--- title: Resolvent Average in Monotone Operator Theory url: https://www.emergentmind.com/topics/resolvent-average type: topic --- # Resolvent Average in Monotone Operator Theory 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 [1505.02718], [2209.06200]. ## 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 \(\sum_{i=1}^n\lambda_i=1\), and a parameter \(\mu>0\), Bartz, Bauschke, Moffat, and Wang define the \(\mu\)-resolvent average by \[ R_\mu(\vec A,\vec\lambda) = \Bigl(\sum_{i=1}^n \lambda_i (A_i+\mu^{-1}\mathrm{Id})^{-1}\Bigr)^{-1} -\mu^{-1}\mathrm{Id}, \] with the fundamental identity \[ J_{\mu\,R_\mu(\vec A,\vec\lambda)} = \sum_{i=1}^n \lambda_i J_{\mu A_i}. \] This identity is the basic reason the construction is stable: convex combinations of firmly nonexpansive maps are again firmly nonexpansive, so \(R_\mu(\vec A,\vec\lambda)\) is again maximally monotone [1505.02718]. Combettes later recast the same object as a special case of a more general resolvent composition. In the product-space formulation, if \[ \bar A = \Bigl(\sum_{k=1}^p\omega_k J_{A_k}\Bigr)^{-1}-\mathrm{Id}, \] then \[ J_{\bar A}=\sum_{k=1}^p\omega_k J_{A_k}. \] 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 [2209.06200]. Several elementary special cases are built into the definition. If \(p=1\), then \(\bar A=A_1\). If \(A_k=A\) for all \(k\), then \(\bar A=A\). In the two-operator case with equal weights, one has \[ \bar A = \Bigl(\tfrac12 J_{A_1}+\tfrac12 J_{A_2}\Bigr)^{-1}-\mathrm{Id}, \qquad J_{\bar A}=\tfrac12 J_{A_1}+\tfrac12 J_{A_2}, \] which is the form most closely tied to classical averaged-projection constructions [2209.06200], [1003.4793]. ## 2. Monotonicity, averagedness, and generalized resolvents The classical correspondence \[ A\ \text{monotone} \quad\Longleftrightarrow\quad J_A\ \text{firmly nonexpansive} \] 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 \(\beta\)-comonotonicity and \(\alpha\)-conically nonexpansive mappings. An operator \(A:X\rightrightarrows X\) is \(\beta\)-comonotone if \[ \forall\,(x,u),(y,v)\in\operatorname{gra}A: \quad \langle x-y,u-v\rangle\ge -\beta\|u-v\|^2, \] and a mapping \(T:X\to X\) is \(\alpha\)-conically nonexpansive if \[ T=(1-\alpha)\mathrm{Id}+\alpha N \quad\text{for some nonexpansive }N. \] For \(0<\alpha<1\), this reduces to the usual notion of \(\alpha\)-averagedness [1902.09827]. Their main characterization states that an operator \(T\) is \(\alpha\)-averaged if and only if there exist \(\mu>0\) and a \(\beta\)-comonotone operator \(A\) such that \[ T=J_{\mu A}=(\mathrm{Id}+\mu A)^{-1}, \qquad \mu=2\alpha, \qquad \beta=\frac{1-2\alpha}{2\alpha}. \] Thus every averaged mapping is a resolvent after appropriate scaling, and the correspondence is quantitative rather than merely qualitative [1902.09827]. This generalized viewpoint gives a direct interpretation of resolvent averages. If \(A_1,\dots,A_n\) are each \(\beta\)-comonotone with common \(\beta>-1\), then each resolvent \(T_i=J_{\mu A_i}\) is \(\alpha\)-averaged for the corresponding parameter relation, and any convex combination \[ T=\sum_{i=1}^n w_i T_i \] is again \(\alpha\)-averaged. Consequently there exists a single \(\beta\)-comonotone operator \[ A=T^{-1}-\mathrm{Id} \] 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 [1902.09827]. The same framework also isolates a sharp boundary phenomenon: the requirement \(\beta>-1\) is essential for single-valuedness and full domain of the resolvent. The paper explicitly notes that when \(\beta\le -1\), one can construct a maximally \((\le -1)\)-comonotone operator whose resolvent fails to be single-valued or onto [1902.09827]. ## 3. Structural properties and inheritance principles A major part of the theory concerns which properties pass from the components \(A_i\) 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 \(A_i\) forces the resolvent average to have it, and recessive if it holds for the resolvent average whenever each \(A_i\) has it [1505.02718]. 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 \(A_{i_0}\) has full domain, then the resolvent average has full domain; if some \(A_{i_0}\) 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 [1505.02718]. In Combettes’s formulation, if one \(A_j\) is \(\alpha\)-strongly monotone, then \(\bar A\) is \(\alpha\)-strongly monotone [2209.06200]. Recessive properties include linearity, affinity, rectangularity, paramonotonicity, \(k\)-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 [1505.02718]. 