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 magnetic resonance device designed via global optimization techniques

In this paper we are concerned with the design of a small low-cost, low-field multipolar magnet for Magnetic Resonance Imaging with a high field uniformity. By introducing appropriate variables, the considered design problem is converted into a global optimization one. This latter problem is solved by means of a new derivative free global optimization method which is a distributed multi-start type algorithm controlled by means of a simulated annealing criterion. In particular, the proposed method employs, as local search engine, a derivative free procedure. Under reasonable assumptions, we prove that this local algorithm is attracted by global minimum points. Additionally, we show that the simulated annealing strategy is able to produce a suitable starting point in a finite number of steps with probability one.This work was supported by CNR/MIUR Research Program Metodi e sistemi di supporto alle decisioni, Rome, Italy.Mathematics Subject Classification (1991):65K05, 62K05, 90C56     

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.     

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.     

A new constructing auxiliary function method for global optimization

A new auxiliary function method based on the idea which executes a two-stage deterministic search for global optimization is proposed. Specifically, a local minimum of the original function is first obtained, and then a stretching function technique is used to modify the objective function with respect to the obtained local minimum. The transformed function stretches the function values higher than the obtained minimum upward while it keeps the ones with lower values unchanged. Next, an auxiliary function is constructed on the stretched function, which always descends in the region where the function values are higher than the obtained minimum, and it has a stationary point in the lower area. We optimize the auxiliary function and use the found stationary point as the starting point to turn to the first step to restart the search. Repeat the procedure until termination. A theoretical analysis is also made. The main feature of the new method is that it relaxes significantly the requirements for the parameters. Numerical experiments on benchmark functions with different dimensions (up to 50) demonstrate that the new algorithm has a more rapid convergence and a higher success rate, and can find the solutions with higher quality, compared with some other existing similar algorithms, which is consistent with the analysis in theory.     

A global optimization method for the molecular replacement problem in X-ray crystallography

The primary technique for determining the three-dimensional structure of a protein molecule is X-ray crystallography, from which the molecular replacement (MR) problem often arises as a critical step. The MR problem is a global optimization problem to locate an optimal position of a model protein so that at this position the model will produce calculated intensities closest to those observed from an X-ray crystallography experiment involving a protein with unknown but similar atomic structure. Improving the applicability and robustness of MR methods is an important research topic because commonly used traditional MR methods, though often successful, have their limitations in solving difficult problems.We introduce a new global optimization strategy that combines a coarse-grid search, using a surrogate function, with extensive multi-start local optimization. A new MR code, called SOMoRe, based on this strategy is developed and tested on four realistic problems, including two difficult problems that traditional MR codes failed to solve directly. SOMoRe was able to solve each test problem without any complication, and SOMoRe solved an MR problem using a less complete model than the models required by three other programs. These results indicate that the new method is promising and should enhance the applicability and robustness of the MR methodology.     

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.     

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.     

线性半向量二层规划问题的全局优化方法

研究了线性半向量二层规划问题的全局优化方法. 利用下层问题的对偶间隙构造了线性半向量二层规划问题的罚问题, 通过分析原问题的最优解与罚问题可行域顶点之间的关系, 将线性半向量二层规划问题转化为有限个线性规划问题, 从而得到线性半向量二层规划问题的全局最优解. 数值结果表明所设计的全局优化方法对线性半向量二层规划问题是可行的.     

A simulated annealing driven multi-start algorithm for bound constrained global optimization

A derivative-free simulated annealing driven multi-start algorithm for continuous global optimization is presented. We first propose a trial point generation scheme in continuous simulated annealing which eliminates the need for the gradient-based trial point generation. We then suitably embed the multi-start procedure within the simulated annealing algorithm. We modify the derivative-free pattern search method and use it as the local search in the multi-start procedure. We study the convergence properties of the algorithm and test its performance on a set of 50 problems. Numerical results are presented which show the robustness of the algorithm. Numerical comparisons with a gradient-based simulated annealing algorithm and three population-based global optimization algorithms show that the new algorithm could offer a reasonable alternative to many currently available global optimization algorithms, specially for problems requiring ‘direct search’ type algorithm.     

