We provide a new global small-gain theorem for feedback interconnections of monotone input-output systems with multi-valued input-state characteristics. This extends a recent small-gain theorem of Angeli and Sontag for monotone systems with singleton-valued characteristics. We prove our theorem using Thieme's convergence theory for asymptotically autonomous systems. An illustrative example is also provided.
In this paper, we propose and study several strongly convergent versions of the forward-reflected-backward splitting method of Malitsky and Tam for finding a zero of the sum of two monotone operators in a real Hilbert space. Our proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. We also develop inertial versions of our methods and strong convergence results are obtained for these metho…
We propose two very simple methods, the first one with constant step sizes and the second one with self-adaptive step sizes, for finding a zero of the sum of two monotone operators in real reflexive Banach spaces. Our methods require only one evaluation of the single-valued operator at each iteration. Weak convergence results are obtained when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous, and strong convergence results are obtained when either one of these two operators is required, in addition, to be strongly monotone. We also obtain the r…
This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we derive two sets of stability conditions based on two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and…
Splitting Algorithms Using Pairs Monotonicity to Solve the Sum of Two Non-monotone Operators Problem
Ton Duc Thang University（胡志明市镏德胜大学）；Sidi Mohammed Ben Abdellah University, National School of Applied Sciences（西迪穆罕默德·本阿卜达拉大学国立应用科学学院）；University of Limoges（利摩日大学）
We consider monotone inclusion problems where the operators may be expectation-valued, a class of problems that subsumes convex stochastic optimization problems as well as subclasses of stochastic variational inequality and equilibrium problems. A direct application of splitting schemes is complicated by the need to resolve problems with expectation-valued maps at each step, a concern that is addressed by using sampling. Accordingly, we propose an avenue for addressing uncertainty in the mapping: Variance-reduced stochastic modified forward-backward splitting scheme (vr-SMFBS). In constrained …
This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in the modeling of certain physical systems. The differential inclusion is described by a time-dependent set-valued mapping having the property that, for a given time instant, the set-valued mapping describes a maximal monotone operator. Under certain mild assumptions on the regularity with respect to the time argument, we construct a sequence of functions parameterized by the sampling time that corresponds to the discretization of the continuous-time system. Using appropriate tools…
Non-Lipschitz Inertial Contraction-Type Method for Monotone Variational Inclusion problems
VIT Bhopal University（维特博帕尔大学）；Aligarh Muslim University（阿利加尔穆斯林大学）；China Medical University Hospital, China Medical University（中国医药大学附设医院，中国医药大学）；Academy of Romanian Scientists（罗马尼亚科学家学院）；Tiangong University（天津工业大学）
