Strongly Monotone Research Articles

We develop a comprehensive dynamic Walrasian framework entirely in L∞ so that prices and allocations are essentially bounded, and market clearing holds pointwise almost everywhere. Utilities are allowed to be locally Lipschitz and quasi-concave; we employ Clarke subgradients to derive generalized quasi-variational inequalities (GQVIs). We endogenize inventories through a capital-accumulation constraint, leading to a differential QVI (dQVI). Existence is proved under either strong monotonicity or pseudo-monotonicity and coercivity. We establish Walras’ law, and the complementarity, stability, and sensitivity of the equilibrium correspondence in L2-metrics, incorporate time-discounting and uncertainty into Ω×[0,T], and present convergent numerical schemes (Rockafellar–Wets penalties and extragradient). Our results close the “in mean vs pointwise” gap noted in dynamic models and connect to modern decomposition approaches for QVIs.

Predictive maintenance of complex mechanical requires robust health monitoring capabilities that can generalize across diverse components and operating conditions. We present a novel component-agnostic framework that unifies Health Indicator (HI) generation and Remaining Useful Life (RUL) prediction through an integrated pipeline comprising: (1) advanced feature engineering, (2) unsupervised health baseline modelling, (3) monotonicity and trendability learning (4) probabilistic degradation detection with confidence-aware RUL estimation. We validate our framework on two distinct Industrial Case Studies: Firstly, Tool wear monitoring in CNC machine using vibration and spindle current data collected from the real production machine. Our framework achieves early degradation detection of tool life with RUL prediction within ±15% of actual failure time. Flank wear (VB) was measured as a standard parameter for evaluating tool wear. Secondly, bearing degradation assessment using the IMS bearing dataset. This validation demonstrates fault detection 40% earlier than traditional threshold methods with 90% confidence intervals. Both case studies show strong HI monotonicity (>85%) and reliable uncertainty quantification, establishing the foundation for scalable, explainable predictive maintenance solutions. The framework's component-agnostic design enables rapid deployment across heterogeneous assets without extensive reconfiguration, while its interpretable architecture facilitates root cause analysis and maintenance decision support. These results demonstrate significant advances in scalable, explainable predictive maintenance, offering practitioners a unified solution for diverse industrial health monitoring challenges.

Abstract We introduce a nonstabilizerness monotone which we name basis-minimized stabilizerness asymmetry (BMSA). It is based on the notion of G-asymmetry, a measure of how much a certain state deviates from being symmetric with respect to a symmetry group G. For pure states, we show that the BMSA is a strong monotone for magic-state resource theory, while it can be extended to mixed states via the convex roof construction. We discuss its relation with other magic monotones, first showing that the BMSA coincides with the recently introduced basis-minimized measurement entropy, thereby establishing the strong monotonicity of the latter. Next, we provide inequalities between the BMSA and other nonstabilizerness measures such as the robustness of magic, stabilizer extent, stabilizer rank, stabilizer fidelity and stabilizer Rényi entropy. We also prove that the stabilizer fidelity, stabilizer Rényi entropy and BMSA with index α ⩾ 2 have the same asymptotic scaling with qubit number. Finally, we present numerical methods to compute the BMSA, highlighting its advantages and drawbacks compared to other nonstabilizerness measures in the context of pure many-body quantum states. We also discuss the importance of additivity and strong monotonicity for measures of nonstabilizerness in many-body physics, motivating the search for additional computable nonstabilizerness monotones.

