We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward-backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as L. Rosasco DIBRIS, Univ...

The concept of a monotone operator — which covers both linear positive semi-definite operators and subdifferentials of convex functions — is fundamental in various branches of mathematics. Over the last few decades, several stronger notions of monotonicity have been introduced: Gossez’s maximal monotonicity of dense type, Fitzpatrick and Phelps’s local maximal monotonicity, and Simons’s monoton...

This paper deals with boundary value problems for nonlinear monotone potential operators. An analysis of the nonlinear (monotone potential) Sturm–Liouville operator Au := − ( k((u′)2)u′(x) )′+q(x)u(x), x ∈ (a, b) shows that the potential of this operator as well as the potential of related boundary value problems play an important role not only for solvability of these problems, but also for li...

Generalized Quasilinearization Method for Coincidences Adriana Buică and Radu Precup Faculty of Mathematics and Computer Science Babeş-Bolyai University 3400 Cluj, Romania [email protected], [email protected] Abstract An abstract unified theory of both monotone iterative and generalized quasilinearization methods is presented for operator equations of coincidence type in ordered Ban...

A unitary family is a family of unitary operators U(x) acting on a finite dimensional hermitian vector space, depending analytically on a real parameter x. It is monotone if 1 i U (x)U(x) is a positive operator for each x. We prove a number of results generalizing standard theorems on the spectral theory of a single unitary operator U0, which correspond to the ’commutative’ case U(x) = eU0. Als...

This tutorial paper presents the basic notation and results of monotone operators and operator splitting methods, with a focus on convex optimization. A very wide variety of algorithms, ranging from classical to recently developed, can be derived in a uniform way. The approach is to pose the original problem to be solved as one of finding a zero of an appropriate monotone operator; this problem...

We consider an abstract second order non-autonomous evolution equation in a Hilbert space H : u ″ + Au γ ( t ) ′ f = 0, where A is self-adjoint and nonnegative operator on , conservative -valued function with polynomial growth (not necessarily to be monotone), time-dependent damping term. How exactly the decay of energy affected by coefficient exponent associated nonlinear term ? There seems li...