Complementarity Problem Research Articles

Rate-independent gradient-enhanced crystal plasticity theory — Robust algorithmic formulations based on incremental energy minimization

Numerically robust algorithmic formulations suitable for rate-independent crystal plasticity are presented. They cover classic local models as well as gradient-enhanced theories in which the gradients of the plastic slips are incorporated by means of the micromorphic approach. The elaborated algorithmic formulations rely on the underlying variational structure of (associative) crystal plasticity. To be more precise and in line with so-called variational constitutive updates or incremental energy minimization principles, an incrementally defined energy derived from the underlying time-continuous constitutive model represents the starting point of the novel numerically robust algorithmic formulations. This incrementally defined potential allows to compute all variables jointly as minimizers of this energy. While such discrete variational constitutive updates are not new in general, they are considered here in order to employ powerful techniques from non-linear constrained optimization theory in order to compute robustly the aforementioned minimizers. The analyzed prototype models are based on (1) nonlinear complementarity problem (NCP) functions as well as on (2) the augmented Lagrangian formulation. Numerical experiments show the numerical robustness of the resulting algorithmic formulations. Furthermore, it is shown that the novel algorithmic ideas can also be integrated into classic, non-variational, return-mapping schemes.

Algorithm for calculating the problem of unilateral frictional contact with an increscent external load parameter

The subject of the study is the contact interaction of deformable elements of linear complementarity problem (LCP). To solve the linear complementarity problem, the Lemke method with the introduction of an increasing parameter of external loading is used. The proposed approach solves the degenerated matrix in a finite number of steps, while the dimensionality of the problem is limited to the area of contact. To solve the problem, the initial table of the Lemke method is generated using the contact matrix of stiffness and the contact load vector. The unknowns in the problem are mutual displacements and interaction forces of contacting pairs of points of deformable solids. The proposed approach makes it possible to evaluate the change in working schemes as the parameter of external load increases. The features of the proposed formulation of the problem are shown, the criteria for stopping the stepwise process of solving such problems are considered. Model examples for the proposed algorithm are given. The algorithm has shown its efficiency in application, including for complex model problems. Recommendations on the use of the proposed approach are given.

A full-Newton step interior-point algorithm for the special weighted linear complementarity problem based on positive-asymptotic kernel function

A full-Newton step interior-point algorithm for the special weighted linear complementarity problem based on positive-asymptotic kernel function

Time will tell: Taxonomic and functional diversity of monilophyte ferns in native forests and abandoned silviculture systems

AbstractDifferent forest systems can influence the function, traits, diversity, and distribution of ferns. By comparing ferns based on their functional traits, we can explore how species loss affects functional diversity, and consequently alter the forest ecosystem. We investigated the impact of abandoned silviculture on fern diversity (taxonomic and functional) and abundance. Our hypothesis suggests a positive effect of nutrients and soil humidity in native forests, while canopy openness in silviculture has a negative effect on monilophyte diversity. This study was conducted in three sites of native forest and three sites of Pinus silviculture. Each site was subdivided into three areas, with three plots per area, resulting in 54 plots (3 sites × 3 areas × 3 plots × 2 forest types). Each plot covered an area of 100 m2 (10 m × 10 m; 5400 m2 in total). In each plot, we estimated abundance, functional attributes (richness, evenness, divergence, and dispersion) based on leaf thickness, leaf width, petiole length, stem length, leaf division, sori shape, indusium presence, and taxonomic diversity. Results indicate that native forest exhibit higher taxonomic and functional diversity (richness—Fric) compared to abandoned Pinus plantations, which may be attributed to high habitat diversity and species specialization in specific microhabitats. In silviculture, diversity homogenization leads to an increase in the redundancy of functional attributes (evenness—Feve), resulting in the filtering out of similar organisms. Abiotic factors such as light and canopy openness negatively influence plant establishment and growth, leading to a decrease in leaf division and indusium presence/absence (by CWMs), supporting niche complementarity theory in natural forest and mass ratio theory in silviculture. Our findings highlight the enduring negative impact of silviculture, even after 20 years, on both the taxonomic and functional diversity of monilophytes, underscoring the crucial role of native forests in preserving fern diversity.

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.

The chemistry between us: Illuminating complementarity patterns in interpersonal role-play assessment via moment-to-moment analyses.

