involve not one, but multiple objectives, usually in conflict. In single-objective optimization there exists a global optimum, while in the multi-objective case no optimal solution is clearly defined but rather a set of optimums, which constitute the so called Pareto-optimal front. Thus, the goal of multi-objective strategies is to generate a set of non-dominated solutions as an approximation to this front. However, most problems of this kind cannot be solved exactly because they have very large and highly complex search spaces. The objective of this work is to compare the performance of a new hybrid method here proposed, with several well-known multi-objective evolutionary algorithms (MOEA). The main attraction of these methods is the integration of selection and diversity maintenance. Since it is very difficult to describe exactly what a good approximation is in terms of a number of criteria, the performance is quantified with adequate metrics that evaluate the proximity to the global Pareto-front. In addition, this work is also one of the few empirical studies that solves three-objective optimization problems using the concept of global Pareto-optimality.     

A new filled function method for unconstrained global optimization

In this paper, a new filled function which has better properties is proposed for identifying a global minimum point for a general class of nonlinear programming problems within a closed bounded domain. An algorithm for unconstrained global optimization is developed from the new filled function. Theoretical and numerical properties of the proposed filled function are investigated. The implementation of the algorithm on seven test problems is reported with satisfactory numerical results.     

Stochastic techniques for global optimization: A survey of recent advances

In this paper stochastic algorithms for global optimization are reviewed. After a brief introduction on random-search techniques, a more detailed analysis is carried out on the application of simulated annealing to continuous global optimization. The aim of such an analysis is mainly that of presenting recent papers on the subject, which have received only scarce attention in the most recent published surveys. Finally a very brief presentation of clustering techniques is given.     

The continuous reactive tabu search: Blending combinatorial optimization and stochastic search for global optimization

A novel algorithm for the global optimization of functions (C-RTS) is presented, in which a combinatorial optimization method cooperates with a stochastic local minimizer. The combinatorial optimization component, based on the Reactive Tabu Search recently proposed by the authors, locates the most promising boxes, in which starting points for the local minimizer are generated. In order to cover a wide spectrum of possible applications without user intervention, the method is designed with adaptive mechanisms: the box size is adapted to the local structure of the function to be optimized, the search parameters are adapted to obtain a proper balance of diversification and intensification. The algorithm is compared with some existing algorithms, and the experimental results are presented for a variety of benchmark tasks.     

A Metropolis algorithm combined with Hooke-Jeeves local search method applied to global optimization

A hybridization of a recently introduced Metropolis algorithm named the Particle Collision Algorithm (PCA) and the Hooke-Jeeves local search method is applied to a testbed of global optimization functions and to real-world chemical equilibrium nonlinear systems. The results obtained by this method, called HJPCA, are compared against those achieved by two state-of-the-art global optimization methods, C-GRASP and GLOBAL. HJPCA performs better than both algorithms, thus demonstrating its potential for other applications.     

On a decomposition method for nonconvex global optimization

A rigorous foundation is presented for the decomposition method in nonconvex global optimization, including parametric optimization, partly convex, partly monotonic, and monotonic/linear optimization. Incidentally, some errors in the recent literature on this subject are pointed out and fixed.     

Convergence analysis of a global optimization algorithm using stochastic differential equations

We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by Π.     

A new genetic algorithm for global optimization of multimodal continuous functions

In this paper a new genetic algorithm is developed to find the near global optimal solution of multimodal nonlinear optimization problems. The algorithm defined makes use of a real encoded crossover and mutation operator. The performance of GA is tested on a set of twenty-seven nonlinear global optimization test problems of variable difficulty level. Results are compared with some well established popular GAs existing in the literature. It is observed that the algorithm defined performs significantly better than the existing ones.     

On the convergence of global methods in multiextremal optimization

A general class of derivative-free optimization procedures is presented including the corresponding convergence theory. This theory turns out to be very constructive, in the sense that the convergence conditions not only can be verified easily for many existing algorithms, but also allow one to construct new procedures. It is shown that popular methods such as branch-and-bound concepts, Pintér's general class of procedures, the algorithms of Pijavskii, Shubert, and Mladineo, and the approach of Zheng and Galperin can not only be subsumed under this class of methods, but also partly be improved by regarding them within the framework presented.     

A convexification method for a class of global optimization problems with applications to reliability optimization

A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimization problems using the proposed convexification schemes. An outer approximation method can then be used to find the global solution of the transformed problem. Applications to mixed-integer nonlinear programming problems arising in reliability optimization of complex systems are discussed and satisfactory numerical results are presented.     

A niche hybrid genetic algorithm for global optimization of continuous multimodal functions

A niche hybrid genetic algorithm (NHGA) is proposed in this paper to solve continuous multimodal optimization problems more efficiently, accurately and reliably. It provides a new architecture of hybrid algorithms, which organically merges the niche techniques and Nelder–Mead's simplex method into GAs. In the new architecture, the simplex search is first performed in the potential niches, which likely contain a global optimum, to locate the promising zones within search space, quickly and reliably. Then another simplex search is used to quickly discover the global optimum in the located promising zones. The proposed method not only makes the exploration capabilities of GAs stronger through niche techniques, but also has more powerful exploitation capabilities by using simplex search. So it effectively alleviates premature convergence and improves weak exploitation capacities of GAs. A set of benchmark functions is used to demonstrate the validity of NHGA and the role of every component of NHGA. Numerical experiments show that the NHGA may, efficiently and reliably, obtain a more accurate global optimum for the complex and high-dimension multimodal optimization problems. It also demonstrates that the new hybrid architecture is potential and can be used to generate more potential hybrid algorithms.     

