A parameter free filled function for unconstrained global optimization

The filled function method is considered as an efficient method to find the global minimum of multidimensional functions. A number of filled functions were proposed recently, most of which have one or two adjustable parameters. However, there is no efficient criterion to choose the parameter appropriately. In this paper, we propose a filled function without parameter. And this function includes neither exponential terms nor logarithmic terms so it is superior to the traditional ones. Theories of the filled function are investigated. And an algorithm which does not compute gradients during minimizing the filled function is presented. Moreover, the numerical experiments demonstrate the efficiency of the proposed filled function.     

A filled function method with one parameter for unconstrained global optimization

The filled function method is considered as an efficient approach to solve the global optimization problems. In this paper, a new filled function method is proposed. Its main idea is as follows: a new continuously differentiable filled function with only one parameter is constructed for unconstrained global optimization when a minimizer of the objective function is found, then a minimizer of the filled function will be found in a lower basin of the objective function, thereafter, a better minimizer of the objective function will be found. The above process is repeated until the global optimal solution is found. The numerical experiments show the efficiency of the proposed filled function method.     

A new filled function method for an unconstrained nonlinear equation

In this paper, we transform an unconstrained system of nonlinear equations into a special optimization problem. A new filled function is constructed by employing the special properties of the transformed optimization problem. Theoretical and numerical properties of the proposed filled function are investigated and a solution of the algorithm is proposed. Under some conditions, we can find a solution or an approximate solution to the system of nonlinear equations in finite iterations. The implementation of the algorithm on six test problems is reported with satisfactory numerical results.     

A new filled function method for nonlinear equations

In this paper, a new global optimization approach based on the filled function method is proposed for solving box-constrained systems of nonlinear equations. We first convert the nonlinear system into an equivalent global optimization problem, and then propose a new filled function method to solve the converted global optimization problem. Several numerical examples are presented and solved by using different local minimization methods, which illustrate the efficiency of the present approach.     

New filled functions for nonsmooth global optimization

The filled function method is an effective approach to find a global minimizer for a general class of nonsmooth programming problems with a closed bounded domain. This paper gives a new definition for the filled function, which overcomes some drawbacks of the previous definition. It proposes a two-parameter filled function and a one-parameter filled function to improve the efficiency of numerical computation. Based on these analyses, two corresponding filled function algorithms are presented. They are global optimization methods which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Numerical results obtained indicate the efficiency and reliability of the proposed filled function methods.     

A filled function method for constrained global optimization

In this paper, a filled function method for solving constrained global optimization problems is proposed. A filled function is proposed for escaping the current local minimizer of a constrained global optimization problem by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a 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.     

Filled functions for unconstrained global optimization

This paper is concerned with filled function techniques for unconstrained global minimization of a continuous function of several variables. More general forms of filled functions are presented for smooth and non-smooth optimization problems. These functions have either one or two adjustable parameters. Conditions on functions and on the values of parameters are given so that the constructed functions have the desired properties of filled functions.     

Finding discrete global minima with a filled function for integer programming

The filled function method is an approach to find the global minimum of multidimensional functions. This paper proposes a new definition of the filled function for integer programming problem. A filled function which satisfies this definition is presented. Furthermore, we discuss the properties of the filled function and design a new filled function algorithm. Numerical experiments on several test problems with up to 50 integer variables have demonstrated the applicability and efficiency of the proposed method.     

A new discrete filled function algorithm for discrete global optimization

A definition of the discrete filled function is given in this paper. Based on the definition, a discrete filled function is proposed. Theoretical properties of the proposed discrete filled function are investigated, and an algorithm for discrete global optimization is developed from the new discrete filled function. The implementation of the algorithms on several 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 new filled function algorithm for constrained global optimization problems

A new filled function with one parameter is proposed for solving constrained global optimization problems without the coercive condition, in which the filled function contains neither exponential term nor fractional term and is easy to be calculated. A corresponding filled function algorithm is established based on analysis of the properties of the filled function. At last, we perform numerical experiments on some typical test problems using the algorithm and the detailed numerical results show that the algorithm is effective.     