Abstract Climate signals, driven by complex interactions and nonlinear relationships, shape weather patterns and long-term trends, complicating the identification of dominant drivers due to collinearity. This study investigates the consistency and uncertainty of machine learning (ML) techniques for feature importance in climate science, comparing SHapley Additive exPlanations (SHAP), Partial Dependence Plots (PDPs), and gain-based feature importance from Extreme Gradient Boosting (XGBoost). SHAP’s integration with Feed Forward Neural Networks (FFNN) and XGBoost is evaluated to assess model-specific uncertainties. Using winter precipitation data from Ohio, USA, as a case study, the relative contributions of global warming (GW) and the Interdecadal Pacific Oscillation (IPO) to precipitation changes are quantified. Results show GW consistently ranks higher than IPO in at least 60% of stations across all methods, with SHAP and PDPs agreeing in 89% of stations. Global SHAP importance from FFNN and XGBoost aligns in 82% of stations, with GW contributing 15% more than IPO on average, though disagreements in 18% of stations highlight model-dependent uncertainties. Temporal analysis using SHAP values indicates a moderate discrepancy in feature importance between FFNN and XGBoost models (Pearson correlation ≈ 0.5), despite their consensus on the increasing dominance of GW in recent decades, contributing to wetter winters. Regression analysis further confirms that GW accounts for approximately 70% of the multi-decadal variability in winter precipitation across Ohio, with PDPs indicating a strong monotonicity (ρ = 0.94) between warming levels and precipitation increase. PDPs visualize marginal effects but struggle with interactions, while gain-based methods tend to favor features with a greater number of effective split points that reduce loss. SHAP, though robust for ranking, varies with the base model. An ensemble framework is proposed, demonstrating the value of combining these ML techniques complementarily to account for uncertainties and enhance interpretability. This study highlights the importance of addressing methodological uncertainties in feature importance rankings to provide robust insights for climate modeling.

This paper pursues two primary objectives. First, we develop a stochastic auxiliary problem principle to address stochastic variational inequalities. We establish the almost sure convergence of the proposed iterative scheme, derived using the stochastic auxiliary problem principle, under the assumptions of strong monotonicity and a growth condition on the involved mapping. Our results demonstrate convergence under highly general conditions on the random noise. While the step lengths α n are assumed to be diminishing, we also provide an alternative result that does not require the step lengths to converge to zero. The second objective is the development of an iteratively regularized stochastic auxiliary problem principle, which allows us to relax the strong monotonicity assumption. The practical relevance and effectiveness of the proposed framework are illustrated through its application to the stochastic inverse problem of estimating coefficients in stochastic partial differential equations. To be precise, we estimate the diffusion coefficient in the stochastic diffusion equation, the flexural rigidity coefficient in the stochastic fourth-order model, and the Lamé parameters in the stochastic linear elasticity. This is achieved by leveraging both a nonconvex output least-squares functional and a convex energy least-squares functional.

This paper introduces and analyzes two novel iterative algorithms for addressing the monotone bilevel split equilibrium problem in real Hilbert spaces. The problem encompasses a general system of variational inequalities and a common fixed point problem involving a countable family of uniformly Lipschitzian pseudocontractive mappings alongside an asymptotically nonexpansive mapping. Our algorithms are predicated on a novel subgradient extragradient implicit method that utilizes the strong monotonicity of one bifunction at the upper-level equilibrium and the monotonicity of another bifunction at the lower level. We establish strong convergence results for the proposed algorithms under mild conditions. A detailed example demonstrates the practicality and effectiveness of our methods.

We consider resolvent splitting algorithms for finding a zero of the sum of finitely many maximally monotone operators. The standard approach to solving this type of problem involves reformulating as a two-operator problem in the product-space and applying the Douglas–Rachford algorithm. However, existing results for linear convergence cannot be applied in the product-space formulation due to a lack of appropriate Lipschitz continuity and strong monotonicity. In this work, we investigate a different approach that does not rely on the Douglas–Rachford algorithm or the product-space directly. We establish linear convergence of the ‘resolvent splitting with minimal lifting’ algorithm due to Malitsky and Tam for monotone inclusions with finitely many operators. Our results are then used to derive linear convergence of a primal–dual algorithm for convex minimization problems involving infimal convolutions. The theoretical results are demonstrated on numerical experiments in image denoising.

The inverse mixed variational inequality problem comes from classical variational inequality, and it has many applications. In this paper, we propose new algorithms to study the inverse mixed variational inequality problems in Hilbert spaces, and these algorithms are based on the generalized projection operator. Next, we establish convergence theorems under inverse strong monotonicity conditions. In addition, we also provide inertial-type algorithms for the inverse mixed variational inequality problems with conditions that differ from the above convergence theorems.

