Research Article | Open Access
Canonical Duality for Box Constrained Nonconvex and Nonsmooth Optimization Problems
This paper presents an application of the canonical duality theory for box constrained nonconvex and nonsmooth optimization problems. By use of the canonical dual transformation method, which is developed recently, these very difficult constrained optimization problems in can be converted into the canonical dual problems, which can be solved by deterministic methods. The global and local extrema can be identified by the triality theory. Some examples are listed to illustrate the applications of the theory presented in the paper.
1. Primal Problems
The methods of solving nonconvex and nonsmooth optimization have been the topic of intense research during the last forty years. As we know, the general methods of nonsmooth optimization are based on some subdifferentials. However, in the general nonconvex and nonsmooth optimization problem, due to the nonconvexity and nonsmoothness of the objective functions and inequality constraints, the computation of subdifferential is rather time-consuming. So the traditional theories and direct methods are very difficult in solving the nonconvex and nonsmooth optimization and global optimality with constraints. Recently, some effective methods have been studied to solve certain box constrained nonconvex minimization problems [1–8].
In this paper, the primary goal is to solve the following box constrained nonconvex and nonsmooth optimization problems (in short, the primal problem ). Considerwhere (the notation denotes integer vectors of with components either or 1) is a feasible space, is a nonzero vector, and is an times matrix. is a nonconvex and nonsmooth function. Here, we simply assume that is defined by where is an times matrix, is a vector of parameter, is an arbitrary constant, and are both nonzero positive constants, is the Euclidean norm, and is an arbitrary positive constant. In constrained global optimization problems, could be the indicator of a feasible set . In particular, if we abandon the box constraint, problem will be turned into unconstrained optimal problem which has been discussed in . The primal problem appears frequently in many applications, such as semilinear nonconvex partial differential equations, structural limit analysis, discretized optimal control problems with distributed parameters, and network communication [6, 11, 12].
In the primal problem , the box constraint is equivalent to . Thus, based on the traditional Lagrangian multiplier method, we have In the case where is positive definite, for a given , the traditional dual function can be defined via the Fenchel-Moreau-Young duality theory: However, due to the nonconvexity of the objective function , the Young-Fenchel inequality can lead to a weak duality relationship in general nonconvex systems: The nonzero value is called the duality gap, which is usually if is indefinite. This nonzero duality gap shows that the Fenchel-Rockafellar duality theory and method can be used mainly in convex systems. In order to eliminate this duality gap, many modified Fenchel-Rockafellar duality theories and methods for nonconvex optimization problems have been proposed [12–20]. Recently, the so-called canonical dual transformation method (without the duality gap) has been developed in general nonconvex systems . This method is a newly useful tool for optimal problem. At present, the method has been successfully used for a large class of nonsmooth or nonconvex minimization [9, 10, 21, 22].
In this paper, we will present the application of the canonical dual transformation method for the solutions of the box constrained nonconvex and nonsmooth optimization problems () in . In the next section, a perfect dual problem is formulated, which is equivalent to the primal problem in the sense that they have the same set of critical points. Section 3 shows the sufficient conditions for global and local minima. In Section 4, some concrete examples for box constrained nonconvex and nonsmooth optimization problems are presented. We state some conclusions in Section 5.
2. Canonical Dual Problem and Complete Solutions
In order to use the canonical dual transformation method to solve the box constrained nonconvex and nonsmooth optimization problem, we need to reformulate the constraint in canonical form .
Following the stand procedure of the canonical dual transformation method developed in , the canonical geometrical operator in the primal problem () can be defined as where is a scale and for . Let be the range of the mapping which can be written asAccording to the canonical transformation method, we define a real-valued function . Thus, the nonconvex and nonsmooth function can be written in a canonical form: Then is a canonical function defined on the subset which can be written as where
We yet assume that matrix is invertible. For each given nonzero vector , the function is defined by where is a canonical function on since its Gâteaux derivative is one-to-one onto mapping. Thus, we can rewrite the primal minimization problem () in the unconstrained canonical form (() in short):where stands for finding all the stationary points of .
Let be a dual variable of y and , where represents a diagonal matrix with as its diagonal entries and the dual variable is also a vector in . Then the -canonical conjugate of the canonical function can be well defined by the -canonical dual transformation :where , , is the Gâteaux derivative of at , and denotes the gradient at .