In assessment and selection, organizations often include interpersonal interactions because they provide insights into candidates' interpersonal skills. These skills are then typically assessed via one-shot, retrospective assessor ratings. Unfortunately, the assessment of interpersonal skills at such a trait-like level fails to capture the richness of how the interaction unfolds at the behavioral exchange level within a role-play assessment. This study uses the lens of interpersonal complementarity theory to advance our understanding of interpersonal dynamics in role-play assessment and their effects on assessor ratings. Ninety-six MBA students participated in four different flash role-plays as part of diagnosing their strengths and weaknesses. Apart from gathering assessor ratings and criterion measures, coders also conducted a fine-grained examination of how the behavior of the two interaction partners (i.e., MBA students and role-players) unfolded at the moment-to-moment level via the Continuous Assessment of Interpersonal Dynamics (CAID) measurement tool. In all role-plays, candidates consistently showed mutual adaptations in line with complementarity principles: Affiliative behavior led to affiliative behavior, whereas dominant behavior resulted in docile, following behavior and vice versa. For affiliation, mutual influence also occurred in that both interaction partners' temporal trends in affiliation were entrained over time. Complementarity patterns were significantly related to ratings of in situ (role-playing) assessors but not to ratings of ex situ (remote) assessors. The effect of complementarity on validity was mixed. Overall, this study highlights the importance of going beyond overall ratings to capture behavioral contingencies such as complementarity patterns in interpersonal role-play assessment. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Neuro‐PINN: A hybrid framework for efficient nonlinear projection equation solutions

AbstractNonlinear projection equations (NPEs) provide a unified framework for solving various constrained nonlinear optimization and engineering problems. This paper presents a deep learning approach for solving NPEs by incorporating neurodynamic optimization and physics‐informed neural networks (PINNs). First, we model the NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization, and the objective becomes solving this ODE system. Second, we use a modified PINN to serve as the solution for the ODE system. Third, the neural network is trained using a dedicated algorithm to optimize both the ODE system and the NPE. Unlike conventional numerical integration methods, the proposed approach predicts the end state without computing all the intermediate states, resulting in a more efficient solution. The effectiveness of the proposed framework is demonstrated on a collection of classical problems, such as variational inequalities and complementarity problems.

Projection and Contraction Method for Pricing American Bond Options

In this paper, an effective numerical method is proposed for a linear complementarity problem (LCP) arising in the valuation of American bond options under the Cox–Ingersoll–Ross (CIR) model. Firstly, a variable substitution is used to simplify the linear complementary model. Secondly, the finite difference method is adopted to discretize the simplified model, and an equivalent variational form is obtained. Based on the positive definiteness of the discretized matrix, a projection and contraction method (PCM) is adopted for the resulting discretized variational problem. Finally, numerical experiments highlight the effectiveness and performance of the proposed algorithm.

Strict feasibility for the polynomial complementarity problem

Strict feasibility for the polynomial complementarity problem

A two-step new modulus-based matrix splitting method for vertical linear complementarity problem

In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516).

Efficient pricing of options in jump–diffusion models: Novel implicit–explicit methods for numerical valuation

This paper presents novel implicit–explicit Runge–Kutta type methods for numerically simulating partial integro-differential equations that arise when pricing options under jump–diffusion models. These methods offer an alternative approach that avoids the need for numerical or analytical inversion of the coefficient matrix. The pricing of European options is formulated as a partial integro-differential equation, while the pricing of American options are treated as a linear complementarity problem. The developed implicit–explicit Runge–Kutta type method is combined with an operator splitting technique to efficiently solve the linear complementarity problem. Stability and convergence analysis of the proposed methods are established using discrete ℓ2-norm. To validate their efficiency and accuracy, the methods are applied to pricing European and American options under Merton’s and Kou’s models, and the computed results are compared with those reported in the literature.

Convergence of a smoothing algorithm for horizontal linear complementarity problem with a nonmonotone line search

In this paper, we present a nonmonotone smoothing Newton type algorithm for solving the horizontal linear complementarity problem. We show that the proposed algorithm is globally convergent under an assumption that has the column -property. Such an assumption is weaker than those required by most other smoothing Newton algorithms. The preliminary numerical results are reported.

The solvability of weighted complementarity problems and a smoothing Newton algorithm under the local error bound

The weighted Complementarity Problem (wCP) can be used for modelling a larger class of problems from science and engineering. In this paper, we first extend the solvability of the Complementarity Problems (CP) obtained by Yoshise [SIAM J. Optim. 17, 1129–1153 (2006)] to the wCP. Especially, we prove that the solution set of the wCP is bounded. We then propose a smoothing Newton algorithm to solve the wCP and prove that it is globally convergent. Moreover, we establish the local quadratic convergence of the proposed algorithm under the local error bound condition which is weaker than the nonsingularity condition used in previous smoothing Newton-type algorithms. Further, we apply the proposed algorithm to solve the weighted horizontal linear complementarity problem and report some numerical results.

On sign-real spectral radii and sign-real expansive matrices

Let Mn be the space of real matrices of order n. The sign-real spectral radius ξ:Mn→R+, introduced in a 1997 paper by S.M. Rump, intervenes for instance in the problem of estimating the componentwise distance to singularity. The function ξ has also a bearing in the analysis of generalized absolute value equations and linear complementarity problems. Although ξ is not a norm, it is at least absolutely homogeneous and continuous. Furthermore, ξ is invariant under transposition, permutation similarity, and a few other linear isomorphisms on Mn. A matrix A∈Mn is called sign-real expansive if ξ(A)≥1. Let Ωn be the set of such matrices. The purpose of this work is to discover new properties of the function ξ and to explore in detail the structure of the set Ωn.

