Subset simulation for unconstrained global optimization

Global optimization problem is known to be challenging, for which it is difficult to have an algorithm that performs uniformly efficient for all problems. Stochastic optimization algorithms are suitable for these problems, which are inspired by natural phenomena, such as metal annealing, social behavior of animals, etc. In this paper, subset simulation, which is originally a reliability analysis method, is modified to solve unconstrained global optimization problems by introducing artificial probabilistic assumptions on design variables. The basic idea is to deal with the global optimization problems in the context of reliability analysis. By randomizing the design variables, the objective function maps the multi-dimensional design variable space into a one-dimensional random variable. Although the objective function itself may have many local optima, its cumulative distribution function has only one maximum at its tail, as it is a monotonic, non-decreasing, right-continuous function. It turns out that the searching process of optimal solution(s) of a global optimization problem is equivalent to exploring the process of the tail distribution in a reliability problem. The proposed algorithm is illustrated by two groups of benchmark test problems. The first group is carried out for parametric study and the second group focuses on the statistical performance.     

求广义几何规划全局最优解的线性化方法

对广义几何规划问题(GGP)提出了一个确定型全局优化算法，这类优化问题能广泛应用于工程设计和非线性系统的鲁棒稳定性分析等实际问题中，使用指数变换及对目标函数和约束函数的线性下界估计，建立了GGP的松弛线性规划(RLP)，通过对RLP可行域的细分以及一系列RLP的求解过程，从理论上证明了算法能收敛到GGP的全局最优解，对一个化学工程设计问题应用本文算法，数值实验表明本文方法是可行的。     

Global solutions to a class of CEC benchmark constrained optimization problems

This paper aims to solve a class of CEC benchmark constrained optimization problems that have been widely studied by nature-inspired optimization algorithms. Based on canonical duality theory, these challenging problems can be reformulated as a unified canonical dual problem over a convex set, which can be solved deterministically to obtain global optimal solutions in polynomial time. Applications are illustrated by some well-known CEC benchmark problems, and comparisons with other methods have demonstrated the effectiveness of the proposed approach.     

Design and simulation of FIR band pass and band stop filters using gravitational search algorithm

In this paper, a new optimization method named gravitational search algorithm (GSA) is adopted for designing optimal linear phase finite impulse response band pass (BP) and band stop (BS) digital filters. Other various population based evolutionary algorithms like real coded genetic algorithm, conventional particle swarm optimization, differential evolution (DE), bee swarm optimization have also been applied for the sake of comparative study of the same optimal designs. In GSA, particles are considered as objects and their performances are measured by their masses. All these objects attract each other by gravity forces, and these forces produce global movements of all objects towards the objects with heavier masses. GSA guarantees the exploitation step of the algorithm and it is apparently free from premature convergence. Extensive simulation results justify superior optimization capability of GSA over the afore-mentioned optimization techniques for the solution of the multimodal, non-differentiable, highly non-linear, and constrained filter design problems.     

An experimental analysis of a population based approach for global optimization

In this paper we perform a computational analysis of a population based approach for global optimization, Population Basin Hopping (PBH), which was proven to be very efficient on very challenging global optimization problems by the authors (see ). The experimental analysis aims at understanding more deeply how the approach works and why it is successful on challenging problems.     

Experimental Testing of Advanced Scatter Search Designs for Global Optimization of Multimodal Functions

Scatter search is an evolutionary method that, unlike genetic algorithms, operates on a small set of solutions and makes only limited use of randomization as a proxy for diversification when searching for a globally optimal solution. The scatter search framework is flexible, allowing the development of alternative implementations with varying degrees of sophistication. In this paper, we test the merit of several scatter search designs in the context of global optimization of multimodal functions. We compare these designs among themselves and choose one to compare against a well-known genetic algorithm that has been specifically developed for this class of problems. The testing is performed on a set of benchmark multimodal functions with known global minima.     

A Lagrangian search method for the <Emphasis Type="Italic">P</Emphasis>-median problem

In this paper, we propose a novel algorithm for solving the classical P-median problem. The essential aim is to identify the optimal extended Lagrangian multipliers corresponding to the optimal solution of the underlying problem. For this, we first explore the structure of the data matrix in P-median problem to recast it as another equivalent global optimization problem over the space of the extended Lagrangian multipliers. Then we present a stochastic search algorithm to find the extended Lagrangian multipliers corresponding to the optimal solution of the original P-median problem. Numerical experiments illustrate that the proposed algorithm can effectively find a global optimal or very good suboptimal solution to the underlying P-median problem, especially for the computationally challenging subclass of P-median problems with a large gap between the optimal solution of the original problem and that of its Lagrangian relaxation.     

