Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity

Abstract

We systematically study the local single-valuedness of the Bregman proximal mapping and local smoothness of the Bregman–Moreau envelope of a nonconvex function under relative prox-regularity—an extension of prox-regularity—which was originally introduced by Poliquin and Rockafellar. As Bregman distances are asymmetric in general, in accordance with Bauschke et al., it is natural to consider two variants of the Bregman proximal mapping, which, depending on the order of the arguments, are called left and right Bregman proximal mapping. We consider the left Bregman proximal mapping first. Then, via translation result, we obtain analogue (and partially sharp) results for the right Bregman proximal mapping. The class of relatively prox-regular functions significantly extends the recently considered class of relatively hypoconvex functions. In particular, relative prox-regularity allows for functions with a possibly nonconvex domain. Moreover, as a main source of examples and analogously to the classical setting, we introduce relatively amenable functions, i.e. convexly composite functions, for which the inner nonlinear mapping is component-wise smooth adaptable, a recently introduced extension of Lipschitz differentiability. By way of example, we apply our theory to locally interpret joint alternating Bregman minimization with proximal regularization as a Bregman proximal gradient algorithm, applied to a smooth adaptable function.

Appendix: Proof of First Part of Theorem 4.1

Lemma A.1

Let the assumptions in Theorem 4.1 hold. Then we have, for the iterates produced by Algorithm 1, that

• (i) A monotonic sufficient decrease over the iterates is guaranteed:

  $$\begin{aligned} F_\lambda (u^{t+1}, x^{t+1}) + D_{\sigma }(x^{t+1}, x^t) + D_{\omega }(u^{t+1}, u^t) \le F_\lambda (u^{t}, x^{t}), \end{aligned}$$

  (36)

• (ii) $\{u^t, x^t\}_{t \in \mathbb {N}}$ is bounded and $x^t \in {{\,\mathrm{int}\,}}(\mathrm{dom}\,\phi )$ for all t.

• (iii) We have that $-\,\infty < \beta \le F_\lambda (u^t, x^t)$ is uniformly bounded from below for all t and $\{F_\lambda (u^{t+1}, x^{t+1})\}_{t\in \mathbb {N}}$ converges.

Proof

In view of the coercivity of $F_\lambda$ and since f, g are proper lsc, the iterates are well-defined.

For part (i) note that by the definition of the $x$-update we have that

$$\begin{aligned} F_\lambda (u^t, x^{t+1}) + D_{\sigma }(x^{t+1}, x^t) \le F_\lambda (u^t, x^t) \end{aligned}$$

and by the definition of the $u$-update

$$\begin{aligned} F_\lambda (u^{t+1}, x^{t+1}) + D_{\omega }(u^{t+1}, u^t) \le F_\lambda (u^t, x^{t+1}). \end{aligned}$$

Summing the two yields (36).

For part (ii) note that the boundedness of $\{u^t, x^t\}_{t \in \mathbb {N}}$ follows from (36) and the coercivity of $F_\lambda$. By the qualification condition and an argument similar to the one in the Proof of Lemma 3.3, we have that $x^t \in {{\,\mathrm{int}\,}}(\mathrm{dom}\,\phi )$.

For part (iii) note that $F_\lambda$ is proper and lsc and the iterates are bounded due to part (ii). In view of [6, Corollary 1.10], $F_\lambda$ is bounded from below over the iterates and the conclusion follows. $\square$

We are now ready to prove the statement from Theorem 4.1:

Proof

We sum the estimate (36) form $t=0$ to T and obtain, in view of Lemma A.1(iii), that

$$\begin{aligned} -\infty < F_\lambda (u^T, x^T) - F_\lambda (u^0, x^0)&= \sum _{t=0}^T F_\lambda (u^{t+1}, x^{t+1}) - F_\lambda (u^{t}, x^{t}) \\&\le -\sum _{t=0}^T \big (D_{\sigma }(x^{t+1}, x^t) +D_{\omega }(u^{t+1}, u^t)\big ). \end{aligned}$$

We take $T \rightarrow \infty$ and deduce that

$$\begin{aligned} D_{\sigma }(x^{t+1}, x^t) +D_{\omega }(u^{t+1}, u^t) \rightarrow 0, \end{aligned}$$

and therefore $D_{\sigma }(x^{t+1}, x^t) \rightarrow 0$ and $D_{\omega }(u^{t+1}, u^t) \rightarrow 0$ and in view of the strict convexity of $\sigma ,\omega$ on ${{\,\mathrm{int}\,}}(\mathrm{dom}\,\phi )$, we also have $\Vert x^{t+1}-x^t\Vert \rightarrow 0$ and $\Vert u^{t+1}-u^t\Vert \rightarrow 0$. In view of the $x$- and $u$-updates and the qualification condition (5), we obtain that:

$$\begin{aligned}&0 \in \partial f(x^{t+1}) + \frac{1}{\lambda } (\nabla \phi (x^{t+1}) - A(u^{t+1})) \\&\qquad +\, \nabla \sigma (x^{t+1}) - \nabla \sigma (x^t) + \frac{1}{\lambda }(A(u^{t+1}) - A(u^t)), \end{aligned}$$

and

$$\begin{aligned} 0 \in \partial g(u^{t+1}) + \frac{1}{\lambda } A^* (\nabla \phi ^*(A(u^{t+1})) - x^{t+1}) + \nabla \omega (u^{t+1}) - \nabla \omega (u^t). \end{aligned}$$

In view of [6, Exercise 8.8(c)] and [6, Proposition 10.5] and since $x^{t+1} \in {{\,\mathrm{int}\,}}(\mathrm{dom}\,\phi )$, this means

$$\begin{aligned} \begin{pmatrix} \nabla \sigma (x^t) - \nabla \sigma (x^{t+1}) + \frac{1}{\lambda }(A(u^t) - A(u^{t+1})) \\ \nabla \omega (u^t) -\nabla \omega (u^{t+1}) \end{pmatrix} \in \partial F_\lambda (u^{t+1}, x^{t+1}). \end{aligned}$$

In view of Lemma A.1(ii), the iterates are bounded and we may consider a convergent subsequence $\{u^{t_j}, x^{t_j}\}_{j \in \mathbb {N}} \subset \{u^t, x^t\}_{t \in \mathbb {N}}$. Let $(u^*, x^*)$ denote the limit point. In view of the closedness of ${{\,\mathrm{gph}\,}}\partial F_\lambda$ under the $F_\lambda$-attentive topology, we have for $j \rightarrow \infty$, since $F_\lambda (u^{t_j}, x^{t_j}) \rightarrow F_\lambda (u^*, x^*)$, the continuity of $\nabla \sigma ,\nabla \omega ,A$ and $\Vert x^{t+1}-x^t\Vert \rightarrow 0$ and $\Vert u^{t+1}-u^t\Vert \rightarrow 0$ that:

$$\begin{aligned} 0\in \partial F_\lambda (u^*, x^*). \end{aligned}$$

It remains to argue that also the limit point $x^* \in {{\,\mathrm{int}\,}}(\mathrm{dom}\,\phi )$ is contained in the interior of $\mathrm{dom}\,\phi$: In view of the qualification condition (5) and an argument similar to the one in the Proof of Lemma 3.3, as well as [6, Proposition 10.5], we obtain that $x^* \in {{\,\mathrm{int}\,}}(\mathrm{dom}\,\phi )$ and conclude that the optimality conditions (32) and (33) hold. $\square$