Resolvent Average in Monotone Operator Theory

• Resolvent average is an operator defined by taking the inverse of the convex combination of resolvents from maximally monotone operators, generalizing proximal averages.
• It preserves key properties like monotonicity and firm nonexpansiveness, enabling stable fixed-point formulations and domain-range analyses.
• The approach unifies monotone operator theory with convex analysis and has practical applications in fixed-point algorithms and structured optimization.

Searching arXiv for recent and foundational papers on resolvent averages and related monotone-operator formulations. arxiv_search(query="resolvent average monotone operators proximal average", max_results=10, sort_by="relevance") The resolvent average is an averaging operation on monotone operators in a real Hilbert space that is defined by averaging resolvents rather than averaging operators directly. For maximally monotone operators $A_1,\dots,A_p$ with positive weights $\omega_k$ summing to $1$, the basic construction is

$$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$

where $J_A=(\mathrm{Id}+A)^{-1}$ is the resolvent. In this form, the construction extends the proximal average from convex subdifferentials to arbitrary maximally monotone operators, and it supports a substantial calculus for fixed points, domains, ranges, monotonicity properties, and algorithmic reformulations in product spaces (Bartz et al., 2015, Combettes, 2022).

1. Definition and canonical formulas

Let $H$ be a real Hilbert space and let $A:H\to 2^H$ be maximally monotone. Its resolvent is the single-valued firmly nonexpansive mapping

$J_A=(\mathrm{Id}+A)^{-1}.$

The resolvent average is obtained by taking a convex combination of such resolvents and then returning to the operator side through the Minty correspondence.

A parameterized form is central in the literature. For finitely many maximally monotone operators $A_1,\dots,A_n$, weights $\lambda_i>0$ with $\omega_k$0, and a parameter $\omega_k$1, Bartz, Bauschke, Moffat, and Wang define the $\omega_k$2-resolvent average by

$\omega_k$3

with the fundamental identity

$\omega_k$4

This identity is the basic reason the construction is stable: convex combinations of firmly nonexpansive maps are again firmly nonexpansive, so $\omega_k$5 is again maximally monotone (Bartz et al., 2015).

Combettes later recast the same object as a special case of a more general resolvent composition. In the product-space formulation, if

$\omega_k$6

then

$\omega_k$7

This form emphasizes that no further splitting is needed: the average is literally the operator whose resolvent is the convex combination of the original resolvents (Combettes, 2022).

Several elementary special cases are built into the definition. If $\omega_k$8, then $\omega_k$9. If $1$0 for all $1$1, then $1$2. In the two-operator case with equal weights, one has

$1$3

which is the form most closely tied to classical averaged-projection constructions (Combettes, 2022, Wang et al., 2010).

2. Monotonicity, averagedness, and generalized resolvents

The classical correspondence

$1$4

is the starting point for most uses of resolvent averages. Since firmly nonexpansive maps form a subclass of averaged mappings, the resolvent average sits naturally at the intersection of monotone operator theory and fixed-point theory.

Bauschke, Moursi, and Wang generalized this picture by introducing $1$5-comonotonicity and $1$6-conically nonexpansive mappings. An operator $1$7 is $1$8-comonotone if

$1$9

and a mapping $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$0 is $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$1-conically nonexpansive if

$$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$2

For $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$3, this reduces to the usual notion of $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$4-averagedness (Bauschke et al., 2019).

Their main characterization states that an operator $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$5 is $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$6-averaged if and only if there exist $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$7 and a $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$8-comonotone operator $$\bar A=\Bigl(\sum_{k=1}^p \omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\sum_{k=1}^p \omega_k J_{A_k},$$9 such that

$J_A=(\mathrm{Id}+A)^{-1}$0

Thus every averaged mapping is a resolvent after appropriate scaling, and the correspondence is quantitative rather than merely qualitative (Bauschke et al., 2019).

This generalized viewpoint gives a direct interpretation of resolvent averages. If $J_A=(\mathrm{Id}+A)^{-1}$1 are each $J_A=(\mathrm{Id}+A)^{-1}$2-comonotone with common $J_A=(\mathrm{Id}+A)^{-1}$3, then each resolvent $J_A=(\mathrm{Id}+A)^{-1}$4 is $J_A=(\mathrm{Id}+A)^{-1}$5-averaged for the corresponding parameter relation, and any convex combination

$J_A=(\mathrm{Id}+A)^{-1}$6

is again $J_A=(\mathrm{Id}+A)^{-1}$7-averaged. Consequently there exists a single $J_A=(\mathrm{Id}+A)^{-1}$8-comonotone operator

$J_A=(\mathrm{Id}+A)^{-1}$9

whose resolvent is exactly that convex combination. In this sense, averages of resolvents can themselves be read as resolvents of an averaged operator in the same comonotonicity class (Bauschke et al., 2019).

