Distributed Coordination for Nonsmooth Convex Optimization via Saddle-Point Dynamics

Abstract

This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as saddle points of an augmented Lagrangian. We show that the associated saddle-point dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous saddle-point-like algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for saddle-point dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex optimization problems subject to equality constraints. Various examples illustrate our discussion.

Appendix

We gather in this appendix various intermediate results used in the derivation of the main results of the paper.

1.1 Auxiliary Results for Performance Characterization

The following two results characterize properties of the generalized Hessian of $\mathcal {C}^{1,1}$ functions. In each case, let $f\in \mathcal {C}^{1,1}({\mathbb R}^{n},\mathbb R)$ and consider the set-valued map $\partial (\nabla f):{\mathbb R}^{n}\rightrightarrows {\mathbb R}^{n \times n}$ as defined in Sect. 2.1. For $x,y\in {\mathbb R}^{n}$, we let $[x,y]=\{x+\theta (y-x)\mid \theta \in [0,1]\}$ and study the set

$$\begin{aligned} \partial (\nabla f([x,y]))=\bigcup _{z\in [x,y]}\partial (\nabla f)(z). \end{aligned}$$

Lemma A.1

(Positive definiteness) Let $f\in \mathcal {C}^{1,1}({\mathbb R}^{n},\mathbb R)$ and suppose $\partial (\nabla f)\succ 0$. Then, ${{\mathrm{co}}}\big \{\partial (\nabla f([x,y]))\big \}\succ 0$ for all $x,y\in {\mathbb R}^{n}$.

Proof

Since $\partial (\nabla f)(x)\succ 0$ for all $x\in {\mathbb R}^{n}$, it follows $\bigcup _{z\in [x,y]}\partial (\nabla f)(z)\succ 0$. Moreover, since the cone of symmetric positive definite matrices is convex itself, it contains all convex combinations of elements in $\bigcup _{z\in [x,y]}\partial (\nabla f)(z)$, i.e., in particular ${{\mathrm{co}}}\big \{\partial (\nabla f([x,y]))\big \}\succ 0$, concluding the proof.$\square$

Lemma A.2

(Compactness) Let $f\in \mathcal {C}^{1,1}({\mathbb R}^{n},\mathbb R)$. Then, the set $\partial (\nabla f([x,y]))$ and its convex closure are both compact.

Proof

Boundedness of the set $\partial (\nabla f([x,y]))$ follows from Rockafellar and Wets (1998, Proposition 5.15). To show that $\partial (\nabla f([x,y]))$ is closed, take $\nu \in {{\mathrm{cl}}}\partial (\nabla f([x,y]))$. By definition, there exists $\{\nu _{n}\}\subset \partial (\nabla f([x,y]))$ such that $\nu _{n}\rightarrow \nu$. Since $\nu _{n}$ belongs to $\partial (\nabla f([x,y]))$, let us denote $\nu _{n}\in \partial (\nabla f)(z_{n})$, i.e., $\nu _{n}$ is a vector based at $z_{n}$. Similarly, let z be the point at which the vector $\nu$ is based. Following the above arguments, we have $z_{n}\rightarrow z$. Since [x, y] is compact and $z_{n}\in [x,y]$, we deduce $z\in [x,y]$. Assume, by contradiction, that $\nu \notin \partial (\nabla f)(z)$. Then, since $\partial (\nabla f)(z)$ is closed, there exists $\varepsilon >0$ such that $\{\nu \}\cap \partial (\nabla f)(z)+\mathbb {B}(0,\varepsilon )=\emptyset$. Using upper semi-continuity, there exists $N\in \mathbb {N}$ such that if $n\ge N$, then $\partial (\nabla f)(z_{n})\subset \partial (\nabla f)(z)+\mathbb {B}(0,\varepsilon )$. This fact is in contradiction with $\nu _{n}\rightarrow \nu$. Therefore, it follows $\nu \in \partial (\nabla f)(z)\subset \partial (\nabla f([x,y]))$, and we conclude $\partial (\nabla f([x,y]))$ is closed. Thus, $\partial (\nabla f([x,y]))$ is compact, and so is ${{\mathrm{co}}}\big \{\partial (\nabla f([x,y]))\big \}$, concluding the proof.$\square$