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: \[ R_\mu(\vec A,\vec\lambda)^{-1} = R_{\mu^{-1}}(\vec A^{-1},\vec\lambda). \] 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 [1505.02718]. Domain and range behavior has been studied in two complementary forms. In the finite-dimensional analysis of Bauschke, Moffat, and Wang, \(\operatorname{dom}A_{\mathrm{avg}}\) and \(\operatorname{ran}A_{\mathrm{avg}}\) are nearly convex and satisfy \[ \operatorname{dom}A_{\mathrm{avg}}\approx \sum_{i=1}^n \lambda_i \operatorname{dom}A_i, \qquad \operatorname{ran}A_{\mathrm{avg}}\approx \sum_{i=1}^n \lambda_i \operatorname{ran}A_i, \] where \(\approx\) denotes equality up to closure and relative interior [1105.0029]. In Combettes’s product-space treatment, the corresponding formulas are presented exactly as \[ \operatorname{dom}\bar A=\sum_k \omega_k \operatorname{dom}A_k, \qquad \operatorname{ran}\bar A=\sum_k \omega_k \operatorname{ran}A_k, \] together with the analogous identities for interiors [2209.06200]. 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 \(A_1\) and \(A_2\) with weights \(\lambda_1=\lambda\) and \(\lambda_2=1-\lambda\), Bauschke and coauthors introduced the cycle set \[ S=\{(x,y)\in H\times H\mid x=J_{A_1/\lambda_2}(y),\ y=J_{A_2/\lambda_1}(x)\}, \] together with the fixed-point sets \[ E=\operatorname{Fix}(J_{A_1/\lambda_2}\circ J_{A_2/\lambda_1}), \qquad F=\operatorname{Fix}(J_{A_2/\lambda_1}\circ J_{A_1/\lambda_2}). \] Their key result is that \[ T:S\to \operatorname{Fix}(J_A), \qquad T(x,y)=\lambda_1 x+\lambda_2 y, \] is a homeomorphism, with inverse \[ T^{-1}(z)=(J_{A_1}(z),J_{A_2}(z)). \] Hence the fixed points of the average resolvent are geometrically equivalent to the limiting cycles of alternating scaled resolvents [1003.4793]. For \(m\) operators, the same idea becomes a product-space fixed-point problem. Given \(A_1,\dots,A_m\) and weights \(\lambda_i\), define \(\mu_i=1-\lambda_i\) and consider \[ S = \Bigl\{\mathbf x=(x_1,\dots,x_m)\in X^m\ \Big|\ x_i = J_{\mu_i^{-1}A_i}\!\Bigl(\sum_{j\neq i}\frac{\lambda_j}{\mu_i}x_j\Bigr) \ \forall i \Bigr\}. \] Then the linear map \[ L:S\to \operatorname{Fix}(J_A), \qquad L(x_1,\dots,x_m)=\sum_{i=1}^m \lambda_i x_i \] is a bijection, Lipschitz-continuous with constant \(1\), and has inverse \[ L^{-1}(x)=(J_{A_1}x,\dots,J_{A_m}x). \] Thus \(\operatorname{Fix}(J_A)=L(S)\), so the fixed-point set of the average of resolvents can be lifted to a structured fixed-point set in a product space [1102.1478]. This reformulation leads directly to algorithms. In the equal-weight case \(\lambda_i=1/m\), the paper defines product-space operators \(R\) and \(J\), proves that \(R\) is \(\alpha\)-averaged with \[ \alpha=\frac{m}{2m-2}, \] that \(J\) is firmly nonexpansive, and therefore that \[ R\circ J \quad\text{is}\quad \frac{2m}{3m-2}\text{-averaged}. \] Algorithm 1, based on the iteration \(x^{n+1}=R(J(x^n))\), has a complete convergence proof: if \(\operatorname{Fix}(J_A)\neq\varnothing\), then the iterates converge weakly to a point in \(S\), and the corresponding weighted sums converge weakly to a point in \(\operatorname{Fix}(J_A)\); if \(\operatorname{Fix}(J_A)=\varnothing\), then \(\|x^n\|\to\infty\) [1102.1478]. 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 \(\mathbb R^{50}\) with \(55\) random hyperplanes indicate significantly faster decay of the proximity residual than direct iteration of the averaged resolvent [1102.1478]. 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 \(f_1,\dots,f_p\in\Gamma_0(H)\) are proper lower semicontinuous convex functions, then \(\partial f_k\) is maximally monotone and \[ J_{\partial f_k}=\operatorname{prox}_{f_k}. \] In Combettes’s formulation, the proximal average \(f\) satisfies \[ \operatorname{prox}_f = \sum_{k=1}^p \omega_k \operatorname{prox}_{f_k}, \] and therefore \[ J_{\overline{\partial f}^{\,\Lambda}} = \sum_{k=1}^p \omega_k J_{\partial f_k} = \sum_{k=1}^p \omega_k \operatorname{prox}_{f_k} = \operatorname{prox}_f = J_{\partial f}. \] Equivalently, \[ \overline{\partial f}^{\,\Lambda}=\partial f. \] This is the operator-theoretic statement that the subdifferential of the proximal average is the resolvent average of the subdifferentials [2209.06200]. In the parameterized theory of Bartz, Bauschke, Moffat, and Wang, if each \(A_i=\partial f_i\), then \[ R_\mu(\partial \vec f,\vec\lambda)=\partial\, p_\mu(\vec f,\vec\lambda), \] so the resolvent average recovers a significant part of the theory of proximal averages within full monotone operator theory [1505.02718]. The two-operator geometry translates directly to minimizers. If \(A_i=\partial f_i\), then \(\operatorname{Fix}(J_{\partial f})=\operatorname*{argmin} f\), and the cycle-set/homeomorphism results become statements about minimizers of proximal-average functionals. In particular, \[ \operatorname*{argmin} p(f_1,f_2;\lambda) = \operatorname{Fix}\bigl(\lambda_1\operatorname{prox}_{f_1}+\lambda_2\operatorname{prox}_{f_2}\bigr), \] and this minimizer set is homeomorphic to the cycle set of the alternating proximal mappings associated with scaled functions [1003.4793]. The generalized resolvent theory also reaches beyond convexity. For a proper lower semicontinuous \(f:X\to(-\infty,+\infty]\) that is \(\rho\)-hypoconvex, Bauschke, Moursi, and Wang show that for any \(0