Strongly Monotone Research Articles

In defense of value incomparability: A reply to Dorr, Nebel, and Zuehl

AbstractCian Dorr, Jacob Nebel, and Jake Zuehl have argued that no objects are incomparable in value. One set of arguments they offer depart from a principle they call ‘Strong Monotonicity’, which states that if x is good and y is not good, then x is better than y. In this article, we respond to those arguments, thereby defending the possibility of value incomparability.

Generalized Nash Equilibrium Seeking for Noncooperative Game With Different Monotonicities by Adaptive Neurodynamic Algorithm.

This article proposes a novel adaptive neurodynamic algorithm (ANA) to seek generalized Nash equilibrium (GNE) of the noncooperative constrained game with different monotone conditions. In the ANA, the adaptive penalty term, which acts as trajectory-dependent penalty parameters, evolves based on the degree of constraints violation until the trajectory enters the action set of noncooperative game. It is shown that the trajectory of the ANA enters the action set in finite time benefited from the adaptive penalty term. Moreover, it is proven that the trajectory exponentially (or polynomially) converges to the unique GNE when the pseudo-gradient of cost function in noncooperative game satisfies strong (or "generalized" strong) monotonicity. To the best of our knowledge, this is the first time to study the polynomial convergence of GNE seeking algorithm. Furthermore, when the pseudo-gradient mentioned above satisfies monotonicity in general, based on Tikhonov regularization method, a new ANA for finding its ε -generalized Nash equilibrium ( ε -GNE) is proposed, and the related exponential convergence of the algorithm is established. Finally, the river basin pollution game and 5G base station location game are given as examples to showcase the algorithm's effectiveness.

The standard symmetrized skew information and its applications

In this paper, we propose two new measures: one is symmetrized skew information and the other one is standard symmetrized skew information. Firstly, we prove their properties, such as non-negativity, convexity, invariance, additivity, monotonicity and strong monotonicity. Next, we conduct research on relationships between standard symmetrized skew information and several well-known measures in one-qubit state, aiming to compare their similarities and differences. In addition, standard symmetrized skew information is used to study quantum uncertainty. We also give the definition of standard symmetrized skew information of assistance, and provide it a straightforward operational explanation for better understanding. Finally, standard symmetrized skew information can be applied to detect entanglement.

Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

Feasible Online Learning with Gradient Feedback Online gradient descent (OGD) is well-known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of [Formula: see text] for strongly convex cost functions, and (2) in the multiagent setting of strongly monotone games with each agent employing OGD we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of [Formula: see text]. Whereas these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In “Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback,” M. Jordan, T. Lin, and Z. Zhou design a fully adaptive OGD algorithm, AdaOGD, that does not require a priori knowledge of these parameters. In the single-agent setting, the algorithm achieves [Formula: see text] regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs AdaOGD in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of [Formula: see text], again optimal up to log factors. The algorithms are illustrated in a learning version of the classic newsvendor problem, in which, because of lost sales, only (noisy) gradient feedback can be observed. The results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multiretailer settings.

Quantum G-coherence factorization law under fully and strictly incoherent operations

Quantum coherence plays an important role in quantum resource theory, which is strongly related with entanglement. In order to quantify the full coherence of qudit states, we define G-coherence and convex roof of G-coherence, and prove that the G-coherence is a strong coherence monotone and the convex roof of G-coherence is a coherence measure under fully and strictly incoherent operation (FSIO), respectively. Similar to the entanglement factorization law, we prove a coherence factorization law for arbitrary d-dimensional quantum pure and mixed states under FSIO channels, which generalizes the entanglement factorization law for bipartite pure states. Our results will play an important role in the simplification of dynamical coherence measure.

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.

Stochastic Bregman extragradient algorithm with line search for stochastic mixed variational inequalities

In this paper, we present a stochastic Bregman extragradient algorithm with line search for solving a class of stochastic mixed variational inequality, which not only does not require any information of the Lipschitz constant but allows to search a potentially larger step size per iteration. Compared with the existing algorithms, the proposed algorithm allows different step sizes in the prediction and correction steps, thus enhancing the algorithm's flexibility. Under the generalized monotonicity, we derive the almost sure convergence, the iteration complexity O ( 1 / ϵ ) and the oracle complexity O ( 1 / ϵ 2 ) for our algorithm. Furthermore, under the generalized strong monotonicity and the sample size increases at a geometric rate, the linear convergence rate of the proposed algorithm with respect to the Bregman distance between the iterations and solution is established. Numerical results demonstrate a favourable comparison of the proposed algorithm with existing ones.

Novel Stability Conditions for Nonlinear Monotone Systems and Consensus in Multiagent Networks

We introduce a novel definition of monotonicity, termed “type-K” in honor of Kamke, and study nonlinear type-K monotone dynamical systems possessing the plus-subhomogeneity property, which we call “K-subtopical” systems after Gunawardena and Keane. We show that type-K monotonicity, which is weaker than strong monotonicity, is also equivalent to monotonicity for smooth systems evolving in continuous-time, but not in discrete-time. K-subtopical systems are proved to converge toward equilibrium points, if any exists, generalizing the result of Angeli and Sontag about convergence of topical systems' trajectories toward the unique equilibrium point when strong monotonicity is considered. The theory provides an new methodology to study the consensus problem in nonlinear multi-agent systems (MASs). Necessary and sufficient conditions on the local interaction rule of the agents ensuring the K-subtopicality of MASs are provided, and consensus is proven to be achieved asymptotically by the agents under given connectivity assumptions on directed graphs. Examples in continuous-time and discrete-time corroborate the relevance of our results in different applications

Poisson stable solutions for stochastic PDEs driven by Lévy noise

This paper is mainly concerned with the existence, uniqueness and Poisson stability (including stationarity, periodicity, almost periodicity and almost automorphy) of solutions for a class of stochastic partial differential equations driven by Lévy noise, where the involved coefficients are assumed to be strictly monotone. Based on the variational method, we establish the well-posedness of L2-bounded solution and then prove that it has the same characters of periodicity, almost periodicity and almost automorphy as the coefficients of the equation. Moreover, we also investigate the additive noise case under strong monotone condition. In particular, we illustrate our results by applying to concrete models such as stochastic reaction-diffusion equations, stochastic porous media equations and stochastic p-Laplace equations.