The same framework also isolates a sharp boundary phenomenon: the requirement $H$0 is essential for single-valuedness and full domain of the resolvent. The paper explicitly notes that when $H$1, one can construct a maximally $H$2-comonotone operator whose resolvent fails to be single-valued or onto (Bauschke et al., 2019).

3. Structural properties and inheritance principles

A major part of the theory concerns which properties pass from the components $H$3 to the resolvent average. Bartz, Bauschke, Moffat, and Wang formalized this by distinguishing dominant properties from recessive properties. A property is dominant if its presence in any one $H$4 forces the resolvent average to have it, and recessive if it holds for the resolvent average whenever each $H$5 has it (Bartz et al., 2015).

Among the dominant properties identified are nonempty interior of domain, full domain, surjectivity, single-valuedness, strict monotonicity, strong monotonicity, cocoercivity, and disjoint injectivity. For example, if some $H$6 has full domain, then the resolvent average has full domain; if some $H$7 is strictly monotone, then so is the resolvent average; and strong monotonicity and cocoercivity also propagate in this dominant sense, with explicitly computable constants in the parameterized theory (Bartz et al., 2015). In Combettes’s formulation, if one $H$8 is $H$9-strongly monotone, then $A:H\to 2^H$0 is $A:H\to 2^H$1-strongly monotone (Combettes, 2022).

Recessive properties include linearity, affinity, rectangularity, paramonotonicity, $A:H\to 2^H$2-cyclic monotonicity, cyclic monotonicity, weak sequential closedness of the graph, certain displacement-mapping structures, and nonexpansiveness when each component operator is both monotone and nonexpansive (Bartz et al., 2015). This dominant/recessive distinction is one of the main organizing principles of the subject: it turns the resolvent average into a controlled synthesis operation rather than a purely formal interpolation.

The inversion formula is another structural pillar: $A:H\to 2^H$3 This identity is particularly useful because it transfers statements between primal and inverse operators and, in the subdifferential setting, yields corresponding identities for proximal averages (Bartz et al., 2015).

Domain and range behavior has been studied in two complementary forms. In the finite-dimensional analysis of Bauschke, Moffat, and Wang, $A:H\to 2^H$4 and $A:H\to 2^H$5 are nearly convex and satisfy

$A:H\to 2^H$6

where $A:H\to 2^H$7 denotes equality up to closure and relative interior (Bauschke et al., 2011). In Combettes’s product-space treatment, the corresponding formulas are presented exactly as

$A:H\to 2^H$8

together with the analogous identities for interiors (Combettes, 2022). This suggests that different formulations emphasize different regularity mechanisms for the same construction.

4. Fixed-point geometry and product-space reformulations

The fixed-point theory of resolvent averages was developed first for two operators and then for finitely many operators. For two maximally monotone operators $A:H\to 2^H$9 and $J_A=(\mathrm{Id}+A)^{-1}.$0 with weights $J_A=(\mathrm{Id}+A)^{-1}.$1 and $J_A=(\mathrm{Id}+A)^{-1}.$2, Bauschke and coauthors introduced the cycle set

$J_A=(\mathrm{Id}+A)^{-1}.$3

together with the fixed-point sets

$J_A=(\mathrm{Id}+A)^{-1}.$4

Their key result is that

$J_A=(\mathrm{Id}+A)^{-1}.$5

is a homeomorphism, with inverse

$J_A=(\mathrm{Id}+A)^{-1}.$6

Hence the fixed points of the average resolvent are geometrically equivalent to the limiting cycles of alternating scaled resolvents (Wang et al., 2010).

For $J_A=(\mathrm{Id}+A)^{-1}.$7 operators, the same idea becomes a product-space fixed-point problem. Given $J_A=(\mathrm{Id}+A)^{-1}.$8 and weights $J_A=(\mathrm{Id}+A)^{-1}.$9, define $A_1,\dots,A_n$0 and consider

$A_1,\dots,A_n$1

Then the linear map

$A_1,\dots,A_n$2

is a bijection, Lipschitz-continuous with constant $A_1,\dots,A_n$3, and has inverse

$A_1,\dots,A_n$4

Thus $A_1,\dots,A_n$5, so the fixed-point set of the average of resolvents can be lifted to a structured fixed-point set in a product space (Bauschke et al., 2011).

This reformulation leads directly to algorithms. In the equal-weight case $A_1,\dots,A_n$6, the paper defines product-space operators $A_1,\dots,A_n$7 and $A_1,\dots,A_n$8, proves that $A_1,\dots,A_n$9 is $\lambda_i>0$0-averaged with

$\lambda_i>0$1

that $\lambda_i>0$2 is firmly nonexpansive, and therefore that

$\lambda_i>0$3

Algorithm 1, based on the iteration $\lambda_i>0$4, has a complete convergence proof: if $\lambda_i>0$5, then the iterates converge weakly to a point in $\lambda_i>0$6, and the corresponding weighted sums converge weakly to a point in $\lambda_i>0$7; if $\lambda_i>0$8, then $\lambda_i>0$9 (Bauschke et al., 2011).

