# A Projection Neural Network for Circular Cone Programming

Authors: Yaling Zhang; Hongwei Liu
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 θi∈0,π/2,i=1,2,···,N, and then the ni-dimensional circular cone denoted by Lθi can be expressed as follows:(1)Lθi=xi=xi1xi0∈Rni-1×R:xi1≤tan⁡θixi0,where · is the standard Euclidean norm.Whenθi=π/4, the circular cone reduces to the well-known second-order cone given by(2)Kni=xi=xi1xi0∈Rni-1×R:xi1≤xi0.In this paper, we consider a linear circular cone programming (LCCP), which is described as follows:(3)mincTxs.t.Ax=b,x∈Lθ,where A∈Rm×n,  c∈Rn,b∈Rm and x=x1,⋯,xN∈Rn1×⋯×RnN, and ∑i=0Nni=n. Here, x=x1,⋯,xN∈Rn1×⋯×RnN is viewed as a column vector in Rn1+⋯+nN. In addition,(4)Lθ=Lθ1×Lθ2×···×LθN,and xi∈Lθi. When θi=π/4,i=1,2,⋯,N, we have(5)K=Kn1×Kn2×···×KnN.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 as(6)minbTys.t.ATy+z=c,z∈Lθ∗,where y∈Rn is the Lagrange multipliers, and Lθ∗ is the dual cone of cone Lθ, which is defined as(7)Lθ∗=Lθ1∗×Lθ2∗×···×LθN∗,where(8)Lθi∗≔zi∣ziTxi≥0,xi∈Lθi.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(9)Ax=b,ATy+z=c,xTz=0,x∈Lθ,z∈Lθ∗.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 Lθ 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 mappingf:Rn→Rn, the following first-order differential equation is(10)dudt=fut,ut0=u0∈Rn.Next, we give the definition of Lyapunov stability [19].Definition 1 ([19]).
(1)
For (10), a point u∗∈Rn is called an equilibrium point of (10) if fu∗=0. (2)
If there is a neighborhoodN⊆Rn of u∗ such thatfu∗=0 and fu≠0 foru∈N, then u∗ is called an isolated equilibrium point.Definition 2 ([19]).
Letu(t) be a solution of (10). For an isolated equilibrium point u∗∈Rn, if, for any u(t0)=u0 and ϵ>0, there exists a δ>0 such that(11)u0-u∗<δ⟹ut-u∗<ϵfort≥t0,then u∗ is Lyapunov stable.Definition 3 (Lyapunov function [19]).
LetN⊆Rn be an open neighborhood of u~. A continuously differential function g:Rn→R is said to be a Lyapunov function (or energy function) at the state u~ for (10) if(12)gu~=0,gu~>0,∀u∈N∖u~;dgutdt≤0,∀u∈N.The relationship between stabilities and a Lyapunov function is given as follows, which is proposed in paper [20].Lemma 4 ([20]).
An isolated equilibrium pointu∗ is Lyapunov stable if there exists a Lyapunov function over some neighborhood N of u∗.


### 2.2. The Properties and Projection on the Circular Cone

Let(13)Ti=Ini-100tan⁡θi,for i=1,2,⋯,N, where Ini-1 is the ni-1 dimension unit matrix. The following lemma describes the relationship between the second-order cone Kni and the circular coneLθi, where i=1,2,⋯,N.Lemma 5 ([2]).
