Search

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem.

The problem of realization of a monotone response map as a monotone initialized system on a finite dimensional Euclidean state space is investigated. The dynamical part of the system is described by a delta differential equation on an arbitrary time scale. This incorporates continuous- and discrete-time systems. The main result states necessary and sufficient conditions for existence of a monotone realization of a particular class. The criterion is expressed in the language of skew differential global universes.

In this paper, we prove a strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the set of solutions of a finite family of variational inclusion problems in Hilbert space. A strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the solution sets of a finite family of zero points of the maximal monotone operator problem in Hilbert space is also obtained. Using our main result, we have some additional results for various types of non-linear problems in Hilbert space.

In this paper, some proximal point algorithms (PPAs) for maximal monotone operators in Banach spaces are considered. We obtain some results on the boundedness and the convergence of sequences generated by the PPAs with some assumptions. We can also get zero of maximal -expansive operator and maximal -monotone at zero operator using this method and then we apply it for finding a solution of the equilibrium problem.

In this paper, we introduce and study the generalized -D-gap function for an equilibrium problem. We give sufficient conditions under which a critical point of the generalized -D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized -D-gap function, we construct an error bound for the equilibrium problem. We support our approach with concrete examples.

In this paper, we introduce two iterative algorithms for finding the solution of the sum of two monotone operators by using hybrid projection methods and shrinking projection methods. Under some suitable conditions, we prove strong convergence theorems of such sequences to the solution of the sum of an inverse-strongly monotone and a maximal monotone operator. Finally, we present a numerical result of our algorithm which is defined by the hybrid method.

By weakening the neutral element condition of semiuninorms, we introduce a new concept called weak-neutral semiuninorms (shortly, wn-semiuninorms). After analyzing their structure, several classes of wn-semiuninorms are presented and discussed. Particularly, based on a kind of monotone unary functions which are not necessarily continuous and strictly monotone, we introduce representable wn-semiuninorms and discuss some of their properties in detail. We show that there is no idempotent proper wn-semiuninorm. Each representable wn-semiuninorm is Archimidean but not strictly monotone, and its additive generator is unique up to a positive multiplicative constant under some conditions. In the discussion about the representable wn-semiuninorms, we also characterize the solutions to a class of Cauchy functional equations on a restricted domain.

In this paper, we introduce and study a new iterative method which is based on viscosity general algorithm and forward-backward splitting method for finding a common element of the set of common fixed points of multivalued demicontractive and quasi-nonexpansive mappings and the set of solutions of variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings in real Hilbert spaces. We prove that the sequence $x_n$ which is generated by the proposed iterative algorithm converges strongly to a common element of two sets above. Finally, our theorems are applied to approximate a common solution of fixed point problems with set-valued operators and the composite convex minimization problem. Our theorems are significant improvements on several important recent results.

We introduce a novel class of Nash equilibrium seeking dynamics for non-cooperative games with a finite number of players, where the convergence to the Nash equilibrium is bounded by a KL function with a settling time that can be upper bounded by a positive constant that is independent of the initial conditions of the players, and which can be prescribed a priori by the system designer. The dynamics are model-free, in the sense that the mathematical forms of the cost functions of the players are unknown. Instead, in order to update its own action, each player needs to have access only to real-time evaluations of its own cost, as well as to auxiliary states of neighboring players characterized by a communication graph. Stability and convergence properties are established for both potential games and strongly monotone games. Numerical examples are presented to illustrate our theoretical results., Comment: Presented at the IEEE Conference on Decision and Control, on Dec. 14-18, 2020

This paper studies an N-coalition non-cooperative game problem, where the players in the same coalition cooperatively minimize the sum of their local cost functions under a directed communication graph, while collectively acting as a virtual player to play a non-cooperative game with other coalitions. Moreover, it is assumed that the players have no access to the explicit functional form but only the function value of their local costs. To solve the problem, a discrete-time gradient-free Nash equilibrium seeking strategy, based on the gradient tracking method, is proposed. Specifically, a gradient estimator is developed locally based on Gaussian smoothing to estimate the partial gradients, and a gradient tracker is constructed locally to trace the average sum of the partial gradients among the players within the coalition. With a sufficiently small constant step-size, we show that all players’ actions approximately converge to the Nash equilibrium at a geometric rate under a strongly monotone game mapping condition. Numerical simulations are conducted to verify the effectiveness of the proposed algorithm. Ministry of Education (MOE) This research was supported by Singapore Ministry of Education Academic Research Fund Tier 1 RG180/17(2017-T1-002-158).

