On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space - Ulrich Kohlenbach (Optimization Letters, 2021)
In a recent paper, Bauschke et al. study ρ[12pt]minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt documentρ document-comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent JA.[12pt]minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt documentJ_A.document In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for ρ[12pt]minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt documentρ document-comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. zerA[12pt]minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt documentzer Adocument we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for ρ[12pt]minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt documentρ document-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.
Keywords: 510; Generalized monotone operators; Proximal point algorithm; Halpern-type proximal point algorithm; Rates of convergence; Metastability; Proof mining