Abstract
This paper focuses on solving the quadratic programming problems with second-order cone constraints (SOCQP) and the second-order cone constrained variational inequality (SOCCVI) by using the neural network. More specifically, a neural network model based on two discrete-type families of SOC complementarity functions associated with second-order cone is proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions of SOCQP and SOCCVI. The two discrete-type SOC complementarity functions are newly explored. The neural network uses the two discrete-type families of SOC complementarity functions to achieve two unconstrained minimizations which are the merit functions of the Karuch-Kuhn-Tucker equations for SOCQP and SOCCVI. We show that the merit functions for SOCQP and SOCCVI are Lyapunov functions and this neural network is asymptotically stable. The main contribution of this paper lies on its simulation part because we observe a different numerical performance from the existing one. In other words, for our two target problems, more effective SOC complementarity functions, which work well along with the proposed neural network, are discovered.
1. Introduction
In optimization community, it is well known that there are many computational approaches to solve the optimization problems such as linear programming, nonlinear programming, variational inequalities, and complementarity problems; see [1–6] and references therein. These approaches include the method using merit function, interior point method, Newton method, nonlinear equation method, projection method, and its variant versions. All the aforementioned methods rely on iterative schemes and usually only provide “approximate” solution(s) to the original optimization problems and do not offer real-time solutions. However, real-time solutions are eager in many applications, such as force analysis in robot grasping and control applications. Therefore, the traditional optimization methods may not be suitable for these applications due to stringent computational time requirements.
The neural network approach has an advantage in solving real-time optimization problems, which was proposed by Hopfield and Tank [7, 8] in the 1980s. Since then, neural networks have been applied to various optimization problems; see [9–30] and references therein. Unlike the traditional optimization algorithms, the essence of neural network approach for optimization is to establish a nonnegative Lyapunov function (or energy function) and a dynamic system that represents an artificial neural network. This dynamic system usually adopts the form of a first-order ordinary differential equation and its trajectory is likely convergent to an equilibrium point, which corresponds to the solution to the considered optimization problem.
Following the similar idea, researchers have also developed many continuous-time neural networks for second-order cone constrained optimization problems. For example, Ko, Chen and Yang [31] proposed two kinds of neural networks with different SOCCP functions for solving the second-order cone program; Sun, Chen, and Ko [32] gave two kinds of neural networks (the first one is based on the Fischer-Burmeister function and the second one relies on a projection function) to solve the second-order cone constrained variational inequality (SOCCVI) problem; Miao, Chen, and Ko [33] proposed a neural network model for efficiently solving general nonlinear convex programs with second-order cone constraints. In this paper, we are interested in employing neural network approach for solving two types of SOC constrained problems, the quadratic programming problems with second-order cone constraints (SOCQP for short) and the second-order cone constrained variational inequality (SOCCVI for short), whose mathematical formats are described as below.
The SOCQP is in the form ofwhere , is an matrix with full row rank, , and is the Cartesian product of second-order cones (SOCs), also called Lorentz cones. In other words, where are positive integers, , and denotes the SOC in defined bywith denoting the nonnegative real number set . A special case of (3) corresponds to the nonnegative orthant cone , i.e., and . We assume that is a symmetric positive semidefinite matrix and problem (1) satisfies a suitable qualification [34], such as the generalized Slater condition that there exists with strictly feasibility, then is a solution to problem (1) if and only if there exists a Lagrange multiplier such thatIn Section 3, we will employ two new families of SOC complementarity functions and use (4) to build up the neural network model for solving SOCQP.
We say a few words about why we assume that is a symmetric positive semidefinite matrix. First, it is clear that the symmetric assumption is reasonable because can be replaced by which is symmetric. Indeed, with being symmetric positive definite matrix, the SOCQP can be recast as a standard SOCP. To see this, we observe that which is done by completing the square. Then, the SOCQP (with ) is equivalent to which is also the same asThis formulation is further equivalent toNow, we let which says and denote Thus, the above reformulation (8) is expressed as a standard SOCP as follows:In view of this reformulation (10), we focus on SOCQP with being symmetric positive semidefinite in this paper.