LetLθi and Kni be defined as in (1) and (2) fori=1,2,⋯,N. Then we obtain(1)
Lθi=Ti-1Kni and Kni=TiLθi,(2)
TiKni=Lπ/2-θi and Lπ/2-θi=Ti2Lθi,(3)
Lθi∗=Lπ/2-θi and Lθi∗∗=Lθi.From Lemma5 and the definition of Lθi, Lθi∗,i=1,2,⋯,N, we have (14)xi∈Lθi⟺Tixi∈Kni,forxi=xi1xi0∈Rni-1×Rand(15)zi∈Lθi∗⟺Ti-1zi∈Kni,forzi=zi1zi0∈Rni-1×R.Let(16)T=T10⋱0TN.Then, from the conclusion above, we have(17)x∈Lθ⟺Tx∈K,forx=x1,⋯,xN∈Rn1×⋯×RnNand(18)x∈Lθ∗⟺T-1x∈K,forz=z1,⋯,zN∈Rn1×⋯×RnN.Letxi=xi1xi0∈Rni-1×R,i=1,2,⋯,N. Then the spectral decomposition of xi on circular cone Lθi can be given as [2](19)xi=λ1xic1xi+λ2xic2xi,where(20)λ1xi=xi0-xi1cot⁡θi,λ2xi=xi0+xi1tan⁡θiand(21)c1xi=11+cot2⁡θicot⁡θiIni-1001-w1,c2xi=11+tan2⁡θitan⁡θiIni-1001w1with w=-xi1/xi12 if xi1≠0 and any vector in Rni-1 satisfying w=1 if xi1=0.Next we introduce the projection on the circular cone [2].Lemma 6 ([2]).
For anyxi=xi1xi0∈Rni-1×R, i=1,2,⋯,N, let PLθixi be the projection of xi on the circular cone Lθi. Then we obtain(22)PLθixi=λ1xi+c1xi+λ2xi+c2xi,where s+:=max⁡0,s for s∈R.Letx=x1,⋯,xN∈Rn1×⋯RnN. Then the projection PLθx of x on cone Lθ is described as(23)PLθx=PLθ1x1,⋯,PLθNxN∈Rn1×···×RnN.In particular, whenθi=π/4, the projection on second-order cone Kni can be obtained, which have been proposed in papers [21–23]:(24)PKnixi=xi0-xi1+12-w1+xi0+xi1+12w1,where w=-x1/xi1 when xi1≠0 and any vector in Rni-1 satisfying w=1 when xi1=0. Then the projection PKx of x on the cone K is described as(25)PKx=PKn1x1,⋯,PKnNxN∈Rn1×···×RnN.It is well known that anyz∈Rn can be written as(26)z=PΘz+P-Θ∗zwhere Θ denotes any closed convex cone and Θ∗ represents the dual cone of Θ. Hence, when Θ=Lθ and Θ=K, respectively, we have(27)z=PLθz+P-Lθ∗z=PLθz-P-Lθ∗-zand(28)z=PKz+P-Kz=PKz-PK∗-z.Since the second-order cone is self-dual, then K=K∗. We have(29)z=PKz+P-Kz=PKz-P-K-z.


## 2.1. Lyapunov Stability of the First-Order Differential Equation

Given a mappingf:Rn→Rn, the following first-order differential equation is(10)dudt=fut,ut0=u0∈Rn.Next, we give the definition of Lyapunov stability [19].Definition 1 ([19]).
(1)
For (10), a point u∗∈Rn is called an equilibrium point of (10) if fu∗=0. (2)
If there is a neighborhoodN⊆Rn of u∗ such thatfu∗=0 and fu≠0 foru∈N, then u∗ is called an isolated equilibrium point.Definition 2 ([19]).
Letu(t) be a solution of (10). For an isolated equilibrium point u∗∈Rn, if, for any u(t0)=u0 and ϵ>0, there exists a δ>0 such that(11)u0-u∗<δ⟹ut-u∗<ϵfort≥t0,then u∗ is Lyapunov stable.Definition 3 (Lyapunov function [19]).
LetN⊆Rn be an open neighborhood of u~. A continuously differential function g:Rn→R is said to be a Lyapunov function (or energy function) at the state u~ for (10) if(12)gu~=0,gu~>0,∀u∈N∖u~;dgutdt≤0,∀u∈N.The relationship between stabilities and a Lyapunov function is given as follows, which is proposed in paper [20].Lemma 4 ([20]).
An isolated equilibrium pointu∗ is Lyapunov stable if there exists a Lyapunov function over some neighborhood N of u∗.