The normal score transformation applied to a multi-univariate method of global optimization

Nonparametric global optimization methods have been developed that determine the location of their next guess based on the rank-transformed objective function evaluations rather than the actual function values themselves. Another commonly-used transformation in nonparametric statistics is the normal score transformation. This paper applies the normal score transformation to the multi-univariate method of global optimization. The benefits of the new method are shown by its performance on a standard set of global optimization test problems. The normal score transformation yields a method that gives equivalent searches for any monotonic transformation of the objective function.     

GASUB: finding global optima to discrete location problems by a genetic-like algorithm

In many discrete location problems, a given number s of facility locations must be selected from a set of m potential locations, so as to optimize a predetermined fitness function. Most of such problems can be formulated as integer linear optimization problems, but the standard optimizers only are able to find one global optimum. We propose a new genetic-like algorithm, GASUB, which is able to find a predetermined number of global optima, if they exist, for a variety of discrete location problems. In this paper, a performance evaluation of GASUB in terms of its effectiveness (for finding optimal solutions) and efficiency (computational cost) is carried out. GASUB is also compared to MSH, a multi-start substitution method widely used for location problems. Computational experiments with three types of discrete location problems show that GASUB obtains better solutions than MSH. Furthermore, the proposed algorithm finds global optima in all tested problems, which is shown by solving those problems by Xpress-MP, an integer linear programing optimizer (21). Results from testing GASUB with a set of known test problems are also provided.     

Accelerating method of global optimization for signomial geometric programming

Signomial geometric programming (SGP) has been an interesting problem for many authors recently. Many methods have been provided for finding locally optimal solutions of SGP, but little progress has been made for global optimization of SGP. In this paper we propose a new accelerating method for global optimization algorithm of SGP using a suitable deleting technique. This technique offers a possibility to cut away a large part of the currently investigated region in which the globally optimal solution of SGP does not exist, and can be seen as an accelerating device for global optimization algorithm of SGP problem. Compared with the method of Shen and Zhang [Global optimization of signomial geometric programming using linear relaxation, Appl. Math. Comput. 150 (2004) 99–114], numerical results show that the computational efficiency is improved obviously by using this new technique in the number of iterations, the required saving list length and the execution time of the algorithm.     

A parallel global optimization method and its implementation on a transputer system

This paper deals with a new parallel method for solving one-dimensional global optimization problems. We present the formulation of the decision rules of this method, the conditions for its nonredundant parallelization, the generalization for solving multi-dimensional problems and its implementation on a transputer system. There is also an account of some numerical experiments.     

Steklov convexification and a trajectory method for global optimization of multivariate quartic polynomials

Mathematical Programming - The Steklov function $$mu _f(cdot ,t)$$ is defined to average a continuous function f at each point of its domain by using a window of size given by $$t>0$$ . It...     

DESA: a new hybrid global optimization method and its application to analog integrated circuit sizing

This paper presents a new hybrid global optimization method referred to as DESA. The algorithm exploits random sampling and the metropolis criterion from simulated annealing to perform global search. The population of points and efficient search strategy of differential evolution are used to speed up the convergence. The algorithm is easy to implement and has only a few parameters. The theoretical global convergence is established for the hybrid method. Numerical experiments on 23 mathematical test functions have shown promising results. The method was also integrated into SPICE OPUS circuit simulator to evaluate its practical applicability in the area of analog integrated circuit sizing. Comparison was made with basic simulated annealing, differential evolution, and a multistart version of the constrained simplex method. The latter was already a part of SPICE OPUS and produced good results in past research.     

SPT: a stochastic tunneling algorithm for global optimization

A stochastic approach to solving unconstrained continuous-function global optimization problems is presented. It builds on the tunneling approach to deterministic optimization presented by Barhen and co-workers (Bahren and Protopopescu, in: State of the Art in Global Optimization, Kluwer, 1996; Barhen et al., Floudas and Pardalos (eds.), TRUST: a deterministic algorithm for global optimization, 1997) by combining a series of local descents with stochastic searches. The method uses a rejection-based stochastic procedure to locate new local minima descent regions and a fixed Lipschitz-like constant to reject unpromising regions in the search space, thereby increasing the efficiency of the tunneling process. The algorithm is easily implemented in low-dimensional problems and scales easily to large problems. It is less effective without further heuristics in these latter cases, however. Several improvements to the basic algorithm which make use of approximate estimates of the algorithms parameters for implementation in high-dimensional problems are also discussed. Benchmark results are presented, which show that the algorithm is competitive with the best previously reported global optimization techniques. A successful application of the approach to a large-scale seismology problem of substantial computational complexity using a low-dimensional approximation scheme is also reported.     

A randomized method for the identification of switched NARX systems

The identification of switched systems is a complex optimization problem that involves both continuous (parametrizations of the local models, a.k.a. modes) and discrete variables (model structures, switching signal). In particular, the combinatorial complexity associated with the estimation of the switching signal grows exponentially with the number of samples, which makes data segmentation (i.e. estimating the number and location of mode switchings, and the mode sequence) a challenging problem. In this work, we extend a previously developed randomized approach for the identification of switched systems to encompass the estimation of the switching locations. The method operates by extracting samples from a probability distribution of switched models, and gathering information from the associated model performances to update the distribution, until convergence to a limit distribution associated to a specific model. A suitable probability distribution is employed to represent the likelihood of a mode switching at a certain time, and the update process is designed to correct the switching locations and remove redundant switchings. The proposed algorithm has been compared to existing state-of-the-art methods and has been tested on various benchmark examples, to demonstrate its effectiveness.