We present a coupling of the Finite Element and the Boundary Element Method in an isogeometric framework to approximate either two-dimensional Laplace interface problems or boundary value problems consisting of two disjoint domains. We consider the Finite Element Method in the bounded domains to simulate possibly non-linear materials. The Boundary Element Method is applied in unbounded or thin domains where the material behavior is linear. The isogeometric framework allows to combine different design and analysis tools: first, we consider the same type of NURBS parameterizations for an exact geometry representation and second, we use the numerical analysis for the Galerkin approximation. Moreover, it facilitates to perform h- and p-refinements. For the sake of analysis, we consider the framework of strongly monotone and Lipschitz continuous operators to ensure well-posedness of the coupled system. Furthermore, we provide a priori error estimates. We additionally show an improved convergence behavior for the errors in functionals of the solution that may double the rate under certain assumptions. Numerical examples conclude the work which illustrate the theoretical results.

In this article we consider the a posteriori error analysis of hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of strongly monotone type. In particular, we employ and analyze a practical solution scheme based on exploiting a discrete Kačanov iterative linearization. The resulting a posteriori error bound explicitly takes into account the three sources of error: discretization, linearization, and linear solver errors. Numerical experiments are presented to demonstrate the practical performance of the proposed hp-adaptive refinement strategy.

We prove that if a non-transferable utility (NTU) game is cardinally balanced and if, at every individually rational and efficient payoff vector, every non-zero normal vector to the set of payoff vectors feasible for the grand coalition is strictly positive, then the inner core is nonempty. The condition on normal vectors is satisfied if the set of payoff vectors feasible for the grand coalition is non-leveled. An NTU game generated by an exchange economy where every consumer has a continuous, concave, and strongly monotone utility function satisfies our sufficient condition. Our proof relies on Qin’s theorem on the nonemptiness of the inner core.

A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work [16] that covers some prominent procedures (including the Zarantonello, Kačanov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main purpose of this paper is the derivation of an abstract convergence theory for the unified ILG approach (based on general adaptive Galerkin discretization methods) proposed in [16]. The theoretical results will be tested and compared for the aforementioned three iterative linearization schemes in the context of adaptive finite element discretizations of strongly monotone stationary conservation laws.

In this paper, we consider the problem of learning a generalized Nash equilibrium (GNE) in strongly monotone games. First, we propose semi-decentralized and distributed continuous-time solution algorithms that use regular projections and first-order information to compute a GNE with and without a central coordinator. As the second main contribution, we design a data-driven variant of the former semi-decentralized algorithm where each agent estimates their individual pseudogradient via zeroth-order information, namely, measurements of their individual cost function values, as typical of extremum seeking control. Third, we generalize our setup and results for multi-agent systems with nonlinear dynamics. Finally, we apply our methods to connectivity control in robotic sensor networks and almost-decentralized wind farm optimization.

We consider a stochastic variational inequality (SVI) problem with a continuous and monotone mapping over a closed and convex set. In strongly monotone regimes, we present a variable sample-size averaging scheme (VS-Ave) that achieves a linear rate with an optimal oracle complexity. In addition, the iteration complexity is shown to display a muted dependence on the condition number compared with standard variance-reduced projection schemes. To contend with merely monotone maps, we develop amongst the first proximal-point algorithms with variable sample-sizes (PPAWSS), where increasingly accurate solutions of strongly monotone SVIs are obtained via (VS-Ave) at every step. This allows for achieving a sublinear convergence rate that matches that obtained for deterministic monotone VIs. Preliminary numerical evidence suggests that the schemes compares well with competing schemes.

In this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which...