A new matrix splitting generalized iteration method for linear complementarity problems

The linear complementarity problems (LCPs) can be encountered in various scientific computing, management science, and operations research. In this study, we introduce and analyze a new generalized accelerated overrelaxation (NGAOR) method for solving LCPs, in which one special case reduces to a new generalized successive overrelaxation (NGSOR) method. Moreover, we prove the convergence of the proposed methods when the system matrix is an H-matrix (irreducible or strictly diagonally dominant matrix). Numerical results for several experiments are present to show the effectiveness and efficiency of the proposed methods.AMS classification: 65F10, 90C33

Young Duality for Variational Inequalities and Nonparametric Method of Demand Analysis in Input–Output Models with Inputs Substitution: Application for Kazakhstan Economy

The global macroeconomic shocks of the last decade entail the restructuring of national production networks and induce processes of input substitution. We suggest mathematical tools of Young duality for variational inequalities for studying these processes. Based on the tools we provide, a new mathematical model of a production network with several final consumers is created. The model is formulated as a pair of conjugated problems: a complementarity problem for optimal resource allocation with neoclassical production functions and the Young dual problem for equilibrium price indices on network products. The solution of these problems gives an equilibrium point in the space of network inter-industry flows and price indices on goods. Based on our previous results, we suggest an algorithm for model identification with an official economic statistic in the case of constant elasticity of substitution production functions. We give an explicit solution to the complementarity problems in this case and develop the algorithm of the inter-industry flows scenario projection. Since the algorithm needs the scenario projection of final sales structure as its input, we suggest a modified methodology that allows the calculation of scenario shifts in final consumer spending. To do this, we employ the generalized nonparametric method of demand analysis. As a result, we develop new technology for scenario calculation of a national input–output table, including shifts in final consumer spending. The technology takes into account a substitution of inputs in the network and is based on officially published national statistics data. The application of the methodology to study tax collection scenarios for Kazakhstan’s production network is demonstrated.

Challenging the status quo in a non‐challenging way: A dominance complementarity view of voice inquiry

AbstractSpeaking up directly promotes voice endorsement because it enhances communication clarity. Yet, voicers may hesitate to engage in direct voice because it is a dominant communication tactic that may upset, impose on, embarrass, or undermine their leader, potentially resulting in a backlash, greater workload, or a tainted image. These concerns present a puzzle regarding whether alternative communication tactics exist whereby voicers can secure endorsement for improvement‐oriented initiatives without directly challenging their leader. To address this puzzle, we introduce voice inquiry—expressing improvement‐oriented suggestions or concerns in the form of a question—as a submissive communication tactic to secure endorsement. Drawing upon dominance complementarity theory, we argue that voice inquiry prompts endorsement because it enhances leader's sense of power. Given the complementary effect of submissiveness and dominance, we further predict that this effect will be stronger when leader dominance is high. We conducted three Pilot Studies to unpack the content, motivation, prevalence, and submissive nature of voice inquiry. Building on this foundation, we conducted a multi‐wave field study with 373 employees and 178 leaders in a transportation company (Study 1) and a vignette experiment with 243 full‐time workers (Study 2). Across studies, our research demonstrates voice inquiry as a theoretically driven communication tactic that increases endorsement by activating leader sense of power, particularly among dominant leaders.

A modified multivariate spectral gradient projection method for nonlinear complementarity problems

A modified multivariate spectral gradient projection method for nonlinear complementarity problems

Enhancing Autonomous Vehicle Lateral Control: A Linear Complementarity Model-Predictive Control Approach

Model-predictive control (MPC) offers significant advantages in addressing constraint-related challenges and plays a pivotal role in self-driving car technology. Its primary goal is to achieve precise trajectory tracking while prioritizing vehicle stability and safety. However, real-time operations often face challenges related to computational demands and low computational efficiency. To address these challenges, this paper introduces a novel lateral control algorithm for self-driving vehicles, which utilizes the linear complementarity problem (LCP) instead of the conventional quadratic programming (QP) method as the MPC optimization solution. This innovative approach incorporates the electric steering system into the vehicle dynamics model, allowing for precise torque regulation of the steering motor and enhancing control accuracy. The MPC algorithm adopts the LCP solution method to calculate control signals based on the vehicle’s state, ensuring both rapid and stable vehicle control. Simulation results demonstrate that the proposed MPC algorithm, utilizing the LCP solution method, effectively addresses efficiency issues in the lateral motion of self-driving cars. This leads to improvements in both driving stability and real-time performance. Overall, this innovative approach lays a solid foundation for the practical implementation of self-driving cars.

Improved Fixed Point Iterative Methods for Tensor Complementarity Problem

Improved Fixed Point Iterative Methods for Tensor Complementarity Problem