Generalized Bregman Envelopes and Proximity Operators

Abstract

Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen.

A Appendix: Closed Forms for Envelopes and Proximity Operators

For all our examples, we use the closed distances as described in Remark 2.4. The function denoted ${\mathcal {W}}$ is the principal branch of the Lambert ${\mathcal {W}}$ function, whose occurrences in variational analysis have been discussed in, for example, [8, 10, 25].

Example A.1

(Left prox and envelope for $\overline{{\mathcal {D}}}^{F_{\log }}$)

Beginning with the smallest member of ${\mathcal {H}}(\log )$, we first consider the left envelope and corresponding proximity operator characterized by

$$\begin{aligned} \,\overleftarrow{{\text {env}}}_{\gamma ,\theta }^{\dagger F_{\log }}(x)&= \underset{y \in {\mathbb {R}}_+}{\inf } \{\theta (y)+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{F_{\log }}(y,x) \}\\&=\theta (\overleftarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger F_{\log }} (x))+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{F_{\log }}\left( \overleftarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger F_{\log }} (x),x \right) , \end{aligned}$$

where $F_{\log }$ is given in [9, Example 3.6] and $\overline{{\mathcal {D}}^{F_{\log }}}$ is in (39). We have that

$$\begin{aligned} \overleftarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger F_{\log }}(x)&= {\left\{ \begin{array}{ll} \left( \frac{1}{\gamma }+1\right) e^{\gamma }\gamma x &{} \text {if } x\le \frac{e^{-\gamma }}{2+2\gamma };\\ \left( \frac{1}{\gamma }-1\right) e^{-\gamma }\gamma x &{} \text {if } \frac{e^\gamma }{2-2\gamma } < x;\\ \frac{1}{2} &{} \text {otherwise}, \end{array}\right. }\\ \text {and}\quad \,\overleftarrow{{\text {env}}}_{\gamma ,\theta }^{\dagger F_{\log }}(x)&={\left\{ \begin{array}{ll} \frac{1}{2}-e^{\gamma }x &{} \text {if } x\le \frac{e^{-\gamma }}{2+\gamma };\\ \gamma \left( \left( \frac{1}{\gamma }-1 \right) e^{-\gamma }x +e^\gamma x \right) -\frac{1}{2} &{} \text {if } x> \frac{e^\gamma }{2-\gamma };\\ \frac{\left( {\mathcal {W}}\left( \frac{e}{2x} \right) -1 \right) ^2}{2\gamma {\mathcal {W}}\left( \frac{e}{2x} \right) } &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The envelope is shown in Fig. 2a.

Example A.2

(Right prox and envelope for $\overline{{\mathcal {D}}}^{F_{\log }}$)

We next consider the right envelope and corresponding proximity operator characterized by

$$\begin{aligned} \,\overrightarrow{{\text {env}}}_{\gamma ,\theta }^{\dagger F_{\log }}(x)&= \underset{y \in {\mathbb {R}}_+}{\inf } \{\theta (y)+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{F_{\log }}(x,y) \}\\&=\theta (\overrightarrow{{\text {s}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger F_{\log }} (x))+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{F_{\log }}\left( x,\overrightarrow{{\text {s}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger F_{\log }} (x)\right) ,\\ \text {where}\quad \overrightarrow{{\text {s}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger F_{\log }} (x)&\in \overrightarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger F_{\log }} (x). \end{aligned}$$

We use the selection operator $\overrightarrow{{\text {s}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{F_{\log }}$ because, in this case, the prox operator $\overrightarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{F_{\log }}$ is set-valued:

The corresponding right envelope is

$$\begin{aligned} \,\overrightarrow{{\text {env}}}_{\gamma ,\theta }^{\dagger F_{\log }}(x)=&{\left\{ \begin{array}{ll} \frac{1}{2} &{} \text {if } x=0 \,\text {and}\, e \le \frac{1}{\gamma };\\ \frac{1}{2\gamma e} &{} \text {if } x=0 \,\text {and}\, \frac{1}{\gamma }< e;\\ \frac{x{\mathcal {W}}(\gamma )}{\gamma ({\mathcal {W}}(\gamma )+1)}-\frac{1}{2}\\ +\frac{x \left( {\mathcal {W}}\left( \frac{e\gamma ({\mathcal {W}}(\gamma )+1)}{{\mathcal {W}}(\gamma )} \right) -1 \right) ^2}{\gamma {\mathcal {W}}\left( \frac{e\gamma ({\mathcal {W}}(\gamma )+1)}{{\mathcal {W}}(\gamma )} \right) } &{} \text {if } \frac{\gamma ({\mathcal {W}}(\gamma )+1)}{2{\mathcal {W}}(\gamma )}<x;\\ \frac{x{\mathcal {W}}(-\gamma )}{\gamma ({\mathcal {W}}(-\gamma )+1)}+\frac{1}{2}\\ +\frac{x \left( {\mathcal {W}}\left( - \frac{e\gamma ({\mathcal {W}}(-\gamma )+1)}{{\mathcal {W}}(-\gamma )} \right) -1 \right) ^2}{\gamma {\mathcal {W}}\left( - \frac{e\gamma ({\mathcal {W}}(-\gamma )+1)}{{\mathcal {W}}(-\gamma )} \right) } &{} \text {if } 0<x<-\frac{\gamma ({\mathcal {W}}(-\gamma )+1)}{2{\mathcal {W}}(-\gamma )} \,\text {and}\, e<\frac{1}{\gamma };\\ \frac{x({\mathcal {W}}(2xe)-1)^2}{\gamma {\mathcal {W}}(2xe)} &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The envelope is shown in Fig. 3a.

Example A.3

(Left prox and envelope for $\overline{{\mathcal {D}}}^{\sigma _{\log }}$)

Turning to the biggest of the representative functions for the logarithm, we next consider the left envelope and corresponding proximity operator characterized by

$$\begin{aligned} \,\overleftarrow{{\text {env}}}_{\gamma ,\theta }^{\dagger \sigma _{\log }}(x)&= \underset{y \in {\mathbb {R}}_+}{\inf } \{\theta (y)+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{\sigma _{\log }}(y,x) \}\\&=\theta (\overleftarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger \sigma _{\log }} (x))+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{\sigma _{\log }}(\overleftarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger \sigma _{\log }} (x),x). \end{aligned}$$

where $\overline{{\mathcal {D}}^{\sigma _{\log }}}$ is as in 41. We have that

$$\begin{aligned} \overleftarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger \sigma _{\log }} (x)= {\left\{ \begin{array}{ll} \frac{x}{e^{-\gamma +1}} &{} \text {if } x \le \frac{1}{2}e^{-\gamma +1} \,\text {and}\, 1\le \gamma ;\\ x &{} \text {if } x \ge \frac{1}{2} \text {or} \gamma <1;\\ \frac{1}{2} &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The corresponding envelope is

$$\begin{aligned} \,\overleftarrow{{\text {env}}}_{\gamma ,\theta }^{\sigma _{\log }}(x) = {\left\{ \begin{array}{ll} 0 &{} \text {if } x=0;\\ x-\frac{1}{2} &{} \text {if } x \ge \frac{1}{2};\\ \frac{1}{2} -x &{} \text {if } x< \frac{1}{2} \,\text {and}\, \gamma < 1;\\ -\frac{x}{\gamma }e^{\gamma -1}\left( -\log \left( xe^{\gamma -1} \right) +\gamma +\log (x) \right) \\ +\frac{1}{2} &{} \text {if } x \le \frac{1}{2} e^{-\gamma +1} \,\text {and}\, 1 \le \gamma ;\\ -\frac{1}{2\gamma }\log (2x) &{} \text {otherwise}. \end{array}\right. }. \end{aligned}$$

The envelope is shown in Fig. 2c.

Example A.4

(Right prox and envelope for $\overline{{\mathcal {D}}}^{\sigma _{\log }}$) We consider the right envelope and corresponding proximity operator characterized by

$$\begin{aligned} \,\overrightarrow{{\text {env}}}_{\gamma , \theta }^{\dagger \sigma _{\log }}(x)&= \underset{y \in {\mathbb {R}}_+}{\inf } \{\theta (y)+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{\sigma _{\log }}(x,y) \}\\&=\theta (\overrightarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger \sigma _{\log }} (x))+\frac{1}{\gamma }\overline{{\mathcal {D}}}^{\sigma _{\log }}\left( x,\overrightarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger \sigma _{\log }} (x)\right) . \end{aligned}$$

