Abstract We consider a two-stage distributionally robust optimization (DRO) model with multimodal uncertainty, where both the mode probabilities and uncertainty distributions could be affected by the first-stage decisions. To address this setting, we propose a generic framework by introducing a $$\phi $$ ϕ -divergence based ambiguity set to characterize the decision-dependent mode probabilities and further consider both moment-based and Wasserstein distance-based ambiguity sets to characterize the uncertainty distribution under each mode. We identify two special $$\phi $$ ϕ -divergence examples (variation distance and $$\chi ^2$$ χ 2 -distance) and provide specific forms of decision dependence relationships under which we can derive tractable reformulations. Furthermore, we investigate the benefits of considering multimodality in a DRO model compared to a single-modal counterpart through an analytical analysis. Additionally, we develop a separation-based decomposition algorithm to solve the resulting multimodal decision-dependent DRO models with finite convergence and optimality guarantee under certain settings. We provide a detailed computational study over two example problem settings, the facility location problem and shipment planning problem with pricing, to illustrate our results, which demonstrate that omission of multimodality or decision-dependent uncertainties within DRO frameworks result in inadequately performing solutions with worse in-sample and out-of-sample performances under various settings. We further demonstrate the speed-ups obtained by the solution algorithm against the off-the-shelf solver over various instances.

Some theoretical results on the finite convergence property and the temporary stalling behavior of Anderson acceleration on linear systems

Robust precision motion control of a piezoelectric microsurgical manipulator using adaptive non-singular terminal sliding mode.

ABSTRACT Accurate state‐of‐charge (SoC) estimation in lithium‐ion batteries is crucial for efficient energy management, safe operation, and extended battery lifespan. Although sliding mode observers (SMOs) are widely used for this purpose, conventional first‐order designs often suffer from chattering and slow convergence, resulting in noisy and less reliable estimation signals. This paper proposes a finite‐time second‐order sliding mode observer (SO‐SMO) for accurate SoC estimation based on an equivalent circuit model of the battery. The proposed observer analytically derives a closed‐form expression for the finite convergence time, enabling predictable estimation dynamics. Moreover, it eliminates chattering and significantly improves estimation smoothness and robustness against modeling uncertainties and measurement noise. A comparative analysis with the Extended Kalman Filter and traditional SMO demonstrates that the proposed method achieves higher estimation accuracy and faster convergence while maintaining lower computational complexity, making it well‐suited for real‐time applications. Theoretical analysis and simulation results confirm that the SO‐SMO offers a superior balance between accuracy, robustness, and efficiency, establishing its potential for next‐generation battery management systems in electric and hybrid vehicles.

This article addresses the time-varying distributed optimization problem (DOP) for networked multiagent systems (NMASs) operating over directed graphs, considering the impact of edge-based additive measurement noise (EBAMN). First, a finite-time stochastic stability framework is established to demonstrate the global stochastic practical finite-time attraction of the origin, enabling robust control design for stochastic nonlinear systems. The proposed method achieves faster convergence rates and provides bounded finite convergence time estimates, outperforming asymptotic methods. Second, a novel distributed optimization algorithm (DOA) is introduced, incorporating consensus-gain function, state-dependent optimization gains, and integral information of the gradient of local objective functions. Using the It $\mathrm {\hat {o}}$ lemma and Lyapunov theory, the continuous-time DOA guarantees the $p$ th moment convergence for all agents, ensures practical finite-time consensus in probability, and drives that states of NMASs converge to the time-varying optimal solution, even in the presence of EBAMN interferences. Furthermore, a new adaptive dynamic event-triggered mechanism (ETM) integrated with the DOA is proposed. This mechanism significantly enhances communication efficiency and reduces resource consumption throughout the process of tracking the optimal solution while preventing Zeno behavior. Finally, numerical simulations in multiuncrewed aerial vehicle (UAV) target tracking validate the effectiveness of the robust continuous-time DOA against random EBAMN.

This paper considers the problem of unconstrained minimization of smooth functions. Despite the high efficiency of quasi-Newton methods such as BFGS, their performance degrades in ill-conditioned problems with unstable or rapidly varying Hessians—for example, in functions with curved ravine structures. This necessitates alternative approaches that rely not on second-derivative approximations but on the topological properties of level surfaces. As a new methodological framework, we propose using a procedure of incomplete orthogonalization in the directions of gradient differences, implemented through the iterative least-squares method (ILSM). Two new methods are constructed based on this approach: a gradient method with the ILSM metric (HY_g) and a modification of the Hestenes–Stiefel conjugate gradient method with the same metric (HY_XS). Both methods are shown to have linear convergence on strongly convex functions and finite convergence on quadratic functions. A numerical experiment was conducted on a set of test functions. The results show that the proposed methods significantly outperform BFGS (2 times for HY_g and 3.5 times for HY_XS in terms of iterations number) when solving ill-posed problems with varying Hessians or complex level topologies, while providing comparable or better performance even in high-dimensional problems. This confirms the potential of using topology-based metrics alongside classical quasi-Newton strategies.