International audience; We discuss an approach to signal recovery in Generalized Linear Models (GLM) in which the signal estimation problem is reduced to the problem of solving a stochastic monotone Variational Inequality (VI). The solution to the stochastic VI can be found in a computationally efficient way, and in the case when the VI is strongly monotone we derive finite-time upper bounds on the expected $\|\cdot\|_2^2$ error converging to 0 at the rate $O(1/K)$ as the number $K$ of observation grows. Our structural assumptions are essentially weaker than those necessary to ensure convexity of the optimization problem resulting from Maximum Likelihood estimation. In hindsight, the approach we promote can be traced back directly to the ideas behind the Rosenblatt’s perceptron algorithm.

The present paper concerns a class of abstract mixed variational problems governed by a strongly monotone Lipschitz continuous operator. With the existence and uniqueness results in the literature for the problem under consideration, we prove a general convergence result, which shows the continuous dependence of the solution with respect to the data by using arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Then we consider an associated optimal control problem for which we prove the existence of optimal pairs. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to mixed variational problems. To provide an example, we illustrate our results in the study of a mathematical model which describes the equilibrium of an elastic body in frictional contact with a foundation.

The forward-backward splitting algorithm is a popular operator-splitting method for solving monotone inclusion of the sum of a maximal monotone operator and a cocoercive operator. In this paper, we present a new convergence analysis of a variable metric forward-backward splitting algorithm with extended relaxation parameters in real Hilbert spaces. We prove that this algorithm is weakly convergent when certain weak conditions are imposed upon the relaxation parameters. Consequently, we recover the forward-backward splitting algorithm with variable step sizes. As an application, we obtain a variable metric forward-backward splitting algorithm for solving the minimization problem of the sum of two convex functions, where one of them is differentiable with a Lipschitz continuous gradient. Furthermore, we discuss the applications of this algorithm to the fundamental of the variational inequalities problem, constrained convex minimization problem, and split feasibility problem. Numerical experimental results on LASSO problem in statistical learning demonstrate the effectiveness of the proposed iterative algorithm., 27 pages, 2 figures

In this paper, iterative algorithms of Krasnoselkii-type and of Halpern-type for approximating an element of the set of common zeros of a countable family of inverse strongly monotone maps, common fixed points of a countable family of totally quasi- $$\phi $$ -asymptotically nonexpansive nonself multi-valued maps, and a solution of a system of generalized mixed equilibrium problems are constructed and studied. Strong convergence of the sequences generated by these algorithms is established in uniformly smooth and 2-uniformly convex real Banach spaces. The theorems obtained extend, improve and generalize several recent important results.

Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.

In this paper, using Halpern type iteration, we prove a strong convergence theorem for finding a common element of the set of common fixed points for a finite family of demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we obtain well-known and new strong convergence theorems in a Hilbert space.

In this paper we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $C$. We assume that the set $C$ can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method the main idea of which is to project at each step onto a particular half-space constructed by using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on $F$. We provide several examples for the case where $C$ is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.

Let \begin{document}$V$\end{document} be a Banach space, \begin{document}$z'\in V'$\end{document} , and \begin{document}$\alpha: V\to {\mathcal P}(V')$\end{document} be a maximal monotone operator. A large number of phenomena can be modelled by inclusions of the form \begin{document}$\alpha(u) \ni z'$\end{document} , or by the associated flow \begin{document}$D_tu + \alpha(u) \ni z'$\end{document} . Fitzpatrick proved that there exists a lower semicontinuous, convex representative function \begin{document}$f_\alpha: V \!\times\! V'\to \mathbb{R}\cup \{+\infty\}$\end{document} such that 0.1 \begin{document}$f_\alpha(v,v') \ge \langle v',v\rangle\quad\;\forall (v,v'), \qquad\quadf_\alpha(v,v') = \langle v',v\rangle\;\;\Leftrightarrow\;\;\; v'\in \alpha(v).$\end{document} This provides a variational formulation for the above inclusions. Here we use this approach to prove two results of existence of a solution, without using the classical theory of maximal monotone operators. This is based on a minimax theorem, and on the duality theory of convex optimization.