## 2.2. The Properties and Projection on the Circular Cone

Let(13)Ti=Ini-100tan⁡θi,for i=1,2,⋯,N, where Ini-1 is the ni-1 dimension unit matrix. The following lemma describes the relationship between the second-order cone Kni and the circular coneLθi, where i=1,2,⋯,N.Lemma 5 ([2]).
LetLθi and Kni be defined as in (1) and (2) fori=1,2,⋯,N. Then we obtain(1)
Lθi=Ti-1Kni and Kni=TiLθi,(2)
TiKni=Lπ/2-θi and Lπ/2-θi=Ti2Lθi,(3)
Lθi∗=Lπ/2-θi and Lθi∗∗=Lθi.From Lemma5 and the definition of Lθi, Lθi∗,i=1,2,⋯,N, we have (14)xi∈Lθi⟺Tixi∈Kni,forxi=xi1xi0∈Rni-1×Rand(15)zi∈Lθi∗⟺Ti-1zi∈Kni,forzi=zi1zi0∈Rni-1×R.Let(16)T=T10⋱0TN.Then, from the conclusion above, we have(17)x∈Lθ⟺Tx∈K,forx=x1,⋯,xN∈Rn1×⋯×RnNand(18)x∈Lθ∗⟺T-1x∈K,forz=z1,⋯,zN∈Rn1×⋯×RnN.Letxi=xi1xi0∈Rni-1×R,i=1,2,⋯,N. Then the spectral decomposition of xi on circular cone Lθi can be given as [2](19)xi=λ1xic1xi+λ2xic2xi,where(20)λ1xi=xi0-xi1cot⁡θi,λ2xi=xi0+xi1tan⁡θiand(21)c1xi=11+cot2⁡θicot⁡θiIni-1001-w1,c2xi=11+tan2⁡θitan⁡θiIni-1001w1with w=-xi1/xi12 if xi1≠0 and any vector in Rni-1 satisfying w=1 if xi1=0.Next we introduce the projection on the circular cone [2].Lemma 6 ([2]).
For anyxi=xi1xi0∈Rni-1×R, i=1,2,⋯,N, let PLθixi be the projection of xi on the circular cone Lθi. Then we obtain(22)PLθixi=λ1xi+c1xi+λ2xi+c2xi,where s+:=max⁡0,s for s∈R.Letx=x1,⋯,xN∈Rn1×⋯RnN. Then the projection PLθx of x on cone Lθ is described as(23)PLθx=PLθ1x1,⋯,PLθNxN∈Rn1×···×RnN.In particular, whenθi=π/4, the projection on second-order cone Kni can be obtained, which have been proposed in papers [21–23]:(24)PKnixi=xi0-xi1+12-w1+xi0+xi1+12w1,where w=-x1/xi1 when xi1≠0 and any vector in Rni-1 satisfying w=1 when xi1=0. Then the projection PKx of x on the cone K is described as(25)PKx=PKn1x1,⋯,PKnNxN∈Rn1×···×RnN.It is well known that anyz∈Rn can be written as(26)z=PΘz+P-Θ∗zwhere Θ denotes any closed convex cone and Θ∗ represents the dual cone of Θ. Hence, when Θ=Lθ and Θ=K, respectively, we have(27)z=PLθz+P-Lθ∗z=PLθz-P-Lθ∗-zand(28)z=PKz+P-Kz=PKz-PK∗-z.Since the second-order cone is self-dual, then K=K∗. We have(29)z=PKz+P-Kz=PKz-P-K-z.


## 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.
Assumex¯∈Lθ,z¯∈Lθ∗. Then x¯Tz¯=0 if and only if ex¯,z¯=0, where ex¯,z¯=Tx¯-PKTx¯-T-1z¯.Proof.
“⇒” The projection on the closed convex setK has an important property [24],(30)x-PKx,Tx¯-PKx≤0,∀x∈Rnwhere · denotes the inner product of two vectors.