Cross invariance, the Shapley value, and the Shapley–Shubik power index

In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.

Hybrid Nash Equilibrium Seeking Under Partial-Decision Information: An Adaptive Dynamic Event-Triggered Approach

This paper is concerned with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hybrid</i> Nash equilibrium (NE) seeking problem over a network in a partial-decision information scenario. Each agent has access to both its own cost function and local decision information of its neighbors. First, an adaptive gradient-based algorithm is constructed in a fully distributed manner with the guaranteed convergence to the NE, where the network communication is required. Second, in order to save communication cost, a novel event-triggered scheme, namely, edge-based adaptive dynamic event-triggered (E-ADET) scheme, is proposed with on-line-tuned triggering parameter and threshold, and such a scheme is proven to be fully distributed and free of Zeno behavior. Then, a hybrid NE seeking algorithm, which is also fully distributed, is constructed under the E-ADET scheme. By means of the Lipschitz continuity and the strong monotonicity of the pseudo-gradient mapping, we show the convergence of the proposed algorithms to the NE. Compared with the existing distributed algorithms, our algorithms remove the requirement on global information, thereby exhibiting the merits of both flexibility and scalability. Finally, two examples are provided to validate the proposed NE seeking methods.

Analysis of the single reference coupled cluster method for electronic structure calculations: the full-coupled cluster equations

The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an L^2-type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.

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&amp;#x2264;n&amp;#x2264;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&amp;#x2265;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.

A generalized proximal point algorithm with new step size update for solving monotone variational inequalities in real Hilbert spaces

The aim of this paper is to study a generalized version of proximal method with a new step size update for solving variational inequality problem in Hilbert spaces. Weak convergence of the proposed scheme is attained under some appropriate assumptions imposed on the operators and parameters. At the same time, we set up an R-linear convergence rate of our method when the operator is strong monotone. Numerical experiments are given to demonstrate the efficacy of our proposed iterative algorithm. Several versions of recently proposed methods in the literature are special cases of our method.

Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

In this paper, a novel modified proximal dynamical system is proposed to compute the solution of a mixed variational inequality problem (MVIP) within a fixed time, where the time of convergence is finite and is uniformly bounded for all initial conditions. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that a solution of the modified proximal dynamical system exists, is uniquely determined, and converges to the unique solution of the associated MVIP within a fixed time. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumption of strong monotonicity is relaxed to that of strong pseudomonotonicity. Finally, it is shown that the solution obtained using the forward-Euler discretization of the proposed modified proximal dynamical system converges to an arbitrarily small neighborhood of the solution of the associated MVIP within a fixed number of time steps, independent of the initial conditions.

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.

Tikhonov regularized iterative methods for nonlinear problems

ABSTRACT We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward–backward-type algorithm and a Douglas–Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward–backward-type primal–dual algorithm and a Douglas–Rachford-type primal–dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.

Some Properties of Interval Shapley Values: An Axiomatic Analysis

Interval games are an extension of cooperative coalitional games, in which players are assumed to face payoff uncertainty. Characteristic functions thus assign a closed interval instead of a real number. This study revisits two interval game versions of Shapley values (i.e., the interval Shapley value and the interval Shapley-like value) and characterizes them using an axiomatic approach. For the interval Shapley value, we show that the existing axiomatization can be generalized to a wider subclass of interval games called size monotonic games. For the interval Shapley-like value, we show that a standard axiomatization using Young’s strong monotonicity holds on the whole class of interval games.

Parameterized coherence measure

Quantifying coherence is an essential endeavor for both quantum mechanical foundations and quantum technologies. We present a bona fide measure of quantum coherence by utilizing the Tsallis relative operator (α,β)-entropy. We first prove that the proposed coherence measure fulfills all the criteria of a well defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. We then study the ordering of the Tsallis relative operator (α,β)-entropy of coherence, Tsallis relative α-entropies of coherence, Rényi α-entropy of coherence and l1 norm of coherence for both pure and mixed qubit states. This provides a new method for defining new coherence measure and entanglement measure, and also provides a new idea for further study of quantum coherence.

Distributed Nash Equilibrium Seeking for Aggregative Games With Quantization Constraints

The problem of seeking Nash equilibrium (NE) based on aggregative games under quantization constraints is full of challenges. Although the NE seeking algorithm in continuous-time systems has been studied, this problem in discrete-time systems still needs to be solved urgently. To address this problem, three distributed algorithms are first proposed under three quantization cases, adaptive, random, and time-varying quantizations, based on doubly stochastic communication topology networks. Then, the actions of players would eventually converge to NE under the conditions of vanishing step size and strong monotonicity are proved. Moreover, the convergence rate of the three quantization cases are analyzed, respectively. Finally, numerical experiments are implemented on plug-in hybrid electric vehicles (PHEVs) to validate the effectiveness of the proposed distributed algorithms. Comparing the convergence rates of the three proposed algorithms, the convergence effect of the adaptive quantization is better than that of the other two quantization cases.