Complementarity Problem Research Articles

On the two-stage multisplitting iteration methods for linear complementarity problems

In this work, the two-stage multisplitting iteration methods based on the equivalent modulus equations are analyzed for solving linear complementarity problems. New convergence results are presented where the convergence domains of the parameter matrices are enlarged compared the existing literatures. Furthermore, the improved domains are validated by numerical examples with parallel computations using OpenACC.

Primal‐dual active set algorithm for valuating American options under regime switching

AbstractThis paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods.

Gradient projection method on the sphere, complementarity problems and copositivity

By using a constant step-size, the convergence analysis of the gradient projection method on the sphere is presented for a closed spherically convex set. This algorithm is applied to discuss copositivity of operators with respect to cones. This approach can also be used to analyse solvability of nonlinear cone-complementarity problems. To our best knowledge this is the first numerical method related to the copositivity of operators with respect to the positive semidefinite cone. Numerical results concerning the copositivity of operators are also provided.

General convergence results of the modulus-based methods for vertical nonlinear complementarity problems

General convergence results of the modulus-based methods for vertical nonlinear complementarity problems

Modulus-based synchronous multisplitting method for horizontal nonlinear complementarity problem

Modulus-based synchronous multisplitting method for horizontal nonlinear complementarity problem

Real option pricing under the regime-switching model with jumps on a finite time horizon

We consider an irreversible investment decision problem on a finite time horizon where an instantaneous cash flow process of a firm follows a regime-switching jump-diffusion (RSJD) model. The value of a project can be derived from a partial integro-differential equation (PIDE) and then we obtain a closed-form solution of the PIDE. It is proved that the value of the project converges to the solution on an infinite time horizon problem as the lifetime of the project tends to infinity. Moreover, the value function of a real option with a finite expiration date can be evaluated by solving a linear complementarity problem (LCP) under the RSJD model. We apply a 2-step backward differentiation formula (BDF2 method) combined with an operator splitting method to solve the LCP and then it has the second-order convergence rate in the discrete ℓ2-norm with respect to the time and spatial variables. The stability of the proposed method to solve the LCP is also proved in the discrete ℓ2-norm. Finally, a variety of numerical experiments are carried out to determine the optimal investment time and the numerical results on the finite time horizon are analyzed.

Why Is One Social Media Platform Not Enough? A Typology of Platform-Swinging Behavior and Associated Affordance Preferences

This study seeks to advance the scholarship on the phenomenon of social media platform-swinging in the context of the polymedia environment in China. Specifically, drawing on the theoretical frameworks of uses and gratifications and channel complementarity theory, we propose that the platform-swinging behavior is driven by users’ various complementary and supplementary needs. Drawing on 32 semi-structured interviews, we identify four distinct types of platform-swinging behavior: social connecting, impression managing, information seeking, and aimless swinging. We further reveal that, depending on various complementary and supplementary needs, each type of platform-swinging behavior has distinct media affordance preferences. By profiling the distinct platform-swinging behaviors and examining their preferred affordances, the study advances our theoretical understanding of the dynamics between the polymedia ecology and users’ gratifications.

A relaxation two-step parallel modulus method without auxiliary variable for solving large sparse vertical linear complementarity problems

A relaxation two-step parallel modulus method without auxiliary variable for solving large sparse vertical linear complementarity problems

On finiteness of the solution set of extended horizontal linear complementarity problem

In this paper, we introduce the concept of column nondegenerate-W property to the set of matrices, and we show that when the involved set of matrices have the column nondegenerate-W property is equivalent to the Extended Horizontal Linear Complementarity Problem have the finiteness property of the solution set.

The projected splitting iterative methods based on tensor splitting and its majorization matrix splitting for the tensor complementarity problem

The projected splitting iterative methods based on tensor splitting and its majorization matrix splitting for the tensor complementarity problem

A novel predefined-time neurodynamic approach for mixed variational inequality problems and applications

In this paper, we propose a novel neurodynamic approach with predefined-time stability that offers a solution to address mixed variational inequality problems. Our approach introduces an adjustable time parameter, thereby enhancing flexibility and applicability compared to conventional fixed-time stability methods. By satisfying certain conditions, the proposed approach is capable of converging to a unique solution within a predefined-time, which sets it apart from fixed-time stability and finite-time stability approaches. Furthermore, our approach can be extended to address a wide range of mathematical optimization problems, including variational inequalities, nonlinear complementarity problems, sparse signal recovery problems, and nash equilibria seeking problems in noncooperative games. We provide numerical simulations to validate the theoretical derivation and showcase the effectiveness and feasibility of our proposed method.

Inertial Algorithms for Bifunction Harmonic Variational Inequalities

In this paper, we introduce and study some new classes of bifunction harmonic variational inequalities. Various new and known classes of variational inequalities and complementarity problems can obtained as special classes of bifunction harmonic variational inequalities. The auxiliary principle technique is applied to suggest and analyze some hybrid inertial iterative methods for finding the approximate solutions of the bifunction harmonic variational inequalities. The convergence analysis of these iterative methods is also considered under some suitable conditions. Results proved in this paper can be viewed as a refinement and improvement of the known results. It is an interesting open problem to develop some implementable numerical methods for solving these problems and to explore the applications in mathematical and engineering sciences.