Letx=Tx¯-T-1z¯ in the inequality above. Then we have(31)0≤ex¯,z¯2≤ex¯,z¯,Tz.Moreover, because(32)ex¯,z¯,T-1z¯=Tx¯-PKTx¯-T-1z¯,T-1z¯=x¯,z¯-PKTx¯-T-1z¯,T-1z¯and(33)x¯,z¯=0,PKTx¯-T-1z¯,z¯≥0,we have(34)ex¯,z¯,Tz¯≤0.By (31) and (34), ex¯,z¯=0 is obtained.
“⇐” It follows from ex¯,z¯=0 that(35)Tx¯=PKTx¯-T-1z¯∈K.From (17), we have x¯∈Lθ.
Based on (29), we have(36)Tx¯-T-1z¯=PKTx¯-T-1z¯+P-KTx¯-T-1z¯.Then, we get(37)T-1z¯=-P-KTx¯-T-1z¯=PK-Tx¯-T-1z¯∈K.From (18), it is easy to obtain z¯∈Lθ∗. For any Tx∈K, we have(38)Tx¯-T-1z¯-PKTx¯-T-1z¯,Tx-PKTx¯-T-1z¯≤0.Since Tx¯=PKTx¯-T-1z¯, we get(39)Tx-Tx¯,T-1z¯≥0.Substituting Tx=0∈K and Tx=2Tx¯∈K in inequality (39), we have(40)x¯,z¯=x¯Tz¯=0.The proof is completed.Let(41)eu≔Wu-PΩWu-W-1Mu+q,where PΩ· denotes the projection on the set Ω=K×Rm and(42)u=xy,W=T00Im,M=0-ATA0,q=c-b.Obviously, M is a block asymmetric matrix; i.e., MT=-M. So, we have(43)eu=Tx-PKTx-T-1c-ATyAx-b.From Lemma 7, we know that the KKT condition (9) is equivalent to(44)eu=0.LetΩθ=Lθ×Rm. Then it is easy to prove that(45)u∈Ωθ⟺Wu∈Ω.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:(46)Hu=12u-u∗2,where u∗∈Ωθ∗.Here, a dynamical system is proposed to solve (44). The dynamical system is given as follows:(47)dudt=Fu=-rρuW-W-1Meu,where r∈0,1 and(48)ρu=eu2W-W-1Meu2.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.
u ∗ ∈ Ω θ ∗ if and only if u∗ is an equilibrium point of network (47), where Ωθ∗ is the solution set of problem (9).Proof.
“⇒” Sinceu∗∈Ωθ∗, we have eu∗=0. It is easy to know that(49)dudtu=u∗=0,so u∗ is an equilibrium point of network (47).
“⇐” Since u∗ is an equilibrium point of network (47), we obtain Fu∗=-rρuW-W-1Meu∗=0.
Consider(50)Fu∗22=r2ρu∗2eu∗TW-W-1MTW-W-1Meu∗=0.
SinceW-W-1MTW-W-1M is positive definite matrix, eu∗=0 is obtained. From (45), we know u∗∈Ωθ∗.The following result guarantees the existence and uniqueness for the solutionut of neural network (47).Lemma 9.
For anyu0∈Ω, there exists a unique continuous solution ut of (47) with u0=u0 for all t≥0.Proof.
Because(51)eu=Tx-PKTx-T-1c-ATyAx-b,from the results in paper [11], we know that eu is semismooth. From all of the above, we conclude that Fu is semismooth. Thus, there exists a unique solution ut for neural network (47).The results of Lemmas8 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 u0, is Lyapunov stable.Proof.
From Lemma9, there exists a unique continuous solution ut of (47) with u0=u0 for all t≥0. Since u∗∈Ωθ∗, we have(52)u∗,Mu∗+q=Wu∗,W-1Mu∗+q=0.In addition, for anyv∈Ωθ, we have(53)v,Mu∗+q=Wv,W-1Mu∗+q≥0.From (52) and (53), we have(54)Wv-u∗,W-1Mu∗+q≥0.
SettingWv=PΩWu-W-1Mu+q in (54), it follows that(55)W-1Mu∗+q,PΩWu-W-1Mu+q-Wu∗≥0for any u∈Rn+m. For any v∈Rn+m and w∈Ωθ, we have (56)v-PΩv,PΩv-Ww≥0.
