Research Article | Open Access
A Filled Function Approach for Nonsmooth Constrained Global Optimization
A novel filled function is given in this paper to find a global minima for a nonsmooth constrained optimization problem. First, a modified concept of the filled function for nonsmooth constrained global optimization is introduced, and a filled function, which makes use of the idea of the filled function for unconstrained optimization and penalty function for constrained optimization, is proposed. Then, a solution algorithm based on the proposed filled function is developed. At last, some preliminary numerical results are reported. The results show that the proposed approach is promising.
Recently, since more accurate precisions demanded by real-world problems, studies on global optimization have become a hot topic. Many theories and algorithms for global optimization have been proposed. Among these methods, filled function method is a particularly popular one. The filled function method was originally introduced in [1, 2] for smooth unconstrained global optimization. Its idea is to construct a filled function via it the objective function leaves the current local minimum to find a better one. The filled function method consists of two phase: local minimization and filling. The two phases are performed repeatedly until no better minimizer could be located. The filled function method was further developed in literature [3–9]. It should be noted that these filled function methods deal only with smooth unconstrained or box constrained optimization problem. However, many practical problems could only be modelled as nonsmooth constrained global optimization problems. To address this situation, in this paper, we generalize the filled function proposed in  and establish a novel filled function approach for nonsmooth constrained global optimization. The key idea of this approach is to combine the concept of filled function for unconstrained global optimization with the penalty function for constrained optimization.
In general, there are two difficulties in global optimization: the first is how to leave the current local minimizer of to go to a better one; the second is how to check whether the current minimizer is a global solution of the problem. Just like other GO methods, the filled function method has some weaknesses discussed in . In particular, the filled function method cannot solve the second issue, so our paper focuses on the former issue.
The rest of this paper is organized as follows. In Section 2, some preliminaries about nonsmooth optimization and filled function are listed. In Section 3, the concept of modified filled function for nonsmooth constrained global optimization is introduced, a novel filled function is given, and its properties are investigated. In Section 4, an efficient algorithm based on the proposed filled function is developed for solving nonsmooth constrained global optimization problem. Section 5 presents some numerical results. Last, in Section 6, the conclusion is given.
2. Nonsmooth Preliminaries
In this section, we first list some definitions and lemmas from , then we make some assumptions on , and finally we define filled function for problem ()
Definition 2.1. Letting be Lipschitz with constant at the point the generalized gradient of at is defined as where is the generalized directional derivative of in the direction at
Lemma 2.2. Let be Lipschitz with constant at the point then (a) is finite, sublinear and satisfies ;(b)for all , , and to any , one has ;(c), for all
Considering problem () throughout the paper, we need the following assumptions: and , , are Lipschitz continuous with a common constant the number of the different value of local minimizer of is finite;, where denotes the interior of denotes the closure of
Definition 2.3. A function is called a filled function of at if all the following conditions are met:(1) is a strict local maximizer of on (2) has no stationary points in the set that is, ;(3)If is not a global minimizer of problem (), then there exists a point such that is a local minimizer of on with
3. A New Filled Function and Its Properties
Define where is a parameter, is a differentiable function such that and for any and indicates the Euclidean vector norm.
Theorem 3.1. is a strict local maximizer of on
Proof. Since is a local minimizer of () there exists a neighborhood of with such that for any We consider the following two cases.Case 1 ( and ). In this case, note that then Case 2 (). In this case, ; moreover, there exists at least one index such that It follows that Therefore, is a strict local maximizer of
Theorem 3.2. For any one has
Proof. For any similar to the proof of Theorem 3.1, we have Since it follows that Therefore, we have that So, to any , one has . Then
Theorem 3.3. Suppose that Assumptions (1)–(3.8) are satisfied. If is not a global minimizer, and is appropriately large, then there exists a point such that is a minimizer of
Proof. Since is not a global minimizer, there exists another local minimizer of () such that By Assumption (3.1), there exists one point such that Thus, we have
On the other hand, to any where denotes the boundary of the set there exists at least one index such that , which yields
Let To any if is chosen to be appropriately large such that
then, we have that
Denotes Then, if is appropriately large such that (3.10) is met, one has Note that is an open bounded set, thus and for Moreover, we can easily prove that In fact, if it is not true, then which contradicts with (3.12).