Investigating a hybrid simulated annealing and local search algorithm for constrained optimization

Constrained Optimization Problems (COP) often take place in many practical applications such as kinematics, chemical process optimization, power systems and so on. These problems are challenging in terms of identifying feasible solutions when constraints are non-linear and non-convex. Therefore, finding the location of the global optimum in the non-convex COP is more difficult as compared to non-convex bound-constrained global optimization problems. This paper proposes a Hybrid Simulated Annealing method (HSA), for solving the general COP. HSA has features that address both feasibility and optimality issues and here, it is supported by a local search procedure, Feasible Sequential Quadratic Programming (FSQP). We develop two versions of HSA. The first version (HSAP) incorporates penalty methods for constraint handling and the second one (HSAD) eliminates the need for imposing penalties in the objective function by tracing feasible and infeasible solution sequences independently. Numerical experiments show that the second version is more reliable in the worst case performance.     

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 branch and bound method for stochastic global optimization

A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem.     

A hybrid multiagent approach for global trajectory optimization

In this paper we consider a global optimization method for space trajectory design problems. The method, which actually aims at finding not only the global minimizer but a whole set of low-lying local minimizers (corresponding to a set of different design options), is based on a domain decomposition technique where each subdomain is evaluated through a procedure based on the evolution of a population of agents. The method is applied to two space trajectory design problems and compared with existing deterministic and stochastic global optimization methods.     

A global optimization problem in portfolio selection

This paper deals with the issue of buy-in thresholds in portfolio optimization using the Markowitz approach. Optimal values of invested fractions calculated using, for instance, the classical minimum-risk problem can be unsatisfactory in practice because they lead to unrealistically small holdings of certain assets. Hence we may want to impose a discrete restriction on each invested fraction y _i such as y _i >  y _min or y _i =  0. We shall describe an approach which uses a combination of local and global optimization to determine satisfactory solutions. The approach could also be applied to other discrete conditions—for instance when assets can only be purchased in units of a certain size (roundlots).     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

An adaptive simulated annealing algorithm for global optimization over continuous variables

A method is presented for attempting global minimization for a function of continuous variables subject to constraints. The method, calledAdaptive Simulated Annealing (ASA), is distinguished by the fact that the fixed temperature schedules and step generation routines that characterize other implementations are here replaced by heuristic-based methods that effectively eliminate the dependence of the algorithm's overall performance on user-specified control parameters. A parallelprocessing version of ASA that gives increased efficiency is presented and applied to two standard problems for illustration and comparison.This research was supported by the University Research Initiative of the U.S. Army Research Office.     

求解弱线性双层规划问题的一种全局优化方法

双层规划在经济、交通、生态、工程等领域有着广泛而重要的应用.目前对双层规划的研究主要是基于强双层规划和弱双层规划.然而,针对弱双层规划的求解方法却鲜有研究.研究求解弱线性双层规划问题的一种全局优化方法,首先给出弱线性双层规划问题与其松弛问题在最优解上的关系,然后利用线性规划的对偶理论和罚函数方法,讨论该松弛问题和它的罚问题之间的关系.进一步设计了一种求解弱线性双层规划问题的全局优化方法,该方法的优势在于它仅仅需要求解若干个线性规划问题就可以获得原问题的全局最优解.最后,用一个简单算例说明了所提出的方法是可行的.     

A stochastic probing algorithm for global optimization

Recently, simulated annealing methods have proven to be a valuable tool for global optimization. We propose a new stochastic method for locating the global optimum of a function. The proposed method begins with the subjective specification of a probing distribution. The objective function is evaluated at a few points sampled from this distribution, which is then updated using the collected information. The updating mechanism is based on the entropy of a move selecting distribution and is loosely connected to some notions in statistical thermodynamics. Examples of the use of the proposed method are presented. These indicate its superior performance as compared with simulated annealing. Preliminary considerations in applying the method to discrete problems are discussed.     

A global optimization approach for solving the convex multiplicative programming problem

We consider a convex multiplicative programming problem of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]where X is a compact convex set of ⁿ and f ₁, f ₂ are convex functions which have nonnegative values over X.Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ² ₂. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f _i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.Partly supported by the Deutsche Forschungsgemeinschaft Project CONMIN.