In this paper we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems by using new fixed point results of mixed monotone operators on cones.

In this work, we extend the fixed point result of Kirk and Xu for asymptotic pointwise nonexpansive mappings in a uniformly convex Banach space to monotone mappings defined in a hyperbolic uniformly convex metric space endowed with a partial order.

In this paper, using Bregman functions, we introduce a new Halpern-type iterative algorithm for finding common zeros of finitely many maximal monotone operators and obtain a strongly convergent iterative sequence to the common zeros of these operators in a reflexive Banach space. Furthermore, we study Halpern-type iterative schemes for finding common solutions of a finite system of equilibrium problems and null spaces of a -inverse strongly monotone mapping in a 2-uniformly convex Banach space. Some applications of our results to the solution of equations of Hammerstein-type are presented. Our scheme has an advantage that we do not use any projection of a point on the intersection of closed and convex sets which creates some difficulties in a practical calculation of the iterative sequence. So the simple construction of Halpern iteration provides more flexibility in defining the algorithm parameters which is important from the numerical implementation perspective. Presented results improve and generalize many known results in the current literature.

A useful iteration method for the approximation of common fixed points of countable families of nonlinear mappings is applied, by which an iterative algorithm is presented for finding common elements of the set of solutions to a system of generalized mixed equilibrium problems, of null spaces of a countable family of $$\delta _i$$ -inverse strongly monotone mappings, and of the set of common fixed points of a countable family of totally quasi- $$\phi $$ -asymptotically nonexpansive mappings. A strong convergence theorem is established in the framework of two-uniformly convex and uniformly smooth real Banach spaces. Since there is no need to impose uniformity assumption on the involved mappings and no need to compute series at each step in the iteration process, the results improve and extend those announced by most authors with related researches. As application, an iterative solution to a system of nonlinear Hammerstein type equations is studied.

This paper first introduces the concept of a monotone set-valued measure and then focuses on the atoms and pseudo-atoms of the monotone set-valued measure space. Moreover, the paper proposes an integral, which is the integral of real-valued function with respect to the monotone set-valued measure, and shows some properties of this integral. Particularly, the paper gives the representations of integrals defined on atoms and pseudo-atoms. Some classical results are extended to the case of the monotone set-valued measure.

We present a unified approach to the monotone convergence theorem for nonlinear integrals such as the Choquet, the Sipos, the Sugeno, and the Shilkret integral. A nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function. We thus formulate our general type of monotone convergence theorem for such a functional. The key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional. Our approach is also applicable to the Lebesgue integral when a nonadditive measure is σ -additive.

Given a continuum X and a positive integer n, let F n ( X ) be the hyperspace of all nonempty subsets of X having at most n points. Given a mapping f : X → Y between continua, we study the induced mapping f n : F n ( X ) → F n ( Y ) given by f n ( A ) = f ( A ) (the image of A under f). In this paper, we prove some relationships among the mappings f and f n for the following classes of mappings: almost open, almost monotone, atriodic, feebly monotone, local homeomorphism, locally confluent, locally weakly confluent, strongly monotone and weakly semi-confluent.

In a Hilbert space we construct a regularized continuous analog of the Newton method for nonlinear equation with a Frechet differentiable and monotone operator. We obtain sufficient conditions of its strong convergence to the normal solution of the given equation under approximate assignment of the operator and the right-hand of the equation.

In this paper, we introduce the concept of monotone α-nonexpansive mappings in an ordered Banach space E with the partial order ź, which contains monotone nonexpansive mappings as special case. With the help of the Mann iteration, we show some existence theorems of fixed points of monotone α-nonexpansive mappings in uniformly convex ordered Banach space. Also, we prove some weak and strong convergence theorems of the Mann iteration for finding an order fixed point of monotone α-nonexpansive mappings under the condition lim sup n ź ∞ β n ( 1 - β n ) 0 or lim inf n ź ∞ β n ( 1 - β n ) 0 .