Building a sustainable and resilient humanitarian logistics system is essential for reducing disaster losses and supporting long-term socio-economic recovery. Following a major disaster, rapidly organizing casualty evacuation while maintaining system robustness is a fundamental component of sustainable emergency management. This study develops a two-stage robust optimization model for designing a sustainable humanitarian logistics network that simultaneously accounts for two critical post-disaster uncertainties: (i) interruption risks at temporary medical points and (ii) uncertain casualty demand. By explicitly differentiating deprivation costs between mild and serious injuries, the model quantifies human suffering in monetary terms, thereby integrating social and economic sustainability considerations into the optimization framework. A customized column-and-constraint generation (C&CG) algorithm with proven finite convergence is proposed to ensure tractability and practical applicability. Using the 2008 Wenchuan earthquake as a real-world case study, involving 10 affected areas and 10 candidate temporary medical points, the results demonstrate that the proposed approach yields evacuation plans that remain feasible under all tested worst-case realizations, substantially reducing deprivation costs compared with existing benchmarks. The findings highlight that strategically increasing the capacity of key temporary medical nodes enhances the sustainability and resilience of the emergency medical system, offering evidence-based insights for designing sustainable and robust disaster-response strategies.

This paper deals with a robust non-convex polynomial optimization problem (RP), where constraint functions involve uncertain data. By using the Archimedean condition, we first establish necessary and sufficient conditions for the optimal solutions of (RP). Subsequently, we also establish optimality conditions within the framework of truncated polynomials systems for (RP). Based on the obtained optimality conditions, we propose a semidefinite programming (SDP) relaxation of (RP). Then, we obtain asymptotic convergence of Lasserre hierarchy of the SDP relaxation for (RP) in terms of the Archimedean condition. Furthermore, we also obtain a finite convergence result of Lasserre hierarchy of the SDP relaxation.

Abstract In this paper, we study a general split variational inequality problem and investigate the weak sharpness property of its solution set. We also discuss the finite convergence property of a sequence converging to the solution set based on the weak sharpness property. In addition, we introduce an iterative method that converges to the solution set of the considered problem. Then, we formulate the traffic network equilibrium of two cities over an extended period in terms of our considered problem and establish some existence results. Finally, we validate our findings with a numerical example.

This article proposes an optimization dynamic movement primitives (ODMP) strategy and terminal sliding mode motion control (TSMC) method for the generalization problem of mobile robot when facing with obstacles. In scenarios where obstacles are present along the reproduced trajectories, an online trajectory planning algorithm based on model predictive optimization ensures that the robot reproduces the demonstrated trajectory as closely as possible without colliding with obstacles, in which the obstacles are modeled as the soft and hard constraint simultaneously, and the model predictive control (MPC) is utilized to search optimal solutions for path planning in the presence of obstacle. Additionally, a dynamic controller based on super-twisting practice terminal sliding mode (ST-PTSM) control ensures rapid and accurate tracking of the trajectories planned by MPC while improve accuracy during tracking, thus ensuring convergence within a finite convergence time. Experiments on a two-wheeled differential mobile robot are carried out to verify the proposed method.

Abstract We examine convergence of finite difference schemes approximating lower semicontinuous (Barron-Jensen [11]) viscosity solutions for Cauchy problems of Hamilton-Jacobi type. Even for merely bounded and lower semicontinuous initial data, we identify general conditions on a finite difference scheme which guarantee that discrete approximations given by the scheme are uniformly close to the lower semicontinuous viscosity solution of the associated PDE, on compact sets where the solution is continuous. We also provide examples of admissible schemes and initial conditions for which our main results apply. Convergence results for finite difference schemes have recently been used to establish limit theorems for discrete random processes [1, 2]. As an application of our finite difference scheme convergence results, we prove a new limit theorem for a random process called the “cooperative leader random walk.”

This study concerns the numerical resolution and simulation of the Lotka-Volterra diffusive prey-predator model. This model is a system consisting of two semi-linear parabolic partial differential equations. After performing the numerical resolution using the explicit finite difference method, whose consistency and convergence are shown, we move onto the numerical simulations, which are done under MATLAB software. We started by doing several simulations in 3 dimension (3D) to see different cases. We complete the work with a 2D simulation to compare the non-diffusive case to the diffusive cases.

