The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Consequences on the calmness of the optimal value function, in the context of parametric optimization, are explored. Some specific tools for the analysis of the sufficient conditions, in the case of set-valued inclusion with concave data, are provided in a Banach space setting.
In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the…
In the present paper, several properties concerning generalized derivatives of multifunctions implicitly defined by set-valued inclusions are studied by techniques of variational analysis. Set-valued inclusions are problems formalizing the robust fulfilment of cone constraint systems, whose data are affected by a "crude knowledge" of uncertain elements, so they can not be casted in traditional generalized equations. The focus of this study in on the first-order behaviour of the solution mapping associated with a parameterized set-valued inclusion, starting with Lipschitzian properties and then…
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 a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monot…
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 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…
The paper studies coincidence points of parameterized set-valued mappings (multifunctions), which provide an extended framework to cover several important topics in variational analysis and optimization that include the existence of solutions of parameterized generalized equations, implicit function and fixed-point theorems, optimal value functions in parametric optimization, etc. Using the advanced machinery of variational analysis and generalized differentiation that furnishes complete characterizations of well-posedness properties of multifunctions, we establish a general theorem ensuring t…
Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions. We propose a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. The properties of warped resolvents are investigated and connections are made with existing notions. Abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also co…
The notion of weak cyclic monotonicity of set-valued maps generalizing the cyclic monotonicity is introduced. The existence of solutions of differential inclusions with compact, upper semi-continuous, not necessarily convex right-hand sides in R^n is proved for weakly cyclic monotone right-hand sides.
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）
The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth functions. The asymptotic analysis of the trajectories generated relies on Lyapunov theory, where the appropriate energy functional plays a decisive role. While the most part of the paper is related …
Starting from a finite family of continuously differentiable positive definite functions, we study conditions under which a function obtained by max-min combinations is a Lyapunov function, establishing stability for two kinds of nonlinear dynamical systems: a) Differential inclusions where the set-valued right-hand-side comprises the convex hull of a finite number of vector fields, and b) Autonomous switched systems with a state-dependent switching signal. We investigate generalized notions of directional derivatives for these max-min functions, and use them in deriving stability conditions w…
In this paper, a strong variant for multivalued mappings of the well-known property of openness at a linear rate is studied. Among other examples, a simply characterized class of closed convex processes between Banach spaces, which satisfies such a covering behaviour, is singled out. Equivalent reformulations of this property and its stability under Lipschitz perturbations are investigated in a metric space setting. Applications to the solvability of set-valued inclusions and to the exact penalization of optimization problems with set-inclusion constraints are discussed.
The ordinary differential equation $\dot{x}(t)=f(x(t)), \; t \geq 0 $, for $f$ measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the function $f$ with its Filippov regularization $F_{f}$ and consider the differential inclusion $\dot{x}(t)\in F_{f}(x(t))$ which always has a solution. It is interesting to know, inversely, when a set-valued map $Φ$ can be obtained as the Filippov regularization of a (single-valued, measurable) function. In this work we give a full characterization of such set-valued maps, hereby calle…
Douglas--Rachford for multioperator comonotone inclusions with applications to multiblock optimization
Center for Advanced Intelligence Project, RIKEN（RIKEN先进智能项目中心）；School of Science, RMIT University（皇家墨尔本理工大学理学院）；Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo（东京大学信息理工学研究科数学信息学专业）
University of Vienna（维也纳大学）；University of Southampton（南安普顿大学）
We introduce a relaxed inertial forward-backward-forward (RIFBF) splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitz continuous operator. This work aims to extend Tseng's forward-backward-forward method by both using inertial effects as well as relaxation parameters. We formulate first a second order dynamical system which approaches the solution set of the monotone inclusion problem to be solved and provide an asymptotic analysis for its trajectories. We provide for RIFBF, which follows by explicit time di…
In this paper, we propose a numerical approach for solving composite primal-dual monotone inclusions with a priori information. The underlying a priori information set is represented by the intersection of fixed point sets of a finite number of operators, and we propose and algorithm that activates the corresponding set by following a finite-valued random variable at each iteration. Our formulation is flexible and includes, for instance, deterministic and Bernoulli activations over cyclic schemes, and Kaczmarz-type random activations. The almost sure convergence of the algorithm is obtained by…