In Hilbert spaces, the inclusion problem with an arbitrary maximal monotone operator is considered. We prove that the nonemptiness of the solution set of the inclusion problem is equivalent to a coercivity condition. Moreover, a sufficient and necessary condition for the boundedness of the solution set is obtained.

In this paper, we revisit the numerical approach to some classical variational inequalities, with monotone and Lipschitz continuous mapping A, by means of a projected reflected gradient-type method. A main feature of the method is that it formally requires only one projection step onto the feasible set and one evaluation of the involved mapping per iteration. Contrary to what was done so far, we establish the convergence of the method in a more general setting that allows us to use varying step-sizes without any requirement of additional projections. A linear convergence rate is obtained, when A is assumed to be strongly monotone. Preliminary numerical experiments are also performed.

In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equation where A is a maximal monotone operator in a real Hilbert space H and {cn} and {θn} are positive real sequences. We do not assume A−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set of A being nonempty. These theorems provide new approximation results for zeroes of monotone operators, as well as significantly unify and extend previously known results by assuming much weaker conditions on the coefficients {cn} and {θn}.

In this paper, we investigate a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level problem and pseudomonotone mappings in the lower-level one. A strongly convergent algorithm for such a BSVIP is proposed and analyzed. In particular, a problem of finding the minimum-norm solution of a split pseudomonotone variational inequality problem is also studied. As a consequence, we get a strongly convergent algorithm for finding the minimum-norm solution to the split feasibility problem, which requires only two projections at each iteration step. An application to discrete optimal control problems is considered.

We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces. The algorithm is a combination between the gradient method and the Mann-Krasnosel’skii iterative scheme, where the projection can be computed onto each set separately rather than onto their intersection. Strong convergence is proved. Some special cases involving bilevel equilibrium problems with inverse strongly monotone variational inequality, monotone equilibrium constraints and maximal monotone inclusions are discussed. An illustrative example involving a system of integral equations is presented.

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function., arXiv admin note: text overlap with arXiv:1306.0352

In this article, we study the approximation of common zeros of non-self inverse strongly monotone operators defined on a closed convex subset C of a Hilbert space H. For a non-self family of operators, we introduce an iterative algorithm without relying on projections. Approximation of common fixed points for finite families of non-self strict pseudo-contractions in the sense of Browder-Petryshyn is also obtained. The novelty of our algorithm is that the coefficients are not given a priori and no assumptions are made on them, but they are constructed step by step in a natural way.

In this paper, we define and introduce some new concepts of the higher order strongly preinvex functions and higher order strongly monotone operators involving an arbitrary bifunction. Some new relationships among various concepts of higher order strongly preinvex functions have been established. We have shown that the optimality conditions for the preinvex functions can be characterized by class of higher order strongly variational-like inequalities, which appears to be new ones. As a novel applications of the higher order strongly preinvex functions, we have obtained the parallelogram-like laws for the uniformly Banach spaces. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

It is shown that if X is a real Banach space with dual X ⁎ and F : X → X ⁎ is a continuous gradient operator that is coercive in a certain sense and proper on closed bounded sets, then it is surjective. Use of the notion of measure of noncompactness enables sufficient conditions for properness to be given. These give rise to a surjectivity theorem for compact perturbations of strongly monotone maps and also facilitate discussion of a Dirichlet boundary-value problem involving the p-Laplacian.

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.

We consider incentive compatible mechanisms for a domain that is very close to the domain of scheduling $n$ unrelated machines: the single exception is that the valuation of just one machine is submodular. For the scheduling problem with such cost functions, we give a lower bound of $\Omega(\sqrt{n})$ on the approximation ratio of incentive compatible deterministic mechanisms. This is a strong information-theoretic impossibility result on the approximation ratio of mechanisms on relatively simple domains. The lower bound of the current work assumes no restriction on the mechanism side, but an expanded class of valuations, in contrast to previous general results on the Nisan-Ronen conjecture that hold for only special classes of mechanisms such as local, strongly monotone, and anonymous mechanisms. Our approach is based on a novel characterization of appropriately selected smaller instances that allows us to focus on particular type of algorithms (linear mechanisms), from which we extract a locality property that gives the lower bound.