The Stackelberg game depicts a leader-follower relationship wherein decisions are made sequentially, and the Stackelberg equilibrium represents an expected optimal solution when the leader can anticipate the rational response of the follower. Motivated by control of network systems with two levels of decision-making hierarchy, such as the management of energy networks and power coordination at cellular networks, a networked multi-leaders and multi-followers Stackelberg game is proposed. Due to the constraint of limited information interaction among players, a clustered information structure is assumed that each leader can only communicate with a portion of overall followers, namely its subordinated followers, and also only with its local neighboring leaders. In this case, the leaders cannot fully anticipate the collective rational response of all followers with its local information. To address Stackelberg equilibrium seeking under this partial information structure, we propose a distributed seeking algorithm based on implicit gradient estimation and network consensus mechanisms. We rigorously prove the convergence of the algorithm for both diminishing and constant step sizes under strict and strong monotonicity conditions, respectively. Furthermore, the model and the algorithm can also incorporate linear equality and inequality constraints into the followers' optimization problems, with the approach of the interior point barrier function. Finally, we present numerical simulations in applications to corroborate our claims on the proposed framework.

This work examines the behaviors of the online projected gradient ascent (OPGA) algorithm and its variant in a repeated oligopoly price competition under reference effects. In particular, we consider that multiple firms engage in a multiperiod price competition, where consecutive periods are linked by the reference price update and each firm has access only to its own first-order feedback. Consumers assess their willingness to pay by comparing the current price against the memory-based reference price, and their choices follow the multinomial logit (MNL) model. We use the notion of stationary Nash equilibrium (SNE), defined as the fixed point of the equilibrium pricing policy, to simultaneously capture the long-run equilibrium and stability. We first study the loss-neutral reference effects and show that if the firms employ the OPGA algorithm—adjusting the price using the first-order derivatives of their log-revenues—the price and reference price paths attain last-iterate convergence to the unique SNE, thereby guaranteeing the no-regret learning and market stability. Moreover, with appropriate step-sizes, we prove that this algorithm exhibits a convergence rate of [Formula: see text] in terms of the squared distance and achieves a constant dynamic regret. Despite the simplicity of the algorithm, its convergence analysis is challenging due to the model lacking typical properties such as strong monotonicity and variational stability that are ordinarily used for the convergence analysis of online games. The inherent asymmetry nature of reference effects motivates the exploration beyond loss-neutrality. When loss-averse reference effects are introduced, we propose a variant of the original algorithm named the conservative-OPGA (C-OPGA) to handle the nonsmooth revenue functions and show that the price and reference price achieve last-iterate convergence to the set of SNEs with the rate of [Formula: see text]. Finally, we demonstrate the practicality and robustness of OPGA and C-OPGA by theoretically showing that these algorithms can also adapt to firm-differentiated step-sizes and inexact gradients. This paper was accepted by Chung Piaw Teo, optimization and decision analytics. Funding: J. Lavaei acknowledges the support from the U.S. Army Research Laboratory and the U.S. Army Research Office under Grant W911NF2010219, Office of Naval Research under Grant N000142412673, AFOSR, NSF, and the UC Noyce Initiative. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.03464 .

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.

In this paper, we consider a class of nonsmooth weak vector variational-hemivariational inequality problems (abbreviated as, WVVHVIP) in the framework of Hadamard manifolds. By employing an analogous to the KKM lemma, we establish the existence of the solutions for WVVHVIP without utilizing any monotonicity assumptions. Moreover, a uniqueness result for the solutions of WVVHVIP is established by using generalized geodesic strong monotonicity assumptions. We formulate Auslender, regularized, and Moreau-Yosida regularized type gap functions for WVVHVIP to establish necessary and sufficient conditions for the existence of the solutions to WVVHVIP. In addition to this, by employing the Auslender, regularized, and Moreau-Yosida regularized type gap functions, we derive the global error bounds for the solution of WVVHVIP under the generalized geodesic strong monotonicity assumptions. Several non-trivial examples are furnished in the Hadamard manifold setting to illustrate the significance of the established results. To the best of our knowledge, this is the first time that the existence results, gap functions, and global error bounds for WVVHVIP have been investigated in the framework of Hadamard manifolds via Clarke subdifferentials.

