投影神经网络求解约束变分不等式问题

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"基于投影的约束变分不等式神经网络的研究" 本文介绍了一种新颖的神经网络模型，该模型专门用于解决受约束的变分不等式问题。该模型通过将解的必要和充分条件转化为非线性投影方程系统，从而对原问题进行转化。在"IEEETRANSACTIONSONNEURALNETWORKS"2009年3月刊的第20卷第3期中，Xing-Bao Gao和Li-Zhi Liao详细阐述了这一方法。首先，文章提出了五条充分条件，确保所提出的神经网络在李雅普诺夫意义下稳定，并能收敛到原始问题的精确解。通过定义适当的凸能量函数，网络能够保证其稳定性和收敛性。这种新的神经网络模型不仅包含了一个已有的模型，而且可以应用于解决一些非单调和非光滑问题，扩展了神经网络在求解复杂优化问题上的应用范围。在"Introduction"部分，作者考虑了具有线性与非线性约束的变分不等式问题（VI），其形式如下： (1) 其中，\(F\)是连续映射，从集合\(C\)到自身，\(C\)是\(R^n\)中的一个非空、闭、凸集，而\(x\)是需要找到的向量，满足特定条件。文章的主体部分可能涵盖了以下几个关键知识点： 1. 变分不等式（Variational Inequalities）理论：这是优化问题的一个分支，处理的是多点交互的优化问题，其中考虑了约束条件下的方向导数。 2. 投影运算：在解决变分不等式时，投影运算用于将点从空间中的一点映射到约束集上最近的点，是求解这类问题的关键工具。 3. 神经网络模型：神经网络作为一种计算模型，能够模拟人脑神经元的连接和信息处理方式，被用来逼近和解决复杂的非线性问题。 4. 稳定性分析：在李雅普诺夫稳定性理论框架下，作者分析了网络的动态行为，证明了网络在运行过程中不会发散，而是会收敛到问题的解。 5. 收敛性证明：作者通过定义的能量函数，展示了神经网络如何逐步接近并最终达到问题的精确解。 6. 实例验证：通过数值例子，作者展示了所提神经网络的有效性和动态行为，进一步证明了模型的可行性。这些内容对于理解和应用神经网络解决实际约束优化问题具有重要的理论价值和实践意义。对于从事机器学习、优化算法或者神经网络研究的读者，这篇文章提供了新的思路和工具。

376 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 3, MARCH 2009

where and

for , and

and for

It should be noted that Xia [27] proposed an extended pro-

jection neural network by using (5), yet it might not be stable

even when

is co-coercive on (see Example 2), or is

symmetric and positive semideﬁnite on

(see [9, Example 4]).

By replacing

and in the ﬁrst equation of (5) with

and , respectively, we know that if

and only if there exist

such that

(6)

Using (6), Gao et al. [11] proposed a neural network for the

special case of (1) with

, but it cannot be applied to

the problem in which

, or some is not deﬁned on (see

Example 3 in Section IV).

Let

, , and

(7)

then (6) becomes

. Thus,

, i.e., is a solution of

(8)

So we get the following lemma, which reveals the relationship

between the solutions of (4) and (8).

Lemma 2: The following are true.

i) If

is a solution of (4), then

deﬁned in (7) is a solution of (8).

ii) If

is a solution of (8),

then

is a solution of (4).

iii) Let

solves (8) and

solves (4) , then there exists a bijective map-

ping between

and .

Proof: See the Appendix.

Lemmas 1–2 show that each solution of (1) can be obtained

by solving (8). For simplicity, we denote

, and solves (8) in the

following. Then,

, and to solve

problem (1), system (8) suggests the neural network shown in

(9)–(10) at the bottom of the page, where

and is a scaling constant.

Obviously, from (7)–(10),

and (9) can be

written as

(11)

Thus,

is just the equilibrium point set of (9), and the model

in [11] is a special case of (9)–(10) with

The simpliﬁed architecture of neural network (9)–(10) is

shown in Fig. 1, where

, , and the architecture of

is the same as [9, Fig. 1(a)]. According to Fig. 1,

the circuit realizing the proposed neural network (9)–(10) con-

sists of

integrators, activation functions for

and , processors for , , and

, analog multipliers for , weighted connections,

and some summers.

To show the advantages of the proposed neural network

(9)–(10), we compare it with four existing neural networks.

First, let us look at the model in [27], where the state equation

can be written in details as

(12)

From (11) and (12), we see that the complexity of the two

models is almost the same. To clarify this issue further, we

consider a special form of problem (1) with

and

, where , , , and

. Then, (11) and (12) for this problem can be simpliﬁed

(13)

and

(14)

state equation: (9)

output equation:

(10)

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