The corresponding proximity operator is

$$\begin{aligned} \overrightarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger \sigma _{\log }} (x)={\left\{ \begin{array}{ll} \frac{1}{2} &{} \text {if } \frac{1}{2}< x \,\text {and}\, x\le \frac{1}{2}\gamma ;\\ \frac{x}{\gamma } &{} \text {if } 1 \le \gamma \,\text {and}\, \frac{1}{2} \gamma < x ;\\ x &{}\text {otherwise}, \end{array}\right. } \end{aligned}$$

while the corresponding envelope is

$$\begin{aligned} \,\overrightarrow{{\text {env}}}_{\gamma , \theta }^{\dagger \sigma _{\log }}(x) = {\left\{ \begin{array}{ll} \frac{1}{2}-x &{} \text {if } x \le \frac{1}{2};\\ x-\frac{1}{2} &{} \text {if } \frac{1}{2}< x \,\text {and}\, \gamma \le 1;\\ -\frac{1}{2}+\frac{x}{\gamma }(1+\log (\gamma )) &{} \text {if } 1\le \gamma \,\text {and}\, \frac{\gamma }{2} < x ;\\ \frac{x}{\gamma }\log (2x) &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The envelope is shown in Fig. 3c.

The operators in Example A.5 (and their computation) may be found in [7], with the minor modification that here we are computing with the lower closure of the distance and so obtain closed forms which differ at zero. We include them here for their comparison with the new GBDs for the logarithm.

Example A.5

(Proxes and envelopes for $\overline{{\mathcal {D}}}^{{\text {ent}}\oplus {\text {ent}}^*}$) We next consider the case when $\overline{{\mathcal {D}}}^{{\text {ent}}\oplus {\text {ent}}^*}$ is the closed GBD for the Fenchel–Young representative of $\log$ (37), a case whose relationship to the Bregman distance for the Boltzmann–Shannon entropy is discussed in Example 2.2. The left proximity operator and envelope are given by

$$\begin{aligned} \overleftarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger {\text {ent}}\oplus {\text {ent}}^*}(y)&={\left\{ \begin{array}{ll} y\exp (\gamma ), &{}\text { if } 0 \le y<\tfrac{1}{2}\exp (-\gamma ); \\ y\exp (-\gamma ), &{}\text { if } y >\tfrac{1}{2}\exp (\gamma ); \\ \tfrac{1}{2}, &{}\text { otherwise}, \end{array}\right. }\\ \,\overleftarrow{{\text {env}}}_{\gamma ,\theta }^{\dagger {\text {ent}}\oplus {\text {ent}}^*}(y)&={\left\{ \begin{array}{ll} \tfrac{y(1-e^\gamma )}{\gamma } + \tfrac{1}{2}, &{}\text { if } 0 \le y<\tfrac{1}{2}\exp (-\gamma ); \\ \tfrac{y(1-e^{-\gamma })}{\gamma } - \tfrac{1}{2}, &{}\text { if } \tfrac{1}{2}\exp (\gamma ) <y; \\ \tfrac{2y- \ln (y) -1 -\ln (2)}{2\gamma }, &{}\text { otherwise}. \end{array}\right. } \end{aligned}$$

For details, see [7, Example 4.1(ii)]. The envelope is shown in Fig. 2b. The right proximity operator and envelope are given by

$$\begin{aligned} \overrightarrow{{\text {P}}}_ {\negthinspace \negthinspace \gamma ,\theta }^{\dagger {\text {ent}}\oplus {\text {ent}}^*}(x)&={\left\{ \begin{array}{ll} \tfrac{x}{1 -\gamma }, &{}\text { if } 0 \le x<\tfrac{1 -\gamma }{2}; \\ \tfrac{x}{1 +\gamma }, &{}\text { if } x>\tfrac{1 +\gamma }{2}; \\ \tfrac{1}{2}, &{}\text { otherwise}. \end{array}\right. }\\ \,\overrightarrow{{\text {env}}}_{\gamma , \theta }^{\dagger {\text {ent}}\oplus {\text {ent}}^*}(x)&={\left\{ \begin{array}{ll} \tfrac{\ln (1 -\gamma )}{\gamma }x + \tfrac{1}{2}, &{}\text { if } 0 \le x <\tfrac{1 -\gamma }{2}; \\ \tfrac{\ln (1 +\gamma )}{\gamma }x - \tfrac{1}{2}, &{}\text { if } x >\tfrac{1 +\gamma }{2}; \\ \tfrac{1}{\gamma }\left( x\ln (2x) -x +\tfrac{1}{2}\right) , &{}\text { otherwise}. \end{array}\right. } \end{aligned}$$

For details, see [7, Example 4.1(ii)]. The envelope is shown in Fig. 3b.