The SOCCVI, our another target problem, is to find satisfyingwhere the set is finitely representable and is given byHere denotes the Euclidean inner product, , , and are continuously differentiable functions, and is a Cartesian product of second-order cones (or Lorentz cones), expressed aswith , , . When is affine, an important special case of the SOCCVI corresponds to the KKT conditions of the convex second-order cone program (CSOCP):where has full row rank, , , and . Furthermore, when is a convex twice continuously differentiable function, problem (14) is equivalent to the following SOCCVI which is to find such that where In fact, the SOCCVI can be solved by analyzing its KKT conditions:where is the variational inequality Lagrangian function, and . We also point out that the neural network approach for SOCCVI was already studied in [32]. Here we revisit the SOCCVI with different neural models. More specifically, in our earlier work [32], we had employed neural network approach to the SOCCVI problem (11) and (13), in which the neural networks were aimed at solving system (17) whose solutions are candidates of SOCCVI problem (11) and (13). There were two neural networks considered in [32]. The first one is based on the smoothed Fischer-Burmeister function, while the other one is based on the projection function. Both neural networks possess asymptotical stability under suitable conditions. In Section 4, in light of (17) again, we adopt new and different SOC complementarity functions to construct our new neural networks.
As mentioned earlier, this paper studies neural networks by using two new classes of SOC complementarity functions to efficiently solve SOCQP and SOCCVI. Although the idea and the stability analysis for both problems are routine, we emphasize that the main contribution of this paper lies on its simulations. More specifically, from numerical performance and comparison, we observe a new phenomenon different from the existing one in the literature. This may suggest update choices of SOC complementarity functions to work with neural network approach.
2. Preliminaries
Consider the first-order differential equations (ODE):where is a mapping. A point is called an equilibrium point or a steady state of the dynamic system (18) if If there is a neighborhood of such that and , then is called an isolated equilibrium point.
Lemma 1. Suppose that is a continuous mapping. Then, for any and , there exists a local solution to (18) with for some If, in addition, is locally Lipschitz continuous at , then the solution is unique; if is Lipschitz continuous in , then can be extended to .
Let be a solution to dynamic system (18). An isolated equilibrium point is Lyapunov stable if for any and any , there exists a such that for all and . An isolated equilibrium point is said to be asymptotic stable if in addition to being Lyapunov stable, it has the property that as for all . An isolated equilibrium point is exponentially stable if there exists such that arbitrary point of (18) with the initial condition and is well defined on and satisfies where and are constants independent of the initial point.
Let be an open neighborhood of . A continuously differentiable function is said to be a Lyapunov function at the state over the set for (18) ifThe Lyapunov stability and asymptotical stability can be verified by using Lyapunov function, which is a useful tool for analysis.
Lemma 2. (a) An isolated equilibrium point is Lyapunov stable if there exists a Lyapunov function over some neighborhood of .
(b) An isolated equilibrium point is asymptotically stable if there exists a Lyapunov function over some neighborhood of such that .
For more details, please refer to any usual ODE textbooks, e.g., [35].
Next, we briefly recall some concepts associated with SOC, which are helpful for understanding the target problems and our analysis techniques. We start with introducing the Jordan product and SOC complementarity function. For any and , we define their Jordan product associated with as The Jordan product , unlike scalar or matrix multiplication, is not associative, which is a main source of complication in the analysis of SOC constrained optimization. There exists an identity element under this product, which is denoted by . Note that means and means the usual componentwise addition of vectors. It is known that for all . Moreover, if , then there exists a unique vector in , denoted by , such that . We also denote .
A vector-valued function is called an SOC complementarity function if it satisfiesThere have been many SOC complementarity functions studied in the literature; see [36–40] and references therein. Among them, two popular ones are the Fischer-Burmeister function and the natural residual function , which are given by Some existing SOC complementarity functions are indeed variants of and . Recently, Ma, Chen, Huang, and Ko [41] explored the idea of “discrete generalization” to the Fischer-Burmeister function which yields the following class of functions (denoted by ):where is a positive odd integer. Applying similar idea, they also extended to another family of SOC complementarity functions, , whose formula is as follows:where is a positive odd integer and means the projection onto . The functions and are continuously differentiable SOC complementarity functions with computable Jacobian, which can be found in [41].
3. Neural Networks for SOCQP
In this section, we first show how we achieve the neural network model for SOCQP and prove various stabilities for it accordingly. Then, numerical experiments are reported to demonstrate the effectiveness of the proposed neural network.
3.1. The model and Stability Analysis
As mentioned in Section 2, the KKT conditions are expressed in (4). With system (4) and using a given SOC complementarity function , it is clear to see that system (4) is equivalent to where . Moreover, we can specifically describe as the following: Here is a continuously differentiable SOC complementarity function such as and introduced in Section 2. It is clear that if solves , then solves . Accordingly, we consider a specific first-order ordinary differential equation as follows:where is a time scaling factor. In fact, letting , then . Hence, it follows from (28) that . In view of this, we set in the subsequent analysis. Next, we show that the equilibrium of the neural network (28) corresponds to the solution to system (4).