Extending Linear Conditioning to Convex-Concave Optimization: Finite Convergence of the Proximal Point Algorithm

A new and improved sliding mode control (NISMC) with a grey linear regression model (GLRM) facilitates the development of high-quality pure sine wave inverters in photovoltaic (PV) energy conversion systems. SMCs are resistant to variations in internal parameters and external load disturbances, resulting in their popularity in PV power generation. However, SMCs experience a slow convergence time for system states, and they may cause chattering. These limitations can result in subpar transient and steady-state performance of the PV system. Furthermore, partial shading frequently yields a multi-peaked power-voltage curve for solar panels that diminishes power generation. A traditional maximum power point tracking (MPPT) algorithm in such a case misclassifies and fail to locate the global extremes. This paper suggests a GLRM-based NISMC for performing MPPT and generating a high-quality sine wave to overcome the above issues. The NISMC ensures a faster finite system state convergence along with reduced chattering and steady-state errors. The GLRM represents an enhancement of the standard grey model, enabling greater accuracy in predicting global state points. Simulations and experiments validate that the proposed strategy gives better tracking performance of the inverter output voltage during both steady state and transient tests. Under abrupt load changing, the proposed inverter voltage sag is constrained to 10% to 90% of the nominal value and the voltage swell is limited within 10% of the nominal value, complying with the IEEE (Institute of Electrical and Electronics Engineers) 1159-2019 standard. Under rectified loading, the proposed inverter satisfies the IEEE 519-2014 standard to limit the voltage total harmonic distortion (THD) to below 8%.

In this paper, we prove the finite convergence of sequences generated by (inexact and exact) proximal point methods for solving pseudomonotone equilibrium problems on Hadamard manifolds under the linear conditioning of the solution set. Keywords: Equilibrium problems; Hadamard manifolds; pseudomonotone bifunction; linear conditioning; finite convergence; proximal point method.

Robust support vector machine (RSVM) using ramp loss provides a better generalization performance than traditional support vector machine (SVM) using hinge loss. However, the good performance of RSVM heavily depends on the proper values of regularization parameter and ramp parameter. Traditional model selection technique with gird search has extremely high computational cost especially for fine-grained search. To address this challenging problem, in this paper, we first propose solution paths of RSVM (SPRSVM) based on the concave-convex procedure (CCCP) which can track the solutions of the non-convex RSVM with respect to regularization parameter and ramp parameter respectively. Specifically, we use incremental and decremental learning algorithms to deal with the Karush-Khun-Tucker violating samples in the process of tracking the solutions. Based on the solution paths of RSVM and the piecewise linearity of model function, we can compute the error paths of RSVM and find the values of regularization parameter and ramp parameter, respectively, which corresponds to the minimum cross validation error. We prove the finite convergence of SPRSVM and analyze the computational complexity of SPRSVM. Experimental results on a variety of benchmark datasets not only verify that our SPRSVM can globally search the regularization and ramp parameters respectively, but also show a huge reduction of computational time compared with the grid search approach.

Finite convergence of the Moment-SOS hierarchy for polynomial matrix optimization

Using Side Information to Improve Decision Making Under Uncertainty In many real-world planning settings, side information (also called covariate information, contextual information, or features) can be used to improve the estimates of uncertain parameters. Over the past decade, there has been growing interest in data-driven approaches to stochastic programming that take advantage of such side information. In “Data-Driven Sample Average Approximation with Covariate Information,” Kannan, Bayraksan, and Luedtke investigate two flexible data-driven frameworks that integrate a machine learning prediction model within a sample average approximation (SAA) of a stochastic programming problem, including a novel framework that leverages leave-one-out residuals for scenario generation. They establish conditions on the data generation process, the prediction model, and the stochastic program under which the solutions of these data-driven contextual SAAs exhibit asymptotic and finite sample convergence guarantees. Furthermore, they provide examples illustrating that these data-driven formulations can outperform existing methods in the limited data regime, even if the prediction model is misspecified.

Finite convergence criteria for normalized Nash equilibrium through weak sharpness and linear conditioning

This study explores the importance of mesh convergence studies in improving the precision and dependability of numerical simulations by delving into their complex reality. With finite element analysis, mesh quality and density must match element size and shape to produce accurate results. The output of a finite element model is tested for sensitivity to changes in the mesh or grid resolution using a numerical simulation technique known as a mesh convergence study. This research focuses on finite element analysis (FEA) and mesh convergence study for a side bracket geometry as an example. The goal of this mesh convergence research is to determine the optimal mesh resolution that yields results with the appropriate level of precision at a reasonable processing cost by considering mesh characteristics such as element quality, aspect ratio, Jacobean ratio and skewness and evaluating with stress and deformation results using ANSYS.