Abstract
A projection neural network method for circular cone programming is proposed. In the KKT condition for the circular cone programming, the complementary slack equation is transformed into an equivalent projection equation. The energy function is constructed by the distance function and the dynamic differential equation is given by the descent direction of the energy function. Since the projection on the circular cone is simple and costs less computation time, the proposed neural network requires less state variables and leads to low complexity. We prove that the proposed neural network is stable in the sense of Lyapunov and globally convergent. The simulation experiments show our method is efficient and effective.
1. Introduction
The circular cone is a pointed, closed, convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation [1–3]. Let its half-aperture angle be , and then the -dimensional circular cone denoted by can be expressed as follows:where is the standard Euclidean norm.
When , the circular cone reduces to the well-known second-order cone given by
In this paper, we consider a linear circular cone programming (LCCP), which is described as follows:where , , and , and . Here, is viewed as a column vector in . In addition,and . When , we have
The circular cone programming arises in some real-life engineering problems, such as the optimal grasping manipulation problems for multifingered robots [4, 5]. In addition, circular cone programming is applied in the perturbation analysis of second-order cone programming problems [6].
By the Lagrangian method, the dual programming to LCCP can be given aswhere is the Lagrange multipliers, and is the dual cone of cone , which is defined aswhere
Under mild constraint qualifications (e.g., Slater condition), strong duality holds for problem (3). Then, by the strong duality theorem for general conic programming problems [7], the KKT condition for (3) is given as
Circular cone programming is a nonlinear convex programming, and the second-order cone programming is a special case [8, 9]. There are many efficient methods to deal with the second-order cone programming [8, 10, 11]. However, different from the second-order cone, the circular cone is nonsymmetric. Some properties holding in the second-order cone can be extended to the circular cone [12]. However, some other second-order cone properties fail to be satisfied for the circular cone [3, 12].
Recently, an interior point method [13] is proposed for the circular cone programming based on self-concordant barrier functions for its cones. However, in many science and engineering applications, real-time solutions of circular cone problems are often desired. For the large-scale problems, the traditional algorithm may not be efficient due to the complexity of the algorithm used. The artificial neural network based circuit implementation is an efficient approach for the real-time problem. At present, two types of neural networks are developed for the second-order cone programming and have shown some computational advantages. One type is the smooth neural network, including the merit function method derived from the Fischer-Burmeister function [14], the smoothed natural residual merit function method [15], and the merit function method based on the generalized Fischer-Burmeister function [16]. The other is the projection neural network by replacing the scalar projection function with the cone projection function [14, 17]. In paper [18], professor He proposed a neural network based on the simple projection and contraction technique for linear asymmetric variational inequalities. Inspired by the ideal projection and contraction technique and the new results about the projection conclusions on the circular cone [2], we can develop the projection neural network for the linear circular cone programming.
In this paper, we focus on neural network approach to the circular cone programming problem. The energy function is constructed by the distance function based on a cone projection function, whose solutions correspond to the KKT points of the circular cone programming. The dynamic differential equation is given by the descent direction of the energy function based on the projection and contraction technique. The proposed neural network requires less state variables and leads to low complexity. Its Lyapunov stability and global convergence are proved under some conditions. Finally, we test the projection neural network by some numerical examples and the optimal grasping manipulation problems for multifingered robots and also compare the neural network with the second neural network for some second-order cone programming problems in paper [14]. Simulation results demonstrate the effectiveness of the proposed neural network.
2. Preliminaries
In this section, firstly, we introduced some concepts about the Lyapunov stability of the first-order differential equation [16, 19]. Then, we briefly introduce some properties of circular cone and the projection on the circular cone, which are proposed in paper [2].
2.1. Lyapunov Stability of the First-Order Differential Equation
Given a mapping , the following first-order differential equation is
Next, we give the definition of Lyapunov stability [19].
Definition 1 ([19]). (1)For (10), a point is called an equilibrium point of (10) if .(2)If there is a neighborhood of such that and for, then is called an isolated equilibrium point.
Definition 2 ([19]). Let be a solution of (10). For an isolated equilibrium point , if, for any and , there exists a such thatthen is Lyapunov stable.
Definition 3 (Lyapunov function [19]). Let be an open neighborhood of . A continuously differential function is said to be a Lyapunov function (or energy function) at the state for (10) if
The relationship between stabilities and a Lyapunov function is given as follows, which is proposed in paper [20].