A second, Gauss–Seidel-style block iteration updates one block at a time. The paper notes that this composite need not be nonexpansive or averaged, and no convergence proof is given, but numerical experiments in $\omega_k$00 with $\omega_k$01 random hyperplanes indicate significantly faster decay of the proximity residual than direct iteration of the averaged resolvent (Bauschke et al., 2011). The contrast between the proved synchronous scheme and the empirically faster block scheme remains one of the more explicit open points in the algorithmic literature.

5. Relation to proximal averages and proximal mappings

The resolvent average was originally motivated by the proximal average of convex functions. If $\omega_k$02 are proper lower semicontinuous convex functions, then $\omega_k$03 is maximally monotone and

$\omega_k$04

In Combettes’s formulation, the proximal average $\omega_k$05 satisfies

$\omega_k$06

and therefore

$\omega_k$07

Equivalently,

$\omega_k$08

This is the operator-theoretic statement that the subdifferential of the proximal average is the resolvent average of the subdifferentials (Combettes, 2022).

In the parameterized theory of Bartz, Bauschke, Moffat, and Wang, if each $\omega_k$09, then

$\omega_k$10

so the resolvent average recovers a significant part of the theory of proximal averages within full monotone operator theory (Bartz et al., 2015).

The two-operator geometry translates directly to minimizers. If $\omega_k$11, then $\omega_k$12, and the cycle-set/homeomorphism results become statements about minimizers of proximal-average functionals. In particular,

$\omega_k$13

and this minimizer set is homeomorphic to the cycle set of the alternating proximal mappings associated with scaled functions (Wang et al., 2010).

The generalized resolvent theory also reaches beyond convexity. For a proper lower semicontinuous $\omega_k$14 that is $\omega_k$15-hypoconvex, Bauschke, Moursi, and Wang show that for any $\omega_k$16,

$\omega_k$17

is Lipschitz and in fact $\omega_k$18-cocoercive, while its displacement

$\omega_k$19

is $\omega_k$20-conically nonexpansive. Equivalently,

$\omega_k$21

so the proximal mapping of a hypoconvex function is again a resolvent in the generalized comonotone sense (Bauschke et al., 2019).

6. Examples, special cases, and later extensions

The normal-cone case connects the theory to projections and feasibility. If $\omega_k$22 for closed convex sets $\omega_k$23, then $\omega_k$24, and the resolvent average becomes the operator whose resolvent is the weighted average of the projections. In the two-set case, when $\omega_k$25, the corresponding proximal average reduces to

$\omega_k$26

so fixed points of the average of projections coincide with least-squares solutions, while alternating-projection cycles encode the same information through the homeomorphism described above (Wang et al., 2010). In the multi-operator theory, averaging normal cones also recovers common-point searches and related convex-feasibility constructions (Bartz et al., 2015).

Finite-dimensional examples show that the resolvent average can behave quite differently from a naive operator average. In $\omega_k$27, if $\omega_k$28 and $\omega_k$29 with equal weights, then

$\omega_k$30

and therefore

$\omega_k$31

This example is used to illustrate the domain/range calculus and the way resolvent averaging can regularize singular geometric data (Bauschke et al., 2011).

Linear-algebraic examples are especially striking. Bartz, Bauschke, Moffat, and Wang show that averaging two rotations by $\omega_k$32 yields the identity, so the resolvent average can “undo” opposing rotations. They also note that in the class of positive semidefinite or positive definite matrices, the resolvent average coincides with the matrix-resolvent average studied earlier, preserves positivity, and interpolates arithmetic and harmonic means (Bartz et al., 2015).

For strictly positive linear operators, Cornejo and Combettes define

$\omega_k$33

and prove the Löwner-order bounds

$\omega_k$34

They also show monotonicity in each argument, the asymptotic relations

$\omega_k$35

and nonexpansiveness with respect to Thompson’s metric: $\omega_k$36 In that setting, resolvent averages also underpin a family of nonlinear fixed-point equations with existence and uniqueness obtained via strict contractivity in the Thompson metric (Cornejo, 8 Sep 2025).

A terminological caution is useful. A distinct use of “resolvent averaging” appears in random matrix theory, where one studies averages of monomials in matrix resolvent entries such as $\omega_k$37, together with fluctuation-averaging bounds for random band matrices (Erdos et al., 2012). This suggests a terminological overlap rather than a shared construction: the monotone-operator resolvent average is an operator defined by

$\omega_k$38

whereas the random-matrix usage concerns probabilistic averages of entries of $\omega_k$39.

In the monotone-operator literature proper, the resolvent average now functions simultaneously as an interpolation device, a structural calculus for monotonicity classes, a bridge to proximal averages, and a source of product-space fixed-point algorithms. Its central feature is that the averaging occurs at the resolvent level, where firm nonexpansiveness and averagedness are directly accessible, and only afterward is the averaged object pulled back to the operator side.