Strong Monotonicity Condition Research Articles

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

Stabilizer entropies and nonstabilizerness monotones

We study different aspects of the stabilizer entropies (SEs) and compare them against known nonstabilizerness monotones such as the min-relative entropy and the robustness of magic. First, by means of explicit examples, we show that, for Rényi index 0≤n≤2, the SEs are not monotones with respect to stabilizer protocols which include computational-basis measurements, not even when restricting to pure states (while the question remains open for n≥2). Next, we show that, for any Rényi index, the SEs do not satisfy a strong monotonicity condition with respect to computational-basis measurements. We further study SEs in different classes of many-body states. We compare the SEs with other measures, either proving or providing numerical evidence for inequalities between them.Finally, we discuss exact or efficient tensor-network numerical methods to compute SEs of matrix-product states (MPSs) for large numbers of qubits. In addition to previously developed exact methods to compute the Rényi SEs, we also put forward a scheme based on perfect MPS sampling, allowing us to compute efficiently the von Neumann SE for large bond dimensions.

Well-posedness and global error bound for generalized mixed quasi-variational-hemivariational inequalities via regularized gap functions

The aim of this paper is to establish new results on a class of generalized mixed quasi-variational-hemivariational inequalities (GMQVHVI, for short) via regularized gap functions. First, we introduce the new regularized gap function of GMQVHVI. Then we establish the criterion of the Levitin-Polyak well-posedness for GMQVHVI under suitable conditions. Further, we provide the equivalence between the Levitin-Polyak well-posedness in the generalized sense for GMQVHVI and that for a quasi-optimization problem using the new regularized gap function. Finally, the global error bound for GMQVHVI in term of the regularized gap function is derived by employing some the properties of the Clarke generalized directional derivative and strong monotonicity conditions.

АЛГОРИТМ ЕКСТРАПОЛЯЦІЇ З МИНУЛОГО ДЛЯ ВАРІАЦІЙНИХ НЕРІВНОСТЕЙ В ГІЛЬБЕРТОВОМУ ПРОСТОРІ

The article considers variational inequalities with operators acting in a Hilbert space. For these problems, variants of the Extrapolation from the Past method have been proposed and studied. A sub-linear efficiency estimate for the gap function is proved. The strong convergence of the Extrapolation from the Past method for variational inequalities with uniformly monotone operators is proved. The linear rate of convergence of the Extrapolation from the Past method for variational inequalities with operators satisfying the generalized strong monotonicity condition is proved. An adaptive version of the algorithm is proposed. Regularized variants of the algorithm are proposed and theorems on their strong convergence are proved.

Global error bounds for mixed Quasi-Hemivariational inequality problems on Hadamard manifolds

ABSTRACT In this paper, we introduce and study a class of mixed quasi-hemivariational inequality problems on Hadamard manifolds (in short, (MQHIP)). Some regularized gap functions for (MQHIP) are proposed under suitable conditions. Finally, global error bounds for (MQHIP) in terms of regularized gap functions are derived by employing some strong monotonicity conditions and using the properties of Clarke's generalized directional derivative. The main results presented in this paper improve and generalize corresponding results in literature.

CONTRACTION-MAPPING ALGORITHM FOR THE EQUILIBRIUM PROBLEM OVER THE FIXED POINT SET OF A NONEXPANSIVE SEMIGROUP

In this paper, we consider the proximal mapping of a bifunction. Under the Lipschitz-type and the strong monotonicity conditions, we prove that the proximal mapping is contractive. Based on this result, we construct an iterative process for solving the equilibrium problem over the fixed point sets of a nonexpansive semigroup and prove a weak convergence theorem for this algorithm. Also, some preliminary numerical experiments and comparisons are presented.

Weak and strong convergences of the generalized penalty Forward–Forward and Forward–Backward splitting algorithms for solving bilevel hierarchical pseudomonotone equilibrium problems

ABSTRACTIn this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward–Forward algorithm (PFFSA) and the Forward–Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field.

