Monotone Mappings Research Articles

Inertial Bilevel Variational Monotone Inclusion Problem in Banach Spaces

In this paper, we study accelerated Halpern-type iterative method for finding zero solution of sum of two monotone operators which solves variational inequality problem of inverse strongly monotone mapping in 2-uniformly convex and uniformly smooth real Banach spaces. The strong convergence of our proposed method is establish under some standard conditions imposed on parameters. Our theorems generalize many recently announced results in the literature.

Optimal hedging with variational preferences under convex risk measures

We expose a theoretical hedging optimization framework with variational preferences under convex risk measures. We explore a general dual representation for the composition between risk measures and utilities. We study the properties of the optimization problem as a convex and monotone map per se. We also derive results for optimality and indifference pricing conditions. We also explore particular examples inside our setup.

Continuity of extensions of Lipschitz maps and of monotone maps

AbstractLet be a subset of a Hilbert space. We prove that if is such that for all and all non‐negative that add up to one, then for any 1‐Lipschitz , with , there exists a 1‐Lipschitz extension of such that the uniform distance on between and is the same as the uniform distance on between and .Moreover, if either or is convex, we prove the converse: We show that a map that allows for a 1‐Lipschitz, uniform distance preserving extension of any 1‐Lipschitz map on a subset of also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite‐dimensional spaces.

A geometrical solution underlies general neural principle for serial ordering

A general mathematical description of how the brain sequentially encodes knowledge remains elusive. We propose a linear solution for serial learning tasks, based on the concept of mixed selectivity in high-dimensional neural state spaces. In our framework, neural representations of items in a sequence are projected along a “geometric” mental line learned through classical conditioning. The model successfully solves serial position tasks and explains behaviors observed in humans and animals during transitive inference tasks amidst noisy sensory input and stochastic neural activity. This approach extends to recurrent neural networks performing motor decision tasks, where the same geometric mental line correlates with motor plans and modulates network activity according to the symbolic distance between items. Serial ordering is thus predicted to emerge as a monotonic mapping between sensory input and behavioral output, highlighting a possible pivotal role for motor-related associative cortices in transitive inference tasks.

On Banach Contraction Principle for Generalized Fuzzy Variational Inequality Problems

The main objective of the paper is to study the existence theorem of the newly defined generalized fuzzy ψ-variational inequality problem, generalized fuzzy ψ-complementarity problem. Uniqueness of the problems are also studied. Existence of these problems are established with the help of unique point condition of η. The existence of the solution of generalized fuzzy ψ-variational inequality problems is studied with the help of Banach contraction principle. The existence of the solution of generalized fuzzy ψ-complementarity problem is established in η-invex space.Conclusion: Equality of the solution sets of generalized fuzzy ψ-variational inequality problem and generalized fuzzy ψ-complementarity problem is shown using unique point condition of the bifunction η. Unique of the solution of generalized fuzzy ψ-variational inequality problem and generalized fuzzy ψ-complementarity problem is studied using strictly monotonicity condition. Existence theorems of the solution of generalized fuzzy ψ-variational inequality problem are shown using Banach contraction principle in the presence of Lipschitz continuous, strongly monotone mappings T_ψ and weakly monotonicity of η. Existence theorems of the solution of generalized fuzzy ψ-complementarity problem in the presence of condition C_0.

Parallel inertial forward–backward splitting methods for solving variational inequality problems with variational inclusion constraints

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm.

Common fixed point and common coupled fixed point theorems for weakly monotone mappings

In this paper we have established two common fixed point theorems which are variants of the fixed point theorem of Boyd and Wong [On nonlinear contractions, Proc. Amer. Math. Soc. 20(1969), 458-464]. By applying the newly established fixed point theorems two common coupled fixed point theorems have been proved for a pair of weakly increasing mappings. Examples are given to substantiate our theorems.

Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense

In this work, we establish the closedness and convexity of the set of fixed points of equally continuous and asymptotically demicontractive mapping in the intermediate sense. We proposed an inertial hybrid projection technique for determining an approximate common solution to three significant problems. The first is the system of generalized mixed equilibrium problems with relaxed monotone mappings, the second is the problem of fixed points of a countable family of equally continuous and asymptotically demicontractive mappings in the intermediate sense, and the third is of determining a point in a null space of a countable family of inverse strongly monotone mappings in Hilbert space. Based on these problems, we formulate a theorem and establish its strong convergence to their common solution. Additionally, we studied the applications of our algorithm to variational inequality problems and convex optimization problems. Finally, we numerically demonstrate the efficiency and robustness of our scheme. Several results available in the literature can be obtained as special cases of our result.

Percentile Risk-Constrained Budget Pacing for Guaranteed Display Advertising in Online Optimization

Guaranteed display (GD) advertising is a critical component of advertising since it provides publishers with stable revenue and enables advertisers to target specific audiences with guaranteed impressions. However, smooth pacing control for online ad delivery presents a challenge due to significant budget disparities, user arrival distribution drift, and dynamic change between supply and demand. This paper presents robust risk-constrained pacing (RCPacing) that utilizes Lagrangian dual multipliers to fine-tune probabilistic throttling through monotonic mapping functions within the percentile space of impression performance distribution. RCPacing combines distribution drift resilience and compatibility with guaranteed allocation mechanism, enabling us to provide near-optimal online services. We also show that RCPacing achieves O(sqrt(T)) dynamic regret where T is the length of the horizon. RCPacing's effectiveness is validated through offline evaluations and online A/B testing conducted on Taobao brand advertising platform.

New existence results for evolutionary quasi-hemi-variational inequalities and their optimal control

This paper investigates evolutionary quasi-variational inequalities that involve multi-valued monotone and pseudo-monotone maps. The purpose of including the pseudo-monotone map is to encompass generalized derivatives of certain nonconvex locally Lipschitz functionals as a specific instance. Consequently, the considered evolutionary quasi-variational inequalities subsume evolutionary quasi-hemi-variational inequalities as a particular case. We present an existence result wherein the key novelty lies in the variational selection that separates the monotone and pseudo-monotone components, thus eliminating the conventional assumption that their sum is monotone. While we continue to employ the Minty formulation, its necessity is limited to evolutionary monotone variational inequalities alone. Furthermore, we offer new applications to optimal control problems governed by evolutionary quasi-variational inequalities.

Fixed points of $G$-monotone mappings in metric and modular spaces

Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.

A Modified Form of Inertial Viscosity Projection Methods for Variational Inequality and Fixed Point Problems

This paper aims to introduce an iterative algorithm based on an inertial technique that uses the minimum number of projections onto a nonempty, closed, and convex set. We show that the algorithm generates a sequence that converges strongly to the common solution of a variational inequality involving inverse strongly monotone mapping and fixed point problems for a countable family of nonexpansive mappings in the setting of real Hilbert space. Numerical experiments are also presented to discuss the advantages of using our algorithm over earlier established algorithms. Moreover, we solve a real-life signal recovery problem via a minimization problem to demonstrate our algorithm’s practicality.

Strong convergent algorithm for finding minimum-norm solutions of quasimonotone variational inequalities with fixed point constraint and application

The class of quasimonotone mappings are known to be more general and applicable than the classes of pseudomonotone and monotone mappings. However, only very few results can be found in the literature on quasimonotone variational inequality problems and most of these results are on weak convergent algorithms. In this paper, we study the quasimonotone variational inequality problem (VIP) with constraint of fixed point problem (FPP) of quasi-pseudocontractive mappings. We introduce a new inertial Tseng’s extragradient method with self-adaptive step size for approximating the minimum-norm solutions of the aforementioned problem in the framework of Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a common (minimum-norm) solution of the quasimonotone VIP and FPP of quasi-pseudocontractive mappings without the knowledge of the Lipschitz constant of the cost operator. We provide several numerical experiments for the proposed method in comparison with existing methods in the literature. Finally, we applied our result to image restoration problem. Our result improves, extends and generalizes several of the recently announced results in this direction.