Letv=Wu-W-1Mu+q,w=u∗ in (30). We obtain(57)eu-W-1Mu+q,PΩWu-W-1Mu+q-Wu∗≥0.Based on (55) and (57), we get(58)eu-W-1Mu-u∗,Wu-u∗-eu≥0.It follows that(59)u-u∗,W-W-1Meu≥eu2+u-u∗,Mu-u∗.
BecauseMT=-M, we have u-u∗,Mu-u∗=0. Thus we get(60)u-u∗,W-W-1Meu≥eu2.By the equation above, we obtain(61)ddtHut=u-u∗Tdudt=-rρuu-u∗TW-W-1Meu≤-rρueu2≤0,which 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.
Letu∗ is the equilibrium point of neural network (47). The solution trajectory of neural network (47) with initial point u0 is globally convergent to u∗ and has finite convergence time.Proof.
From (61), we know that the level set(62)Lu0≔u∣Hu≤Hu0is bounded. Then, based on the Invariant Set Theorem [25], we have that the solution trajectory ut converges to u¯ as t→+∞, where u¯ is the largest invariant set in(63)Λ=u∈Lu0∣dHutdt=0.Now, we prove the result thatdu/dtu=u~=0 if and only if dHut/dtu=u~=0, and then we can obtain the globe convergence of neural network (47).Ifdu/dtu=u~=0, then dHut/dtu=u~=u~-u∗Tdu/dtu=u~=0.IfdHut/dtu=u~=0, from (61), we have(64)0=dHutdtu=u~=u~-u∗Tdudtu=u~≤-rρu~eu2≤0.So eu~=0. Then(65)dudtu=u~=-rρu~W-W-1Meu~=0.From the conclusions above, ut converges globally to the equilibrium point u∗. 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,r=0.8,RelTol=10-6, and AbsTol=10-9 for the test examples.Example 12.
Consider the following problem with two 3-dimensional circular cones:(66)minx1+x2+x3+x4+x5+x6s.t.x1+2x2+x6=9x1+x4+4x5=20x2+x3+x5=6x1+x2=4x3+2x5=8x2,x3T≤tan⁡θ1x1x5,x6T≤tan⁡θ2x4,where(67)A=120001100140001110110000001020,b=920648,c=111111,θ1=π3,θ2=π3.The optimal solution of Example 12 is(68)x∗=2.288,1.712,0.575,2.863,3.712,3.288Tand(69)y∗=1.628,-0.640,-2.267,0.012,3.267T.Figures1 and 2 depict the trajectories of neural network (47) with the initial values x0=0,0,0,0,0,0T and y0=0,0,0,0,0T converging to its solutions x∗ and y∗, respectively.Figure 1
Thex trajectories followed from Example 12.Figure 2
They trajectories followed from Example 12.Example 13.
Consider the following problem with three 3-dimensional circular cones:(70)minx2+2x3+3x4+x5+2x6+2x7+x8+x9s.t.x1+x2+2x3+2x5+x7=154x1+6x2+3x3+x5+2x7+x8=34x3+x4+x5+x6+3x7+x8+x9=242x2+x3+x6+x7+x8+x9=10x2+x5+2x6+2x7+x8+2x9=16x2,x3T≤tan⁡θ1x1x5,x6T≤tan⁡θ2x4x8,x9T≤tan⁡θ3x7,where(71)A=112020100463010210001111311021001111010012212,b=1534241016,c=012312211,θ1=0.4π,θ2=0.4π,θ3=0.4π.The optimal solution of Example 13 is(72)x∗=0.987,3.025,-0.297,5.721,2.945,-17.359,5.692,-1.536,17.450Tand(73)y∗=0.714,-0.314,0.228,1.268,-0.192T.Figures3 and 4 depict the trajectories of neural network (47) with the initial values x0=0,0,0,0,0,0,0,0,0T and y0=0,0,0,0,0T converging to its solutions x∗ and y∗, respectively.Figure 3
Thex trajectories followed from Example 13.Figure 4
They trajectories followed from Example 13.Example 14.
