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.
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 [
The neural network approach has an advantage in solving real-time optimization problems, which was proposed by Hopfield and Tank [
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 [
The SOCQP is in the form of
We say a few words about why we assume that
The SOCCVI, our another target problem, is to find
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.
Consider the first-order differential equations (ODE):
Suppose that
Let
Let
(a) An isolated equilibrium point
(b) An isolated equilibrium point
For more details, please refer to any usual ODE textbooks, e.g., [
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
A vector-valued function
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.
As mentioned in Section
Let
Since
Besides, the following results address the existence and uniqueness of the solution trajectory of the neural network (
(a) For any initial point
(b) If the level set
This proof is exactly the same as the one in [
Now, we are ready to analyze the stability of an isolated equilibrium
Let If If
The desired results can be proved by using Lemma
In order to demonstrate the effectiveness of the proposed neural network, we test three examples for our neural network (
Consider the following SOCQP problem:
After suitable transformation, it can be recast as an SOCQP with
Transient behavior of the neural network with
Convergence comparison of
Transient behavior of the neural network with
Convergence comparison of
For case of
For case of
Figures
Consider the following SOCQP problem:
For this SOCP, we have
Transient behavior of the neural network with
Convergence comparison of
Transient behavior of the neural network with
Convergence comparison of
Figures
Consider the following SOCQP problem:
Here, we have
Figures
Transient behavior of the neural network with
Convergence comparison of
Transient behavior of the neural network with
Convergence comparison of
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.
Let
Let
The proof is straightforward.
Let The function where If
Now, we are ready to analyze the behavior of the solution trajectory of neural network (
(a) If
(b) If
(a) For any initial state
(b) If the level set
A natural question arises here. When are the level sets
Next, we investigate the convergence of the solution trajectory and stability of neural network (
(a) Let
(b) If
With Proposition
Let If
Again, the arguments are similar to those in [
To study the conditions for nonsingularity based on
When SOCCVI problem corresponds to the KKT conditions of a convex second-order cone program (CSOCP) problem as (
Let
Suppose
We know that
Suppose Assumption
The proof can be done by following the similar arguments as in Theorem
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 [
Block diagram of the proposed neural network with
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
Consider the SOCCVI problem where
This problem has an approximate solution
Transient behavior of neural network with
Convergence comparison of
Transient behavior of the neural network with
Convergence comparison of
Consider the problem where
This problem has an approximate solution
Transient behavior of neural network with
Convergence comparison of
Transient behavior of the neural network with
Convergence comparison of
Figures
We consider the following SOCCVI problem:
The problem has an solution
Figures
Transient behavior of neural network with
Convergence comparison of
Transient behavior of the neural network with
Convergence comparison of
We summarize some observations based on the above experiments. First, we provide the simulation diagrams of convergence comparison of
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
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 [
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
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.