An Optimization Problem with a Separable Non-Convex Objective Function and a Linear Constraint

For a class of global optimization (maximization) problems, with a separable non-concave objective function and a linear constraint a computationally efficient heuristic has been developed.The concave relaxation of a global optimization problem is introduced. An algorithm for solving this problem to optimality is presented. The optimal solution of the relaxation problem is shown to provide an upper bound for the optimal value of the objective function of the original global optimization problem. An easily checked sufficient optimality condition is formulated under which the optimal solution of concave relaxation problem is optimal for the corresponding non-concave problem. An heuristic algorithm for solving the considered global optimization problem is developed.The considered global optimization problem models a wide class of optimal distribution of a unidimensional resource over subsystems to provide maximum total output in a multicomponent systems.In the presented computational experiments the developed heuristic algorithm generated solutions, which either met optimality conditions or had objective function values with a negligible deviation from optimality (less than 1/10 of a percent over entire range of problems tested).     

Efficient crossover designs for comparing test treatments with a control treatment

Within a large family of crossover designs this paper characterizes the mathematical structures of A-optimal and A-efficient crossover designs for the purpose of statistical comparison between t experimental treatments with a control (standard) treatment. It further guides the user how to go about the construction of these designs and if needed doing the last minute modifications. To demonstrate the ideas some very interesting optimal and efficient small designs are constructed. The mathematical and statistical tools developed here could be very useful in other areas of design of experiments. Many interesting and not yet solved design problems for further research are implicitly stated throughout the paper.     

Exactness and algorithm of an objective penalty function

Penalty function is an important tool in solving many constrained optimization problems in areas such as industrial design and management. In this paper, we study exactness and algorithm of an objective penalty function for inequality constrained optimization. In terms of exactness, this objective penalty function is at least as good as traditional exact penalty functions. Especially, in the case of a global solution, the exactness of the proposed objective penalty function shows a significant advantage. The sufficient and necessary stability condition used to determine whether the objective penalty function is exact for a global solution is proved. Based on the objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the objective penalty function is proved for a local solution. An algorithm is presented in the paper in finding a local solution, with its convergence proved under some conditions. Finally, numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm.     

Numerical instabilities in structural optimization – analogy between topology & shape design problems

The formulation of structural optimization problems on the basis of the finite–element–method often leads to numerical instabilities resulting in non–optimal designs, which turn out to be difficult to realize from the engineering point of view. In the case of topology optimization problems the formation of designs characterized by oscillating density distributions such as the well–known “checkerboard–patterns” can be observed, whereas the solution of shape optimization problems often results in unfavourable designs with non–smooth boundary shapes caused by high–frequency oscillations of the boundary shape functions. Furthermore a strong dependence of the obtained designs on the finite–element–mesh can be observed in both cases. In this context we have already shown, that the topology design problem can be regularized by penalizing spatial oscillations of the density function by means of a penalty–approach based on the density gradient. In the present paper we apply the idea of problem regularization by penalizing oscillations of the design variable to overcome the numerical difficulties related to the shape design problem, where an analogous approach restricting the boundary surface can be introduced. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Copula‐based robust optimal block designs

Blocking is often used to reduce known variability in designed experiments by collecting together homogeneous experimental units. A common modeling assumption for such experiments is that responses from units within a block are dependent. Accounting for such dependencies in both the design of the experiment and the modeling of the resulting data when the response is not normally distributed can be challenging, particularly in terms of the computation required to find an optimal design. The application of copulas and marginal modeling provides a computationally efficient approach for estimating population‐average treatment effects. Motivated by an experiment from materials testing, we develop and demonstrate designs with blocks of size two using copula models. Such designs are also important in applications ranging from microarray experiments to experiments on human eyes or limbs with naturally occurring blocks of size two. We present a methodology for design selection, make comparisons to existing approaches in the literature, and assess the robustness of the designs to modeling assumptions.     

An ant colony system for enhanced loop-based aisle-network design

The purpose of this paper is to develop a global optimization model, simplification schemes, and a heuristic procedure for the design of a shortcut-enhanced unidirectional loop aisle-network with pick-up and drop-off stations. The objective is to minimize the total loaded and empty trip distances. This objective is the main determinant for the fleet size of the vehicles, which in turn is the driver of the total life-cycle cost of vehicle-based unit-load transport systems. The shortcut considerably reduces the length of the trips while maintaining the simplicity of the system. The global model solves simultaneously for the loop design, stations’ locations and shortcut design. We then develop two simplifications each containing two serial phases. Phase-1 of the first simplification step focuses on both loaded and empty trips, while that of the second simplification focuses only on loaded trips. In phase-2, both designs are enhanced with a shortcut to minimize both loaded and empty trip distances. The quality and efficiency of the three alternative designs are tested for a set of problems with different layout size and product mix. While the solution time of the second simplification procedure is a small percentage of the global formulation, it generates satisfactory solutions. On this foundation, we then develop a heuristic procedure to replace phase-1 of the second simplification. The heuristic procedure is using ant colony system to generate feasible solutions and then we implement a local search algorithm to improve the results. The heuristic algorithm quickly generates close to optimal solutions for phase-1 of the second simplification. By applying phase-2 of the this second simplification on a set of loops generated by the heuristic, close to optimal solutions are also quickly obtained for the global model.     