Testing Read-Once Formula Satisfaction

We study the query complexity of testing for properties defined by read-once formulas, as instances of massively parametrized properties , and prove several testability and nontestability results. First, we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in ϵ, doubly exponential in the arity, and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm that outputs an approximation of the distance of the input from satisfying the property. For formulas only involving And/Or gates, we provide a more efficient test whose query complexity is only quasipolynomial in ϵ. On the other hand, we show that such testability results do not hold in general for formulas over non-Boolean alphabets. Specifically, we construct a property defined by a read-once arity 2 (non-Boolean) formula over an alphabet of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size. We also present such a formula over an alphabet of size 5 that also satisfies a strong monotonicity condition.

Characterizing the resolute part of monotonic social choice correspondences

The resolute domain of a social choice correspondence is the set of profiles at which only one alternative is selected. The Condorcet, plurality, approval voting, and maximin correspondences—restricted to their respective resolute domains—satisfy anonymity, neutrality, and a strong monotonicity condition, but are not characterized by those three conditions. We present examples illustrating this and provide additional conditions that do suffice for characterization.

On ergodic algorithms for equilibrium problems

In this paper, we present a new iteration method for solving monotone equilibrium problems. This new method is based on the ergodic iteration method Ronald and Bruck in (J Math Anal Appl 61:159---164, 1977) and the auxiliary problem principle Noor in (J Optim Theory Appl 122:371---386, 2004), but it includes the usage of symmetric and positive definite matrices. The proposed algorithm is very simple. Moreover, it simplifies the assumptions necessary in order to converge to the solution. Specifically, whereas previous methods require strong monotonicity and Lipschitz-type continuous conditions, our proposed method only requires weak monotonicity conditions. Applications to the generalized variational inequality problem and some numerical results are reported.

Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

In this paper we present the relaxed inertial proximal algorithm for Ky Fan minimax inequalities. Based on Opial lemma, we propose a weak convergence result to a solution of the problem by eliminating in the algorithm (RIPAFAN) the Browder–Halpern’s factor of contraction. We present after, a first result of strong convergence by adding a strong monotonicity condition. Secondly, we eliminate the strong monotonicity and add a Browder–Halpern’s contraction factor in the algorithm (RIPAFAN) and then ensure the strong convergence to a selected solution with respect to the contraction factor. Some examples are proposed. The first one concerns the convex minimization where the objective function is only controlled with a provided well conditioning. In the second one, we propose monotone set-valued variational inequalities. The last example deals with the problem of fixed point for a nonexpansive set-valued operator.

On the Characterization of Solution Sets of Smooth and Nonsmooth Convex Stochastic Nash Games

Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous-strategy sets. We examine whether similar statements, particularly pertaining to existence and uniqueness, may be made when player objectives are given by expectations. For instance, in deterministic regimes, a suitable coercivity condition associated with the gradient map is sufficient for the existence of a Nash equilibrium; in stochastic regimes, the application of this condition requires being able to analytically evaluate the expectation and its gradients. Our interest is in developing a framework that relies on the analysis of merely the integrands of the expectations; in the context of existence statement, we consider whether the satisfaction of a suitable coercivity condition in an almost-sure sense may lead to statements about the original stochastic Nash game. Notably, this condition also guarantees the existence of an equilibrium of the scenario-based Nash game. We consider a range of such statements for claiming the existence of stochastic Nash equilibria when payoff functions are both smooth and nonsmooth and when strategy sets are possibly coupled through a shared convex constraint. Notably the sufficiency conditions are less stringent, when one either imposes appropriate monotonicity requirements or requires that strategy sets be decoupled. Uniqueness, however, can be claimed by requiring that a strong monotonicity condition holds over a set of positive measure, rather than in an almost-sure sense. When strategy sets are coupled by shared convex expected-value constraints, a suitable regularity condition allows for claiming existence and uniqueness in the primal-dual space. We illustrate our approach by examining two extensions of stochastic Nash–Cournot games, of which the first allows for nonsmooth payoffs through the introduction of risk-measures, while the second allows for shared stochastic constraints.