1.2 Auxiliary Results for Convergence Analysis of Saddle-Point-Like Dynamics

Here, we investigate the explicit computation of the projection operator $P_{T_G}$, which plays a key role in the dynamics (SPLD). Recall that $T_{G}(x)$ and $N_{G}(x)$ denote the tangent and normal cone of $G\subset {\mathbb R}^{n}$ at $x\in G$, respectively. The following geometric interpretation of $P_{T_G}$ is well known in the literature of locally projected dynamical systems (Nagurney and Zhang 1996; Pappalardo and Passacantando 2002):

• (i) if $(x,\lambda )\in {{\mathrm{int}}}G\times {\mathbb R}^{p}$, then $P_{T_G(x)}(F(x,\lambda ))=F(x,\lambda )$;

• (ii) if $(x,\lambda )\in {{\mathrm{bd}}}G\times {\mathbb R}^{p}$, then

  $$\begin{aligned} P_{T_G(x)}(F(x,\lambda ))=\bigcup _{\xi \in F(x,\lambda )}\xi -\max \big \{0, \langle \xi ,n^{\star }(x,\xi )\rangle \big \}n^{\star }(x,\xi ), \end{aligned}$$

  where

  $$\begin{aligned} n^{\star }(x,\xi )\in \mathop \mathrm{argmax}_{n\in N_{G}^{\sharp }(x)}\langle \xi ,n\rangle . \end{aligned}$$

  (12)

Note that if $\{\xi \}\cap T_{G}(x)\ne \emptyset$ for some $(x,\lambda )\in {{\mathrm{bd}}}G\times {\mathbb R}^{p}$ and $\xi \in F(x,\lambda )$, then $\sup _{n\in N_{G}^{\sharp }(x)}\langle \xi ,n\rangle \le 0$, and by definition of $P_{T_G}$, no projection needs to be performed. The following result establishes the existence and uniqueness of the maximizer $n^{\star }(x,\xi )$ of (12) whenever $\{\xi \}\cap T_{G}(x)=\emptyset$.

Lemma A.3

(Computation of the projection operator) Let $(x,\lambda )\in {{\mathrm{bd}}}G\times {\mathbb R}^{p}$. If there exists $\xi \in F(x,\lambda )$ such that $\sup _{n\in N_{G}^{\sharp }(x)}\langle \xi ,n\rangle >0$, then the maximizer $n^{\star }(x,\xi )$ of (12) exists and is unique.

Proof

Let $(x,\lambda )\in {{\mathrm{bd}}}G\times {\mathbb R}^{p}$ and suppose there exists $\xi \in F(x,\lambda )$ such that $\sup _{n\in N_{G}^{\sharp }(x)}\langle \xi ,n\rangle >0$. By definition, the normal cone $N_{G}(x)$ of G at $x\in {{\mathrm{bd}}}G$ is closed and convex. Existence of $n^{\star }(x,\xi )$ follows from compactness of the set $N_{G}^{\sharp }(x)$. Now, let $\tilde{n}^{\star }(x,\xi )$ and $\hat{n}^{\star }(x,\xi )$ be two distinct maximizers of (12) such that $\langle \xi ,\tilde{n}^{\star }(x,\xi )\rangle >0$ and $\langle \xi ,\hat{n}^{\star }(x,\xi )\rangle >0$. Convexity implies $(\tilde{n}^{\star }(x,\xi ) + \hat{n}^{\star }(x,\xi ))/||\tilde{n}^{\star }(x,\xi )+\hat{n}^{\star }(x,\xi )||\in N_{G}^{\sharp }(x)$. Therefore, it follows

$$\begin{aligned} \frac{\langle \xi ,\tilde{n}^{\star }(x,\xi )+\hat{n}^{\star }(x,\xi )\rangle }{||\tilde{n}^{\star }(x,\xi )+\hat{n}^{\star }(x,\xi )||}= \frac{2\langle \xi ,\tilde{n}^{\star }(x,\xi )\rangle }{||\tilde{n}^{\star } (x,\xi )+\hat{n}^{\star }(x,\xi )||}>\langle \xi ,\tilde{n}^{\star }(x,\xi ) \rangle , \end{aligned}$$

which contradicts the fact that $\tilde{n}^{\star }(x,\xi )$ maximizes (12).$\square$

We note that the computational complexity of solving (12) depends not only on the problem dimensions $n,p,m>0$, but also on the convexity and regularity assumptions of the problem data, i.e., on f, h and g.