Therefore, one has This completes the proof.
4. Solution Algorithm
In the previous section, several properties of the proposed filled function are discussed. Now a solution algorithm based on these properties is described as follows.
(1)Choose a disturbance constant ; for example, set .(2)Choose an upper bound of such that ; for example, set .(3)Choose a constant ; for example, .(4)Choose direction with integer , where is the number of variable.(5)Set .
() Start from an initial point , minimize the primal problem () by implementing a nonsmooth local search procedure, and obtain the first local minimizer of .() Let () Construct the filled function: ()If , then go to (7). Else set as an initial point, minimize the filled function problem by implementing a nonsmooth local search procedure, and obtain a local minimizer denoted .()If , then set , go to (). Else go to next step.()If satisfies , then set and , start from as a new initial point, minimize the primal problem () by implementing a local search procedure, and obtain another local minimizer of such that , set , go to (). Else go to next step.() Increase by setting .()If , then set , go to (). Else the algorithm is incapable of finding a better local minimizer. The algorithm stops and is taken as a global minimizer.
The motivation and mechanism behind the algorithm are explained as below.
A set of initial points is chosen in Step of the Initialization step to minimize the filled function. We set the initial points symmetric about the current local minimizer. For example, when the initial points are: For example, when , the directions can be chosen as
In Step and Step of the Main step, we minimize the primal problem () by nonsmooth constrained local optimization algorithms such as penalty function method, bundle method, quasi-newton method and composite optimal method. In Step of the Main step, we minimize the filled function problem by nonsmooth unconstrained local optimization algorithms such as cutting-planes method, powell method, and Hooke-Jeeve method. They are all effective methods.
Recall from Theorem 3.3 that the value of should be selected large enough. Otherwise, there could be no minimizer of in set Thus, is increased successively in Step of the solution process if no better solution is found when minimizing the filled function. If all the initial points have been used and reaches its upper bound , but no better solution is found, then the current local minimizer is taken as a global one.
The proposed filled function method can also apply to smooth constrained global optimization.
5. Numerical Experiment
In this section, we perform a numerical test to give an initial feeling of the potential application of the proposed function approach in real-world problems. In our programs, the filled function is of the form The proposed algorithm is programmed in Fortran 95. The composite optimal method is used to find local minimizers of the original constrained problem, and the Hooke-Jeeve method is used to search for local minimizers of the filled function problems.
The main iterative results of Algorithm NFFA applying on four test examples are listed in Tables 1–4. The symbols used in the tables are given as follows::the iteration number in finding the th local minimize;:the parameter to find the th local minimize;:the th initial point to find the th local minimize;:the th local minimize;:the function value of the th initial point;:the function value of the th local minimizer.
Problem 1. We have
Algorithm NFFA succeeds in finding a global minimizer with . The numerical results are listed in Table 1.
Problem 2. We have
Algorithm NFFA successfully finds an approximate global solution with . Table 2 records the numerical results of Problem 2.
Problem 3. We have where
Algorithm NFFA successfully finds a global solution with . The computational results are listed in Table 3.
Problem 4. We have
The proposed algorithm successfully finds a global solution with . The main iterative results are listed in Table 4.
In this paper, we extend the concept of the filled function for unconstrained global optimization to nonsmooth constrained global optimization. Firstly, we give the definition of the filled function for constrained optimization and construct a new filled function with one parameter. Then, we design a solution algorithm based on this filled function. Finally, we perform some numerical experiments. The preliminary numerical results show that the new algorithm is promising.
The paper was supported by the National Natural Science Foundation of China under Grant nos. 10971053, 11001248.
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