Axiomatics for the Hirsch index and the Egghe index

The Hirsch index and the Egghe index are both numbers that synthesize a researcher's output. The h-index associated with researcher r is the maximum number h such that r has h papers with at least h citations each. The g-index is the maximum number g of papers by r such that the average number of citations of the g papers is at least g. Both indices are characterized in terms of four axioms. One identifies outputs deserving index at most one. A second one establishes a strong monotonicity condition. A third one requires the index to satisfy a property of subadditivity. The last one consists of a monotonicity condition, for the h-index, and an aggregate monotonicity condition, for the g-index.

Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems

The D-gap function approach has been adopted for solving variationalinequality problems. In this paper, we extend the approach forsolving equilibrium problems. From the theoretical point, we studythe convergence and global error bound of a D-gap function basedNewton method.   A general equilibrium problem is first formulated as an equivalentunconstrained minimization problem using a new D-gap function. Thenthe conditions of 'strict monotonicity' and 'strong monotonicity'for equilibrium problems are introduced. Under the strictmonotonicity condition, it is shown that a stationary point of theunconstrained minimization problem provides a solution to theoriginal equilibrium problem. Without the assumption of Lipschitzcontinuity, we further prove that strong monotonicity conditionguarantees the boundedness of the level sets of the new D-gapfunction and derive error bounds on the level sets. Combining thestrict monotonicity and strong monotonicity conditions, we show theexistence and uniqueness of a solution to the equilibrium problem,and establish the global convergence property of the proposedalgorithm with a global error bound.

A quasi-variational approach to a competitive economic equilibrium problem without strong monotonicity assumption

This paper is focused on the investigation of the Walrasian economic equilibrium problem involving utility functions. The equilibrium problem is here reformulated by means of a quasi-variational inequality problem. Our goal is to give an existence result without assuming strong monotonicity conditions. To this end, we make use of a perturbation procedure. In particular, we will consider suitable perturbed utility functions whose gradient satisfies a strong monotonicity condition and whose associated equilibrium problem admits a solution. Then, we will prove that the limit solution solves the unperturbed problem. We stress out that our result allows us to consider a wide class of utility functions in which the Walrasian equilibrium problem may be solved.

Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations

Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rates of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.

Descent method for nonsmooth variational inequalities

A descent method with a gap function is proposed for a finite-dimensional variational inequality with nonintegrable and nonsmooth mapping. The convergence of the method with line search is established under strong monotonicity conditions on the underlying mapping.

New convergence results of iterative methods for set-valued mixed variational inequalities

An iterative method for solving set-valued variational inequalities is considered and its convergence properties are studied under strong monotonicity and coercivity conditions. The results obtained in this paper include, as a special case, some known results in this field. Mathematics subject classification (1991): 49M45, 90C30, 65K10.

Unconstrained minimization approaches to nonlinear complementarity problems

The nonlinear complementarity problem can be reformulated as unconstrained minimization problems by introducing merit functions. Under some assumptions, the solution set of the nonlinear complementarity problem coincides with the set of local minima of the corresponding minimization problem. These results were presented by Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow. In this note, we generalize some results of Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow to the case where the considered function is only directionally differentiable. Some results are strengthened in the smooth case. For example, it is shown that the strong monotonicity condition can be replaced by the P-uniform property for ensuring a stationary point of the reformulated unconstrained minimization problems to be a solution of the nonlinear complementarity problem. We also present a descent algorithm for solving the nonlinear complementarity problem in the smooth case. Any accumulation point generated by this algorithm is proved to be a solution of the nonlinear complementarity under the monotonicity condition.

AUXILIARY PROBLEM PRINCIPLE EXTENDED TO MONOTONE VARIATIONAL INEQUALITIES


 
 
 In the present work, the use of the operator method of regularization in the sense of Tikhonov, which makes it possible to develop an iterative scheme via auxiliary problem principle, converging strongly towards the solution of multi- valued monotone variational inequality within the absence of strong monotonicity condition of involved operator, is sustained.