A genetic algorithm for designing microarray experiments

Heuristic techniques of optimization can be useful in designing complex experiments, such as microarray experiments. They have advantages over the traditional methods of optimization, particularly in situations where the search space is discrete. In this paper, a search procedure based on a genetic algorithm is proposed to find optimal (efficient) designs for both one- and multi-factor experiments. A genetic algorithm is a heuristic optimization method that exploits the biological evolution to obtain a solution of the problem. As an example, optimal designs for \(3\times 2\) factorial microarray experiments are presented for different numbers of arrays and for various sets of research questions. Comparisons between different operators of the genetic algorithm are performed by simulation studies.     

Integrating Interval Estimates of Global Optima and Local Search Methods for Combinatorial Optimization Problems

The problem of estimating the global optimal values of intractable combinatorial optimization problems is of interest to researchers developing and evaluating heuristics for these problems. In this paper we present a method for combining statistical optimum prediction techniques with local search methods such as simulated annealing and tabu search and illustrate the approach on a single machine scheduling problem. Computational experiments show that the approach yields useful estimates of optimal values with very reasonable computational effort.     

An effective algorithm for generation of factorial designs with generalized minimum aberration

Fractional factorial designs are popular and widely used for industrial experiments. Generalized minimum aberration is an important criterion recently proposed for both regular and non-regular designs. This paper provides a formal optimization treatment on optimal designs with generalized minimum aberration. New lower bounds and optimality results are developed for resolution-III designs. Based on these results, an effective computer search algorithm is provided for sub-design selection, and new optimal designs are reported.     

Reducing costs of backhaul networks for PCS networks using genetic algorithms

Designing cost-effective telecommunications networks often involves solving several challenging, interdependent combinatorial optimization problems simultaneously. For example, it may be necessary to select a least-cost subset of locations (network nodes) to serve as hubs where traffic is to be aggregated and switched; optimally assign other nodes to these hubs, meaning that the traffic entering the network at these nodes will be routed to the assigned hubs while respecting capacity constraints on the links; and optimally choose the types of links to be used in interconnecting the nodes and hubs based on the capacities and costs associated with each link type. Each of these three combinatorial optimization problems must be solved while taking into account its impacts on the other two. This paper introduces a genetic algorithm (GA) approach that has proved effective in designing networks for carrying personal communications services (PCS) traffic. The key innovation is to represent information about hub locations and their interconnections as two parts of a chromosome, so that solutions to both aspects of the problem evolve in parallel toward a globally optimal solution. This approach allows realistic problems that take 4–10 hours to solve via a more conventional branch-and-bound heuristic to be solved in 30–35 seconds. Applied to a real network design problem provided as a test case by Cox California PCS, the heuristics successfully identified a design 10% less expensive than the best previously known design. Cox California PCS has adopted the heuristic results and plans to incorporate network optimization in its future network designs and requests for proposals.     

An incremental clustering algorithm based on hyperbolic smoothing

Clustering is an important problem in data mining. It can be formulated as a nonsmooth, nonconvex optimization problem. For the most global optimization techniques this problem is challenging even in medium size data sets. In this paper, we propose an approach that allows one to apply local methods of smooth optimization to solve the clustering problems. We apply an incremental approach to generate starting points for cluster centers which enables us to deal with nonconvexity of the problem. The hyperbolic smoothing technique is applied to handle nonsmoothness of the clustering problems and to make it possible application of smooth optimization algorithms to solve them. Results of numerical experiments with eleven real-world data sets and the comparison with state-of-the-art incremental clustering algorithms demonstrate that the smooth optimization algorithms in combination with the incremental approach are powerful alternative to existing clustering algorithms.     

A robustness approach to uncapacitated network design problems

In this paper, we address uncapacitated network design problems characterised by uncertainty in the input data. Network design choices have a determinant impact on the effectiveness of the system. Design decisions are frequently made with a great degree of uncertainty about the conditions under which the system will be required to operate. Instead of finding optimal designs for a given future scenario, designers often search for network configurations that are “good” for a variety of likely future scenarios. This approach is referred to as the “robustness” approach to system design. We present a formal definition of “robustness” for the uncapacitated network design problem, and develop algorithms aimed at finding robust network designs. These algorithms are adaptations of the Benders decomposition methodology that are tailored so they can efficiently identify robust network designs. We tested the proposed algorithms on a set of randomly generated problems. Our computational experiments showed two important properties. First, robust solutions are abundant in uncapacitated network design problems, and second, the proposed algorithms performance is satisfactory in terms of cost and number of robust network designs obtained.     

Optimal Noncollapsing Space-Filling Designs for Irregular Experimental Regions

Space-filling and noncollapsing are two important properties in designing computer experiments. We study how the noncollapsing, space-filling designs for irregular experimental regions can be generated efficiently by the proposed metaheuristic methods. We solve this optimal design problem using variants of the discrete particle swarm optimization (DPSO) approaches. Numerical results, including an application in data center thermal management, are used to illustrate the performances of the proposed algorithms. Based on these numerical results, we assert that the most efficient approach is to reformulate the target optimal design problem as a constrained optimization problem and then use a modified DPSO to solve the constrained optimization problem.