In the case of , by the definition of the canonical function, is Gâteaux differentiable, and the duality relation is invertible, where denotes the Gâteaux derivative of at . Thus, the canonical conjugate , where of , can be obtained by the Legendre transformation: The dual feasible spaces are three subsets of :
By replacing in by the complementary form , we obtain the Gao-Strang total complementary function : Thus, on the dual feasible spaces, the canonical dual function can be formulated as The canonical dual problem can be formulated as problem ():
In the case of , the canonical conjugate of can be written as It is similar to that in the case of where the dual feasible spaces are also three subsets of : Hence, on the dual feasible spaces, the canonical dual problem () can be formulated as
Theorem 1 (perfect duality theorem). Suppose that vector is a stationary point of the canonical dual problem (); then the vector defined by is a stationary point of the primal problem (), and
Proof. Without loss of generality, we suppose that is a stationary point of (); then we have the following canonical dual equation: In terms of and (24), one has that By the condition , it is obtained that . On the other hand, as is the stationary point of (), then where is the th component of the vector . It is obvious that is also a stationary point of the primal problem (). Thus, by and we have This proves the theorem.
Theorem 1 shows that, by use of the canonical dual transformation, the primal problem can be converted into a canonical dual problem, which can be solved to obtain the stationary points. However, the stationary conditions are only necessary for nonconvex optimization problem. In the next section we will present the optimality criteria.
3. Optimality Criteria
The two subsets of the set are, respectively, defined by
Theorem 2 (triality theorem). For each dual solution , one lets If , then is a global maximizer of on , while is a global minimizer of on , and If , then, on the neighborhoods and of and the associated , respectively, we have that either or holds.
Proof. Without loss of generality, we suppose that ; that is, . The extended Lagrangian can be written as It can be easily proved that the critical points of solve the primal problem, and By Theorem 1, we know that the vector is a stationary point of problem () if and only if is a stationary point of problem () and If is definite, the extended Lagrangian is convex on and concave in each and ; that is, the extended Lagrangian is a saddle function on . By formula (12) and the definition of , one has thatThus, we have If , the matrix is indefinite; the function is a so-called super-Lagrangian ; that is, it is locally concave in each of its variables and on the neighborhood . In this case, by the triality theory developed in , we have either or The theorem is proved.
Theorem 2 shows that if , then is global minimizer of , while the stationary point , then is either a local minimizer or a local maximizer of the primal problem . The optimality criteria provides the sufficient conditions of both global and local minina, which can be used to develop for solving the nonconvex and nonsmooth problem with box constraints.
4. Numerical Tests
The canonical duality theory can be applied to solve many constrained nonconvex and nonsmooth problems in engineering and science. Now we list some examples to illustrate the applications of the theory presented in this paper.
Example 1. In the case of , the nonconvex and nonsmooth function whereIf we choose , then ; the function can be written asAnd its canonical dual function can be written asIn the effective domain , the canonical dual function has one real root: In the effective domain , the canonical dual function does not have any real roots. In the effective domain , the canonical dual function has two real roots: Since , by Theorem 2, we know that is a global minimizer of and , while is a local maximizer and is a local minimizer (see Figure 1(a)). The graphs of the box constrained nonconvex and nonsmooth function and its canonical dual function are shown in Figure 1.
Example 2. In the case of , if we choose , , , , , then , and the function can be written asIn the effective domain , the canonical dual function has one real root Neither in the effective domain nor in the effective domain , the canonical dual function have any real roots.
Since , by Theorem 2, we know that is a global minimizer of and (see Figure 2).
It is very difficult to solve the nonconvex and nonsmooth optimization problem with box constraints via the traditional methods. However, using the canonical dual transformation method, which is presented in the paper, we can easily obtain the global minimizer of by solving its canonical dual problem. As the canonical dual function is concave on the dual feasible space , when is nonempty, the canonical dual function has one stationary point in at least. Thus, we can solve the canonical dual problem by well-developed deterministic optimization methods. The presented paper shows again that the canonical dual transformation method may possess important computational impacts on global optimization.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was supported by Natural Science Foundation of Guangdong Province (s20120100815) and Youth Foundation of Wuyi University (2013zk14 and 2013zk04).
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Copyright © 2015 Jing Liu and Huicheng Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.