Lemma 3. Let be an equilibrium of the neural network (28) and suppose that is nonsingular. Then solves system (4).
Proof. Since and is nonsingular, it is clear to see that if and only if .
Besides, the following results address the existence and uniqueness of the solution trajectory of the neural network (28).
Theorem 4. (a) For any initial point , there exists a unique continuously maximal solution with for the neural network (28).
(b) If the level set is bounded, then can be extended to .
Proof. This proof is exactly the same as the one in [32, Proposition 3.4], so we omit it here.
Now, we are ready to analyze the stability of an isolated equilibrium of the neural network (28), which means and for , with being a neighborhood of .
Theorem 5. Let be an isolated equilibrium point of the neural network (28). (a)If is nonsingular, then the isolated equilibrium point is asymptotically stable and hence Lypunov stable.(b)If is nonsingular for all , then the isolated equilibrium point is exponentially stable.
Proof. The desired results can be proved by using Lemma 3 and mimicking the arguments as in [32, Theorem 3.1].
3.2. Numerical Experiments
In order to demonstrate the effectiveness of the proposed neural network, we test three examples for our neural network (28). The numerical implementation is coded by Matlab 7.0 and the ordinary differential equation solver adopted here is ode23, which uses Ruge-Kutta formula. As mentioned earlier, in general the parameter is set to be . For some special examples, the parameter is set to be another value.
Example 6. Consider the following SOCQP problem:
After suitable transformation, it can be recast as an SOCQP with , , , and . This problem has an optimal solution . Now, we use the proposed neural network (28) with two cases and , respectively, to solve the above SOCQP and their trajectories are depicted in Figures 1–4. For the sake of coding needs and check, the following expressions are presented.
For case of , we have Note that the element can also be expressed as where and if ; otherwise with being any vector in satisfying .
For case of , we replace as , Hence, and have the forms as follows:
Figures 1 and 3 show the transient behaviors of Example 6 for neural network model (28) based on smooth SOC complementarity functions and with initial states , respectively. In Figure 2, we see the convergence comparison of the neural network model using function with different values of . Figure 4 depicts the influence of the parameter on the value of norm of error for neural network model using function.
Example 7. Consider the following SOCQP problem:
For this SOCP, we have This problem has an approximate solution . Note that the precise solution is . Indeed, we have We also report numerical experiments for two cases when and ; see Figures 5–8.
Figures 5 and 7 show the transient behaviors of Example 7 for neural network model (28) based on and with initial states , respectively. Figure 6 provides the convergence comparison by using function with different values of . Figure 8 shows the convergence of neural network model using function, which indicates that this class of functions performs not well for this problem.
Example 8. Consider the following SOCQP problem:
Here, we have and , . This problem has an optimal solution .
Figures 9 and 11 show the transient behaviors of Example 8 for neural network model (28) based on and with initial states , respectively. Figure 10 shows that there are no difference between the neural networks using function with . Figure 12 elaborates that when the neural network based on function produces fast decrease of norm of error. We point out that the neural network does not converge when for both cases.
4. Neural Networks for SOCCVI
This section is devoted to another type of SOC constrained problem, SOCCVI. Like what we have done for SOCQP, in this section, we first show how we build up the neural network model for SOCCVI and prove various stabilities for it accordingly. Then, numerical experiments are reported to demonstrate the effectiveness of the proposed neural network.
4.1. The Model and Stability Analysis
Let be a SOC complementarity function like and defined as in (24) and (25), respectively. Mimicking the arguments described as in [42], we can verify that the KKT system (17) is equivalent to the following unconstrained smooth minimization problem:where and is given by with . In other words, is a smooth merit function for the KKT system (17). Hence, based on the above smooth minimization problem (38), it is natural to propose a neural network for solving the SOCCVI as follows:where is a scaling factor. To prove the stability of neural network (40), we need to present some properties of .
Proposition 9. Let be defined as in (38). Then, for . Moreover, if and only if solves the KKT system (17).
Proof. The proof is straightforward.
Proposition 10. Let be defined as in (38). Then, the following results hold. (a)The function is continuously differentiable everywhere with where(b)If is nonsingular, then for any stationary point of , is a KKT triple of the SOCCVI problem.(c) is nonincreasing with respect to .
Proof. It follows from the chain rule immediately.