Characterizing equipment performance degradation and predicting remaining useful life (RUL) are critical aspects of predictive maintenance in mechanical systems. The foundation of effective RUL prediction lies in constructing health indicator (HI) based on condition monitoring signals that accurately reflect equipment degradation and health status. In addition, the individual variability and uncertainty in the degradation process often make it challenging for a single degradation path to represent the entire process fully. To address these issues, this article introduces a novel framework for performance degradation characterization and RUL prediction. Initially, we constructed the HI using the Wasserstein distance and the Cumulative sum (CUMSUM) control chart. This approach not only captures changes in the signal probability distribution during degradation but also exhibits strong monotonicity, trendability, and robustness. Next, we propose a dynamic first prediction time (FPT) dynamic identification method based on Chebyshev’s inequality, which effectively mitigates the influence of outliers and minor fluctuations. Additionally, we develop a dynamic path matching and multipath adaptive drift linear multifractional Lévy stable motion (DPM-MPALMLSM) model for RUL prediction. The MPALMLSM model incorporates multiple degradation paths that accurately capture the non-Gaussian characteristics, long-range dependence features, and multifractal properties of the degradation process, with drift coefficients dynamically updated as monitoring data evolves. The dynamic path matching method, grounded in performance evaluation, facilitates efficient switching between degradation paths, enhancing RUL prediction accuracy. The effectiveness and precision of the proposed framework are demonstrated using full-life testing data from heavy truck transmissions, the XJTU-SY and IMS benchmark bearing datasets.

This paper introduces an inertial subgradient-type algorithm for solving equilibrium problems with strong monotonicity, constrained over the fixed point set of a nonexpansive mapping in the framework of a real Hilbert space. The proposed method integrates inertial and subgradient strategies to enhance convergence properties while avoiding the computational challenges of metric projections onto complex sets. A strong convergence theorem is established under appropriate constraint qualifications for the scalar sequences. Numerical experiments in both finite and infinite dimensional settings, including applications to Nash–Cournot oligopolistic market equilibrium models, highlight the efficacy and computational advantages of the algorithm. These results demonstrate the potential for broader applications in optimization and variational analysis.

The Russell graph measure is a non-radial efficiency measure for non-oriented Data Envelopment Analysis (DEA) models. It is strongly monotonic, but its projection point is not the closest one. Prior studies attempted to reverse the optimization of DEA models from a minimization problem to a maximization one for finding closer targets; however, this modification fails to satisfy strengthen the monotonicity of he efficiency measure. To resolve the conflict between the closer targets and strong monotonicity of efficiency measures, this study proposes a maximum Russell graph measure DEA model based on an extended facet production possibility set. It provides the closest target with only a single improvement in either an output or input term for the assessed DMU and avoids the free-lunch issue. Moreover, the maximum Russell graph measure satisfies strong monotonicity. Further practical advantages of the proposed efficiency measure are demonstrated numerically in comparison to other existing non-radial efficiency measures.

In this paper, we propose a general forward-reflected-backward splitting method for solving monotone inclusions in Hilbert spaces. Our method extends and improves the one of Malitsky and Tam (SIAM J. Optim., 2020). The weak convergence of the proposed algorithm is established under standard conditions. The linear convergence of the proposed method is derived under an additional condition like the strong monotonicity. We also give some theoretical comparisons to demonstrate the efficiency of the proposed algorithm.

This paper presents a computational implementation of a dynamic Walrasian equilibrium model in continuous time, formulated via variational inequality (VI) and quasi-variational inequality (QVI) frameworks. Building on the theoretical existence and stability results of the underlying equilibrium model, we discretize the time interval and implement a projected extragradient algorithm on the price simplex to compute equilibrium prices and allocations. We illustrate convergence behavior in a stylized Cobb-Douglas example and discuss the effects of discretization mesh size, stepsize selection, and algorithm performance. Numerical experiments demonstrate the practicality of the method, bridging the gap between theory and implementation. We also comment on mesh-independence, per-iteration complexity, and the role of strong monotonicity in achieving linear convergence. The computational study extends the purely theoretical framework by showing how real-world discretization and algorithmic choices impact convergence.

With a little help from my friends: Essentiality vs opportunity in group criticality

The concepts presented in this paper pertain to the development and examination of AB-coupled fixed point results for mapping in partially ordered S-metric spaces that possess the strong mixed monotone property. The existence and uniqueness of AB-coupled fixed points are also demonstrated. We generalize the main theorems of Gnana Bhaskar and Lakshmikantham (2006) in [15] and Virendra Singh Chouhan and Richa Sharma (2015) [4].

A property of prefix codes called strong monotonicity is introduced, and it is proven that for a given source, a prefix code is optimal if and only if it is complete and strongly monotone.