Consider the problem with two 4-dimensional circular cones as follows:(74)min6x1+x3+x4+x5+x6+x7+x8s.t.3x2+x4+2x8=162x2+x3+x4+x5=12x1+x5+x6+x7+x8=14x1+x4+x5+x7=5x2+x5+2x7+x8=6x2,x3,x4T≤tan⁡θ1x1x6,x7,x8T≤tan⁡θ2x5,where(75)A=0301000202111000100011111001101001001021,b=16121456,c=60111111,θ1=π4,θ2=π3.The optimal solution of Example 14 is(76)x∗=0,0,0,0,12,1,-7,8Tand(77)y∗=-0.505,1.010,1.000,-2.019,1.010T.Figures5 and 6 depict the trajectories of neural network (47) with the initial values x0=0,0,0,0,0,0,0,0,0T and y0=0,0,0,0,0T converging to its solutions x∗ and y∗, respectively.Figure 5
Thex trajectories followed from Example 14.Figure 6
They trajectories followed from Example 14.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 Example12, let θ1=π/4,θ2=π/4. 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 x∗=3,1,2,5,3,4T.Figures7 and 8 depict the trajectories obtained using neural network (47) and the neural network with CP function in [14], respectively.Figure 7
Transient behavior of the neural network (47) in Example 15.Figure 8
Transient behavior of the neural network with CP function in Example15.The simulation results show that both trajectories are convergent tox∗, 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 Example14, let θ1=π/4,θ2=π/4. Then a second-order cone problem is obtained, which has an optimal solution x∗=0,0,0,0,12,1,-7,8T.Figures9 and 10 depict the trajectories obtained using neural network (47) and the neural network with CP function in [14], respectively.Figure 9
Transient behavior of the neural network (47) in Example 16.Figure 10
Transient behavior of the neural network with CP function in Example16.The simulation results show that both trajectories are convergent tox∗. 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 m fingers, the optimization problem can be formulated as(78)min12fTfs.t.Gf=-fextfi1,fi2T≤μfi3i=1,2,⋯,m,where f=f11,f12,⋯,fm3 is the grasping force, G is the grasping transformation matrix, fext is the time-varying external wrench, and μ 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 0,1,0T,  1,0.5,0T, and 0,-1,0T, and the robot hand moves along a vertical circular trajectory of radius r with a constant velocity v. Let x=f13,f11,f12,f23,f21,f22,f33,f31,f32T. Then problem (78) is reformulated as(79)min12xTQxs.t.Ax=bx2,x3≤tan⁡θ1x1x5,x6≤tan⁡θ2x4x8,x9∥≤tan⁡θ3x7,where Q=diag1,1,1,1,1,1,1,1,1 and(80)A=001-100010-10000-11000-100-1000-10-100-0.5000100001000000-10.50-1010,b=0-fcsin⁡θtM1g-fccos⁡θt000,c=000000000T,θ1=θ2=θ3=actanμ-1.Here, M1 is the mass of the polyhedral, g=9.8m/s2, fc=Mv2/r the centripetal force, t the time, and θt=vt/r∈0,2π. In this example, we set the data as follows: M1=0.1kg,r=0.2m,n=0.4πm/s, and μ=0.6. In the time-varying grasping force, we only test numerical examples when t=0s and t=0.5s. The other results are easily given when t∈0,0.5.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 eu and the matrix M in neural network (47) as follows:(81)M=Q-ATA0,eu=Tx-PKTx-T-1Qx+c-ATyAx-b.The results of the projection neural networks (47) to solve Example 17 are shown in Figures 11 and 12 when t=0s and t=0.5s, respectively. The simulation results demonstrate that the neural networks are convergent for the grasping force optimization problem.Figure 11
Transient behavior of the neural network (47) when t=0.5s.Figure 12
Transient behavior of the neural network (47) when t=0s.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.



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*Source: 4607853-2018-06-10.xml*