From and the fact that matrix is nonsingular, it is clear that if and only if . Hence, we see that is a KKT triple of the SOCCVI problem provided is a stationary point of .
From the definition of and (40), it is easy to verify thatwhich says is a monotonically decreasing function with respect to .
Now, we are ready to analyze the behavior of the solution trajectory of neural network (40) and establish three kinds of stabilities for an isolated equilibrium point.
Proposition 11. (a) If is a KKT triple of the SOCCVI problem, then is an equilibrium point of neural network (40).
(b) If is nonsingular and is an equilibrium point of (40), then is a KKT triple of the SOCCVI problem.
Proof. From Proposition 9 and being a KKT triple of SOCCVI problem, it is clear that , which implies . Besides, by Proposition 10, we know that . This shows that is an equilibrium point of neural network (40).
It follows from being an equilibrium point of neural network (40) that . In other words, is the stationary point of . Then, the result is a direct consequence of Proposition 10(b).
Proposition 12. (a) For any initial state , there exists exactly one maximal solution with for the neural network (40).
(b) If the level set is bounded, then .
Proof. Since is continuous differentiable, it says that is continuous. This means is bounded on a local compact neighborhood of , which implies that is locally Lipschitz continuous. Thus, applying Lemma 1 leads to the desired result.
This proof is similar to the proof of Case(i) in [10, Proposition 4.2], so we omit it.
Remark 13. A natural question arises here. When are the level sets bounded for all ? For the time being, we are not able to answer this question yet. We suspect that there needs more subtle properties of , , and to finish it.
Next, we investigate the convergence of the solution trajectory and stability of neural network (40), which are the main results of this section.
Theorem 14. (a) Let with be the unique maximal solution to the neural network (40). If and is bounded, then .
(b) If is nonsingular and is the accumulation point of the trajectory , then is a KKT triple of the SOCCVI problem.
Proof. With Proposition 10(b) and 10(c) and Proposition 12, the arguments are exactly the same as those for [19, Corollary 4.3]. Thus, we omit them.
Theorem 15. Let be an isolated equilibrium point of the neural network (40). Then, the following results hold. (a) is asymptotically stable and hence is also Lyapunov stable.(b)If is nonsingular, then it is exponentially stable.
Proof. Again, the arguments are similar to those in [32, Theorem 3.1] and we omit them.
To study the conditions for nonsingularity based on and , we need the following assumptions.
Assumption 16. The gradients are linear independent.
is positive definite on the null space of the gradients .
When SOCCVI problem corresponds to the KKT conditions of a convex second-order cone program (CSOCP) problem as (14) where both and are linear, the above Assumption 16(b) is indeed equivalent to the well-used condition of being positive definite, e.g., [22, Corollary 1].
Assumption 17. Let and , where and . For , we have (a) or ;(b) or .
Theorem 18. Suppose for and that Assumptions 16 and 17 hold. Then, the matrixis nonsingular.
Proof. We know that is nonsingular if and only if the following equation only has zero solution:To reach the conclusion, we need to prove . First, plugging the components of into (46), we havewhere From (47) and (48), we see thatwhile from (49), we haveNext, we will claim thatTo see this, we note that In view of this, to prove inequality (53), it suffices to show thatfor . For convenience, we denote , , , and . With these notations, we have which says that it is enough to show is semipositive definite in order to prove inequality (55). To this end, we compute thatIt can be verified that and are positive semidefinite. Then, from Assumption 17 and (57), we conclude that is semipositive definite; and hence inequality (55) holds. Thus, inequality (53) also holds accordingly. Now, tt follows from (51), (52), and (53) that which implies that . Then, (47) and (48) becomeIn light of Assumption 16(a) and (58), we knowCombining (59) and (60) together, it is clear to obtainNote that and are strict complementary. Hence, it yields . In summary, from equation (46), we deduce , which says the matrix is nonsingular.
Theorem 19. Suppose Assumption 16 holds andThen, the matrix is nonsingular.
Proof. The proof can be done by following the similar arguments as in Theorem 18.
To close this subsection, we say a few words about the complexity of the proposed neural network. Since SOCQP can be transformed into an SOCCVI problem, we only take SOCCVI as an example to illustrate the complexity of the proposed neural network model. In light of the main ideas for constructing neural network (see [25] for details), we establish a specific first-order ordinary differential equation, i.e., an artificial neural network. More specifically, based on the gradient of the objective function, we employ the neural network for solving the KKT system (17) of SOCCVI with the differential equation (40), where is a time scaling factor. In fact, if , then . Hence, it follows from (40) that . In view of this, for simplicity and convenience, we set in this paper. Indeed, the dynamic system (40) can be realized by an architecture with the two discrete-type classes of SOC complementarity functions and shown in Figure 13. Moreover, the architecture of this artificial neural network is categorized as a “recurrent” neural network according to the classifications of artificial neural networks as in [25]. The circuit for (40) requires integrators, processors for , processors for , processors for , processors for , processors for , 1 processor for and , processors for and , processors for and , connection weights, and some summers.