Lemma 4 ([20]). An isolated equilibrium point is Lyapunov stable if there exists a Lyapunov function over some neighborhood of .
2.2. The Properties and Projection on the Circular Cone
Letfor , where is the dimension unit matrix. The following lemma describes the relationship between the second-order cone and the circular cone, where .
Lemma 5 ([2]). Let and be defined as in (1) and (2) for. Then we obtain(1) and ,(2) and ,(3) and .
From Lemma 5 and the definition of , ,, we have and
LetThen, from the conclusion above, we haveand
Let ,. Then the spectral decomposition of on circular cone can be given as [2]whereandwith if and any vector in satisfying if .
Next we introduce the projection on the circular cone [2].
Lemma 6 ([2]). For any , , let be the projection of on the circular cone . Then we obtainwhere for .
Let . Then the projection of on cone is described as
In particular, when , the projection on second-order cone can be obtained, which have been proposed in papers [21–23]:where when and any vector in satisfying when . Then the projection of on the cone is described as
It is well known that any can be written aswhere denotes any closed convex cone and represents the dual cone of . Hence, when and , respectively, we haveandSince the second-order cone is self-dual, then . We have
3. A Projection-Based Neural Network Model for Circular Cone Programming
In this section, we build a projection-based neural network model for circular cone programming.
Firstly, an equivalent projection equation is built for the KKT condition (9). Since the circular cone is a non-self-dual cone, our analysis is based on the relationship of circular cone and second-order cone.
Lemma 7. Assume . Then if and only if , where .
Proof. “⇒” The projection on the closed convex set has an important property [24],where denotes the inner product of two vectors.
Let in the inequality above. Then we haveMoreover, becauseandwe haveBy (31) and (34), is obtained.
“” It follows from thatFrom (17), we have
Based on (29), we haveThen, we getFrom (18), it is easy to obtain For any , we haveSince , we getSubstituting and in inequality (39), we haveThe proof is completed.
Letwhere denotes the projection on the set andObviously, is a block asymmetric matrix; i.e., . So, we haveFrom Lemma 7, we know that the KKT condition (9) is equivalent to
Let . Then it is easy to prove that
Next, we give the following neural network model for circular cone programming, which consists of the following energy function and dynamical system.
The energy function is given as follows:where .
Here, a dynamical system is proposed to solve (44). The dynamical system is given as follows:where and
The system described by (47) can be realized by a recurrent neural network with two-layer structure.
4. Stability Analysis
In this section, the stability analysis of projection neural network for circular cone programming is given.
For the dynamical system (47), we have the following result.
Lemma 8. if and only if is an equilibrium point of network (47), where is the solution set of problem (9).
Proof. “⇒” Since , we have . It is easy to know thatso is an equilibrium point of network (47).
“” Since is an equilibrium point of network (47), we obtain .
ConsiderSince is positive definite matrix, is obtained. From (45), we know .
The following result guarantees the existence and uniqueness for the solution of neural network (47).
Lemma 9. For any , there exists a unique continuous solution of (47) with for all .
Proof. Becausefrom the results in paper [11], we know that is semismooth. From all of the above, we conclude that is semismooth. Thus, there exists a unique solution for neural network (47).
The results of Lemmas 8 and 9 indicate that neural network model (47) is well defined. Now, we give the Lyapunov stability of neural network (47).
Theorem 10. The solution of neural network (47), with initial point , is Lyapunov stable.
Proof. From Lemma 9, there exists a unique continuous solution of (47) with for all . Since , we haveIn addition, for any, we haveFrom (52) and (53), we haveSetting in (54), it follows thatfor any . For any and , we have Let in . We obtainBased on (55) and (57), we getIt follows thatBecause , we have Thus we getBy the equation above, we obtainwhich shows that the solution of neural network (47) is Lyapunov stable.
Next, we give the globally convergent result of the trajectory of neural network (47).
Theorem 11. Let is the equilibrium point of neural network (47). The solution trajectory of neural network (47) with initial point is globally convergent to and has finite convergence time.
Proof. From (61), we know that the level setis bounded. Then, based on the Invariant Set Theorem [25], we have that the solution trajectory converges to as , where is the largest invariant set in
Now, we prove the result that if and only if , and then we can obtain the globe convergence of neural network (47).
If , then .
If , from (61), we haveSo . ThenFrom the conclusions above, converges globally to the equilibrium point . Moreover, from Lemma 9 and the same argument as in [14, 26], the neural network (47) has finite convergence time.