Forcing minimal patterns of triods

Rotation numbers for some maps of triods were introduced in [9]. The goal of this paper is to study patterns of triods which don't force other patterns with the same rotation number which we name triod twists. We obtain their complete characterization and show that these patterns can be conjugated to circle rotations by a piecewise monotone map. We also describe the dynamics of unimodal triod twist patterns with a given rational rotation number.

СТИСНЕННЯ ПРИРОДНОМОВНИХ ТЕКСТІВ РЕВЕРСНИМИ МУЛЬТИРОЗДІЛЬНИКОВИМИ КОДАМИ

We study a class of binary reverse multi-delimiter (RMD) data compression codes in application to natural language text compression. The RMD-codewords start with delimiters, i.e., prefixes of the form that cannot occur in other places of the codeword. The position of the delimiter in an RMD codeword differs from its position in “direct” multi-delimiter (MD) codes, where delimiters are codeword suffixes. RMD and MD codes possess many useful properties, such as unique decodability, completeness, universality, synchronizability, asymptotic densities, and finite automaton acceptability. For RMD-codes, we construct a monotonic mapping from the set of natural numbers to the set of codewords. For original MD-codes, hitherto, this was an open question. The discovered mapping and the byte quantification of a decoding automaton allow us to develop very fast byte-aligned algorithms for decoding and direct Boyer–Moore style search in compressed files. Compared with the known byte (SCDC) and Fibonacci codes, RMD codes demonstrate the best compression ratio on natural language texts (more than four times closer to the entropy bound than that of SCDC). Computer experiments demonstrate that RMD codes can be decoded almost as fast as SCDC and times faster than Fibonacci codes. In natural language text compression, we also practiced the RMD-encoding as a preprocessing tool, which improves the performance of the known modern powerful archivers.

Bhaskar-Lakshmikantham fixed point theorem vs Ran-Reunrings one and some possible generalizations and applications in matrix equations

<p>We provided a generalization of the existence and uniqueness of fixed points in partially ordered metric spaces for a monotone map. We applied the major results in the investigation of coupled fixed points for ordered pairs of two maps that met various monotone features, which included a mixed monotone property or a total monotone property. To ascertain necessary requirements for the existence and uniqueness of solutions to systems of matrix equations, the results regarding coupled fixed points for ordered pairs of maps were utilized. These results are illustrated with numerical examples. Some of the known results are a consequence of the results we obtained.</p>

Wiener Tauberian theorem and half-space problems for parabolic and elliptic equations

<abstract><p>For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem.</p></abstract>

APPROXIMATION METHOD FOR COMMON FIXED POINT OF A COUNTABLE FAMILY OF MULTI-VALUED QUASI-ϕ-NONEXPANSIVE MAPPINGS IN BANACH SPACES

In this paper, we study a new iterative algorithm to approximate a common fixed point of acountable family of multi-valued quasi-φ nonexpansive mappings in a uniformly smooth and uniformlyconvex real Banach space. We prove strong convergence theorem which improve and generalize recentlyannounced results in the literature. We also give some example to illustrate our theorem and we alsogive application to zero point problem of maximal monotone mappings

Density of periodic measures and large deviation principle for generalised mod one transformations

We introduce a new class of piecewise monotonic maps, called generalised mod one transformations, which include all (α,β) and generalised β -transformations, and prove that all transitive generalised mod one transformations with positive topological entropy satisfy the level-2 large deviation principle with a unique measure of maximal entropy. This is obtained by our result on the density of periodic measures for more general class of piecewise mototonic maps, which is a partial answer to the open problem posed by Hofbauer and Raith (1998 Fund. Math. 157 221–34); Raith (www.math.iupui.edu/~mmisiure/open/PR1.pdf).

An Incremental Gradient Method for Optimization Problems With Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive non-asymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. The proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.