A subgradient-based neural network to constrained distributed convex optimization

Abstract

As artificial intelligence and large data develop, distributed optimization shows the great potential in the research of machine learning, particularly deep learning. As an important distributed optimization problem, the nonsmooth distributed optimization problem over an undirected multi-agent system with inequality and equality constraints frequently appears in deep learning. To deal with this optimization problem cooperatively, a novel neural network with lower dimension of solution space is presented. It is demonstrated that the state solution of proposed approach can enter the feasible region. Also, it can also prove that the state solution achieves consensus and finally converges to the optimal solution set. Moreover, the proposed approach here does not depend on the boundedness of the feasible region, which is a necessary assumption in some simplified neural network. Finally, some simulation results and a practical application are given to reveal the efficacy and practicability.

Appendix

Proof

Combining $\lim _{k\rightarrow +\infty }{\mathbf {x}}(t_{k})=\bar{{\mathbf {x}}}$ with Theorem 3, we have

$$\begin{aligned} \lim _{k\rightarrow +\infty }{\mathbf {x}}^{\mathrm {T}}(t_{k}){\mathbf {L}}{\mathbf {x}}(t_{k})=0, \end{aligned}$$

which means that $\lim _{k\rightarrow +\infty }{\mathbf {L}}{\mathbf {x}}(t_{k})={\mathbf {L}}\bar{{\mathbf {x}}}=0$. Therefore, combined with Theorems 2, $\bar{{\mathbf {x}}}\in \varOmega$ is a feasible solution of problem (2).

By $\lim _{k\rightarrow +\infty }H\left( t_{k}, {\mathbf {x}}(t_{k})\right) =0,$ there exist $\eta (t_{k})\in \partial G({\mathbf {x}}(t_{k}))$, $\gamma (t_{k})\in \partial {\mathbf {f}}({\mathbf {x}}(t_{k}))$ and $\xi (t_{k}) \in \partial D({\mathbf {x}}(t_{k}))$ satisfying

$$\begin{aligned}&\lim _{k\rightarrow +\infty }\big (\gamma (t_{k})+(t_{k}+1)^{2}\eta (t_{k}) +(t_{k}+1){\mathbf {L}}{\mathbf {x}}(t_{k})+(t_{k}+1)^{3}\xi (t_{k})\big )=0. \end{aligned}$$

(39)

Moreover, combined with the u.s.c. of $\partial {\mathbf {f}}$, one has

$$\begin{aligned}&\lim _{k\rightarrow +\infty } \gamma (t_{k})={\bar{\gamma }} \in \partial {\mathbf {f}}(\bar{{\mathbf {x}}}). \end{aligned}$$

(40)

Meanwhile, based on the convexity of $G(\cdot )$ and $D(\cdot )$ on $\varOmega$, and the properties of positive semidefinite of ${\mathbf {L}}$, for any ${\mathbf {y}}\in \varOmega$, we have

$$\begin{aligned} \left\{ \begin{array}{ll} ({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\eta (t_{k})\le G({\mathbf {y}})-G({\mathbf {x}}(t_{k}))= 0,\\ ({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\xi (t_{k})\le D({\mathbf {y}})-D({\mathbf {x}}(t_{k}))= 0,\\ ({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}{\mathbf {L}}{\mathbf {x}}(t_{k})=-{\mathbf {x}}^{\mathrm {T}}(t_{k}){\mathbf {L}}{\mathbf {x}}(t_{k})\le 0. \end{array} \right. \end{aligned}$$

(41)

Hence, from (39)-(41), for any ${\mathbf {y}}\in \varOmega =S_{1}\cap S_{2}\cap S_{3}$, one has

$$\begin{aligned} 0=&\lim _{k\rightarrow +\infty }({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\Big \{\gamma (t_{k})+(t_{k}+1)^{2}\eta (t_{k})+(t_{k}+1){\mathbf {L}}{\mathbf {x}}(t_{k})\\&+(t_{k}+1)^{3}\xi (t_{k})\Big \}\\ \le&\limsup _{k\rightarrow +\infty }({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\Big \{(t_{k}+1){\mathbf {L}}{\mathbf {x}}(t_{k})+\gamma (t_{k})\Big \}\\ \le&\limsup _{k\rightarrow +\infty }({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\gamma (t_{k})\\ =&({\mathbf {y}}-\bar{{\mathbf {x}}})^{\mathrm {T}}{\bar{\gamma }}. \end{aligned}$$

By the convexity of ${\mathbf {f}}$ on $\varOmega$, one has

$$\begin{aligned} {\mathbf {f}}({\mathbf {y}}) \ge {\mathbf {f}}(\bar{{\mathbf {x}}}), \forall {\mathbf {y}}\in \varOmega , \end{aligned}$$

(42)

which means that $\bar{{\mathbf {x}}}$ is an optimal solution of problem (2). $\square$