4.2. Numerical Experiments
In this subsection, to demonstrate effectiveness of the proposed neural networks, some illustrative SOCCVI problems are tested. The numerical implementation is coded by Matlab 2014b and the ordinary differential equation solver adopted is ode23, which uses Runge-Kutta formula. In the subsequent tests, the parameter in neural networks of Example 22 is set to be 10 and others are set to be 1.
Example 20. Consider the SOCCVI problem where
This problem has an approximate solution . Figures 14 and 16 show the transient behaviors of Example 20 for neural network model (40) based on and with initial states , respectively. Figure 15 depicts the convergence comparison of the neural network model using function with different values of . From this, we see that the performance of is significantly better than in other cases. Figure 17 depicts the influence of the parameter on the value of norm of error for neural network model using function. Again, when , the neural network based on function produces fast decrease of norm of error.
Example 21. Consider the problem where
This problem has an approximate solution . In light of the analysis of previous section, this CSOCP in Example 21 can be transformed into an SOCCVI. We use the proposed neural networks to solve the problem with the trajectory obtained by it shown in Figures 18–21.
Figures 18 and 20 show the transient behaviors of Example 21 for neural network model (40) based on and with initial states , respectively. From Figure 19, we see the convergence comparison of the neural network model using function with . There is no significant difference when perturbing the values of . Figure 21 depicts the influence of the parameter on the value of norm of error for neural network model using function.
Example 22. We consider the following SOCCVI problem: where is an symmetric matrix and, , , . Clearly, and are full row rank and .
The problem has an solution . It can be verified that the Lagrangian function for this example is Note that is positive definite.
Figures 22 and 24 show the transient behaviors of Example 22 for neural network model (40) based on and with initial states , respectively. Figures 23 and 25 depict the influence of the parameter on the value of norm of error for neural network model based on and functions. We see that the influence of perturbing parameter is tiny.
We summarize some observations based on the above experiments. First, we provide the simulation diagrams of convergence comparison of and functions with different value for 6 examples. It can be seem from the Figures 2, 4, 6, 8, 10, 12, 15, 17, and 21 that the convergence speed is the fastest when . Under the same calculation time, the norm error in the case of is generally smaller than that in the cases of and . But from Figures 19, 23, and 25, we find that the convergence of case , , and is similar. In other words, we observe that, for our examples, is basically the best choice. However, the theoretical evidence is not clear yet, which will be left for future study.
5. Conclusions
In this paper, we propose a unified neural network model for solving two types of second-order cone optimization problems. We implement the neural network model with two classes of SOC complementarity functions and , which are freshly discovered. Overall, when is larger, the performance of the neural network based on and gets poorer. This suggests that the parameter is the best choice to work with the neural network no matter or is employed. On the other hand, we observe that the neural network using the family of functions performs better than the one using , in general. This is a very interesting phenomenon and new discovery, which is different from the observation in [31]. Note that in [31], for standard SOCP (second-order cone program), it was observed that the neural network based on the NR function has better performance than the one based on the FB function in most cases (except for some oscillating cases). Our numerical experiments show that for SOCQP and SOCCVI, the family of (a variant of FB function) may be considered to be employed in the neural network approach.
Another point that we want to clarify is the significance of this paper. By using two new SOC complementarity functions, this paper can be viewed as a follow-up of our previous works (see [31, 32]). From the natural feature of neural network, we pay more attention to the convergence path. This kind of convergence path varies with different SOC complementarity functions. In engineering, an important application of neural network is the design of robot arm. Therefore, compared with previous works, we achieve different convergence paths and the convergence paths are very good. In addition, we provide more detailed graphs regarding error analysis in this paper.
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 they have no conflicts of interest.
Acknowledgments
Juhe Sun’s work is supported by National Natural Science Foundation of China (Grant no. 11301348). B. Saheya’s work is supported by Natural Science Foundation of Inner Mongolia (Award no. 2017MS0125) and research fund of IMNU (Award no. 2017YJRC003). Jein-Shan Chen’s work is supported by Ministry of Science and Technology, Taiwan.