全局优化问题的一个新的无参数填充函数

求全局最优化问题的填充函数算法被提出以来,参数的选取和调整一直是制约算法有效性的因素.如何在实际的计算过程中选取合适的参数,直接影响和决定了运算速度和效率.因此,构造不含参数的填充函数就显得极为重要.提出一个新的无参数的填充函数,对其理论性质进行了分析,并给出相应的填充函数算法,数值计算验证了算法的有效性.     

A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems

In this paper, a discrete filled function algorithm embedded with continuous approximation is proposed to solve max-cut problems. A new discrete filled function is defined for max-cut problems, and properties of the function are studied. In the process of finding an approximation to the global solution of a max-cut problem, a continuation optimization algorithm is employed to find local solutions of a continuous relaxation of the max-cut problem, and then global searches are performed by minimizing the proposed filled function. Unlike general filled function methods, characteristics of max-cut problems are used. The parameters in the proposed filled function need not to be adjusted and are exactly the same for all max-cut problems that greatly increases the efficiency of the filled function method. Numerical results and comparisons on some well known max-cut test problems show that the proposed algorithm is efficient to get approximate global solutions of max-cut problems.     

A DERIVATIVE-FREE ALGORITHM FOR UNCONSTRAINED OPTIMIZATION

In this paper a hybrid algorithm which combines the pattern search method and the genetic algorithm for unconstrained optimization is presented. The algorithm is a deterministic pattern search algorithm,but in the search step of pattern search algorithm,the trial points are produced by a way like the genetic algorithm. At each iterate, by reduplication,crossover and mutation, a finite set of points can be used. In theory,the algorithm is globally convergent. The most stir is the numerical results showing that it can find the global minimizer for some problems ,which other pattern search algorithms don＇t bear.     

A filled function method for finding a global minimizer of a function of several variables

The concept of a filled function is introduced. We construct a particular filled function and analyze its properties. An algorithm for global minimization is generated based on the concept and properties of the filled function. Some typical examples with 1 to 10 variables are tested and computational results show that in most cases this algorithm works better than the tunneling algorithm. The advantages and disadvantages are analyzed and further research directions are discussed.     

A discrete filled function algorithm for approximate global solutions of max-cut problems

A discrete filled function algorithm is proposed for approximate global solutions of max-cut problems. A new discrete filled function is defined for max-cut problems and the properties of the filled function are studied. Unlike general filled function methods, using the characteristic of max-cut problems, the parameters in proposed filled function need not be adjusted. This greatly increases the efficiency of the filled function method. By combining a procedure that randomly generates initial points for minimization of the filled function, the proposed algorithm can greatly reduce the calculation cost and be applied to large scale max-cut problems. Numerical results on different sizes and densities test problems indicate that the proposed algorithm is efficient and stable to get approximate global solutions of max-cut problems.     

A conjugate gradient method with descent direction for unconstrained optimization

A modified conjugate gradient method is presented for solving unconstrained optimization problems, which possesses the following properties: (i) The sufficient descent property is satisfied without any line search; (ii) The search direction will be in a trust region automatically; (iii) The Zoutendijk condition holds for the Wolfe–Powell line search technique; (iv) This method inherits an important property of the well-known Polak–Ribière–Polyak (PRP) method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. The global convergence and the linearly convergent rate of the given method are established. Numerical results show that this method is interesting.     

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.     

Filled functions for unconstrained global optimization

The paper is concerned with the filled functions for global optimization of a continuous function of several variables. More general forms of filled functions are presented for smooth and nonsmooth optimizations. These functions have either two adjustable parameters or one adjustable parameter. Conditions on functions and on the values of parameters are given so that the constructed functions are desired filled functions.