5. Simulation Experiments
In this section, we give some examples to test the simulation performance of neural network (47). The neural network is run in the MATLAB 7.0 environment on an Intel Core processor 1.80GHZ personal computer with 4.0GB of RAM. The numerical examples are solved by ODE23 in the ode solver, which is a nonstiff medium order method. In the ode solver,,, and for the test examples.
Example 12. Consider the following problem with two 3-dimensional circular cones:whereThe optimal solution of Example 12 isand
Figures 1 and 2 depict the trajectories of neural network (47) with the initial values and converging to its solutions and , respectively.
Example 13. Consider the following problem with three 3-dimensional circular cones:whereThe optimal solution of Example 13 isand
Figures 3 and 4 depict the trajectories of neural network (47) with the initial values and converging to its solutions and , respectively.
Example 14. Consider the problem with two 4-dimensional circular cones as follows:whereThe optimal solution of Example 14 isand
Figures 5 and 6 depict the trajectories of neural network (47) with the initial values and converging to its solutions and , respectively.
In paper [14], two neural networks are proposed for the second-order cone programs. The first neural network uses the Fischer-Burmeister function to achieve an unconstrained minimization with a merit function. The second neural network utilizes the natural residual function with the cone projection (CP) function to achieve low computation complexity. The second neural network is a projection neural network. Here we compared our projection neural network method with the neural network based on the cone projection function in paper [14] by the next two test examples.
Example 15. In Example 12, let . Then the circular cone programming problem is converted into a second-order cone programming problem. This test problem is from Example 5.2 in paper [14]. This problem has an optimal solution .
Figures 7 and 8 depict the trajectories obtained using neural network (47) and the neural network with CP function in [14], respectively.
The simulation results show that both trajectories are convergent to , but the neural network with the CP function yields the oscillating trajectory and has longer convergence time than the neural network (47).
Example 16. In Example 14, let . Then a second-order cone problem is obtained, which has an optimal solution .
Figures 9 and 10 depict the trajectories obtained using neural network (47) and the neural network with CP function in [14], respectively.
The simulation results show that both trajectories are convergent to . In addition, neural network with the CP function yields the oscillating trajectory and has longer convergence time than the neural network (47).
Example 17. In this example, we use the grasping force optimization problem for the multifingered robotic hand [4, 8, 14] to demonstrate the effectiveness of the proposed neural network. The force optimization is to minimize the magnitude of grasping force from every finger applied to an object. For the robotic hand with fingers, the optimization problem can be formulated aswhere is the grasping force, is the grasping transformation matrix, is the time-varying external wrench, and is the friction coefficient. Problem (78) is a convex quadratic circular cone programming problem. In [14], problem (78) is formulated as a convex quadratic second-order cone programming problem by variable transformation.
In this example, we consider a three-fingered grasping force optimization example [14]. The three-finger robot hand grasps a polyhedral with the grasp points , , and , and the robot hand moves along a vertical circular trajectory of radius with a constant velocity . Let . Then problem (78) is reformulated aswhere andHere, is the mass of the polyhedral, , the centripetal force, the time, and . In this example, we set the data as follows: , and . In the time-varying grasping force, we only test numerical examples when and . The other results are easily given when .
Because the neural network (47) is designed to solve the linear circular cone programming problem, it cannot be used to solve problem (34) directly. But our neural network can be extended to convex quadratic circular cone programming problem easily. The extension method is only to change the residual function and the matrix in neural network (47) as follows:The results of the projection neural networks (47) to solve Example 17 are shown in Figures 11 and 12 when and , respectively. The simulation results demonstrate that the neural networks are convergent for the grasping force optimization problem.
For these simulation examples, the proposed network in (47) with other initial points always converges to the theoretical optimal solution. The simulation results demonstrate that the neural networks are effective for the circular cone programming.
6. Conclusion
In the paper, a projection neural network method is proposed for the circular cone programming. The projection equation and the contraction technique are used to construct the energy function and dynamical system. In the method, the projection and contraction technique accelerates the convergence of the neural network method. In addition, the proposed neural network requires less state variables, so the neural network method is simple and efficient.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Science Foundations for Young Scientists of China (11101320, 61201297), National Science Basic Research Plan in Shaanxi Province of China (2015JM1031), and the Fundamental Research Funds for the Central Universities (JB150713).