Decentralized Optimization Over Tree Graphs

Abstract

This paper presents a decentralized algorithm for non-convex optimization over tree-structured networks. We assume that each node of this network can solve small-scale optimization problems and communicate approximate value functions with its neighbors based on a novel multi-sweep communication protocol. In contrast to existing parallelizable optimization algorithms for non-convex optimization, the nodes of the network are neither synchronized nor assign any central entity. None of the nodes needs to know the whole topology of the network, but all nodes know that the network is tree-structured. We discuss conditions under which locally quadratic convergence rates can be achieved. The method is illustrated by running the decentralized asynchronous multi-sweep protocol on a radial AC power network case study.

Appendix A Proof of Theorem 1.1

Let us introduce the shorthands

$$\begin{aligned} z^{k+1} = \left( \begin{array}{c} z_1^{k+1} \\ z_2^{k+1} \end{array} \right) = \left( \begin{array}{c} x^{k+1} \\ \kappa ^{k+1} \end{array} \right) \qquad \text {and} \qquad z^\star = \left( \begin{array}{c} x^\star \\ \kappa ^\star \end{array} \right) \end{aligned}$$

to denote, respectively, the primal dual minimizer of (6) at the kth iteration of the algorithm and the primal-dual minimizer of (5). Due to the regularity of $x^\star$, the LICQ condition must be satisfied in a neighborhood of $x^\star$, which implies that the first order necessary KKT conditions

$$\begin{aligned} R(x^k,z^{k+1}) = 0 \qquad \text {and} \qquad R(x^\star ,z^\star ) = \widetilde{R}(z^\star ) = 0 \end{aligned}$$

(24)

with shorthands

$$\begin{aligned} R( \xi , \zeta )= & {} \nabla _z \left[ \varPhi (\xi ,\zeta _1) + \zeta _2^\top C(\zeta _1) \right] \qquad \text {and} \\ \widetilde{R}(\zeta )= & {} R(\zeta _1,\zeta ) = \nabla _z \left[ F(\zeta _1) + \zeta _2^\top C(\zeta _1) \right] \end{aligned}$$

are satisfied recalling that $\varPhi$ is a locally accurate approximation of F. Now, because the derivative of R with respect to its second argument, $\nabla _z R(x,\cdot )$, is uniformly Lipschitz continuous function in a neighborhood of $z^\star$, the first equation in (24) yields

$$\begin{aligned} 0= & {} R(x^k,z^{k+1}) = R ( x^k, z^k ) + \int _{0}^1 \nabla _z R(x^k, z^k + s (z^{k+1}-z^k) ) (z^{k+1}-z^k) \, \mathrm {d}s \nonumber \\\end{aligned}$$

(25)

$$\begin{aligned}= & {} \widetilde{R} ( z^k ) + M(z_k) (z^{k+1}-z^k) + \mathbf {O}\left( \Vert z^{k+1}-z^k \Vert ^2 \right) \; , \end{aligned}$$

(26)

where we have set $M(z^k) = \nabla _z R(x^k, z^k) = \nabla _z \widetilde{R}(z^k)$ and used that $\widetilde{R}(z^k) = R(x^k,z^k)$. Notice that the KKT matrix $M(z_k)$ is invertible for all $z^k$ in an open neighborhood of $z^\star$ as we assume that the LICQ and SOSC condition are satisfied at $z^\star$. Consequently, because we have $\widetilde{R}(z^k) = \mathbf {O}( \Vert z^k - z^\star \Vert )$, the above equation implies that

$$\begin{aligned} z^{k+1} = z^k - M(z^k)^{-1} \widetilde{R}(z^k) + \mathbf {O}( \Vert z^k - z^\star \Vert ^2 ) \; . \end{aligned}$$

(27)

From here on, the proof is very similar to the standard proof of quadratic convergence of Newton’s method (see, e.g., [37, Thm. 3.5]); that is we use (27) to establish the inequality

$$\begin{aligned} \Vert z^{k+1} - z^\star \Vert= & {} \left\| z^k - z^\star - M(z^k)^{-1} \widetilde{R}(z^k) \right\| + \mathbf {O}( \Vert z^k - z^\star \Vert ^2 ) \nonumber \\= & {} \left\| z^k - z^\star - M(z^k)^{-1} \left( \widetilde{R}(z^k) - \widetilde{R}(z^\star ) \right) \right\| + \mathbf {O}( \Vert z^k - z^\star \Vert ^2 ) \nonumber \\= & {} \left\| \left( I - M(z^k)^{-1} \int _0^{1} \nabla _z \widetilde{R}(z^k+s(z^k-z^\star )) \, \mathrm {d}s \right) (z^k-z^\star ) \right\| \nonumber \\&+ \mathbf {O}( \Vert z^k - z^\star \Vert ^2 ) \nonumber \\= & {} \underbrace{\left\| I - M(z^k)^{-1} \nabla _z \widetilde{R}(z^k) \right\| }_{=0} \Vert z^k - z^\star \Vert + \mathbf {O}( \Vert z^k - z^\star \Vert ^2 ) \; . \end{aligned}$$

(28)

Because the LICQ condition holds the iterates of the multiplier sequence $\kappa ^k$ is uniquely determined by the sequence $x^k$ (since $x^{k+1}$ depends only on $x^k$, but not on $\kappa ^k$), the above equation also implies that

$$\begin{aligned} \Vert x^{k+1} - x^\star \Vert = \mathbf {O}( \Vert x^k - x^\star \Vert ^2 ) \; . \end{aligned}$$

The latter equation corresponds to the statement of the theorem establishing local quadratic convergence.