Ecosymbiotic Complementarity, an Old Theory Applicable in Today’s Ethnobiological Studies

Ethnobiology analyzes the interactions between people and their surrounding environments from various perspectives. Some studies have been criticized by social scientists, who argue that ethnobiologists have insufficiently considered the conflicts between the dominant economic and political model and rural communities’ lives, which are often idealized. However, several ethnobiological studies have increasingly considered these aspects, and important initiatives in this direction have emerged from interactions with other research fields and frameworks, such as sustainability science, political ecology, agroecology, and social-ecological systems. To address criticism on what has been considered ethnobiological utopianism, it is valuable to theorize, develop methodological tools, and construct explanatory models. From such perspective, this work aims to recover an old theory called ecosymbiotic complementarity, which we propose can strengthen analytical approaches in the social-ecological systems framework. We recover the original proposal of the ecosymbiotic complementarity theory, emphasizing the relationships between rural communities and economic and political variables. We consider that the ecosymbiotic complementarity theory can contribute to the contextualization of rural societies analyzed in ethnobiological research.

Reflections on China-India Economic and Trade Relations from the Perspective of Industrial Structure

China and India are adjacent to each other and are the top two developing countries in the world in terms of population, and the different industrial structures of the two countries have a specific impact on the economic and trade cooperation between the two countries. The article firstly analyzes the industrial structure of China's strong industry and India's strong service industry and the status quo of bilateral trade, and summarizes the problems of small trade volume, unidirectional dependence between India and China, political hostility hindering economic and trade cooperation, and the two countries' industries' complementarities are greater than competitiveness through calculations and comparisons. Based on the theory of industrial complementarities, the article proposes accordingly to increase intra-industry trade, improve the investment environment, encourage investment and technological cooperation, participate in the global value chain, set up a free trade zone and strengthen the cooperation in multilateral mechanisms, and establish a free trade zone. The establishment of free trade zones and the strengthening of economic and trade cooperation in multilateral mechanisms are six proposals.

Operator Splitting Method to Solve the Linear Complementarity Problem for Pricing American Option: An Approximation of Error

Operator Splitting Method to Solve the Linear Complementarity Problem for Pricing American Option: An Approximation of Error

Price-Responsive Prosumers in Transmission-Constrained Power Markets: Pricing, Investment, and Social Welfare

The emergence of prosumers with renewable distributed energy resources has led to a paradigm shift in the electricity market. We formulate a complementarity problem to analyze a transmission-constrained power market in the presence of price-responsive prosumers who invest in photovoltaics (PVs) coupled with battery storage under two pricing schemes: net metering and net billing. Our study has three main findings. First, both investments in PVs and transmission tariffs are greater under net metering than under net billing over the range of PV capital costs. Second, the prosumer investments in PVs increase when coupled with battery systems under both pricing schemes. Third, the total social surplus under net billing tends to be larger than that under net metering. Moreover, the gap in the total social surplus between the pricing schemes widens when battery capacities are larger. Overall, we find that net billing could be a more efficient regulatory scheme, possibly avoiding over-investment in PVs.

Effects of maltodextrin on water adsorption and thermodynamic properties of Codonopsis Radix spray-dried powder

This study aims to reveal the effects of maltodextrin(MD) on the water adsorption and thermodynamic properties of Codonopsis Radix(DS) spray-dried powder by determining the moisture and energy changes of the powder in the process of moisture absorption. The static weighing method was used to obtain the isothermal water adsorption data of the spray-dried powder in 6 saturated salt solutions(KAc, MgCl_2·6H_2O, K_2CO_3, NaBr, NaCl, and KCl) at 3 temperatures(25, 35, and 45 ℃). Six models were used for fitting of the water adsorption process, and the most suitable model was selected based on the model performance. Furthermore, the corresponding net equivalent adsorption heat and differential entropy were calculated, and the adsorption entropy change was integrated. The linear relationship between net equivalent adsorption heat and differential entropy was drawn based on the entropy-enthalpy complementarity theory. The results showed that the water adsorption properties of DS and DS-MD spray-dried powder followed the type Ⅲ isotherm and was well fitted by the GAB model. The monolayer water content M_0 decreased with the increase in temperature. At the same temperature, the M_0 of DS spray-dried powder decreased after the addition of MD. The net equivalent adsorption heat and differential entropy of DS and DS-MD spray-dried powder decreased with the increase in water content, which presented a linear relationship. The addition of MD decreased the water activity corresponding to the lowest integral adsorption entropy of the powder, and the system became more stable. The results indicated that the spray-dried powder became more stable after the addition of MD.

Electricity sector impacts of water taxation for natural gas supply under high renewable generation

The future of energy development could be soon restrained by water scarcity. Shale gas, which is highly water-consuming in its extraction, is even more vulnerable. Thus, understanding the power system’s response to water constraints considering uncertainties from variable renewable energy generation becomes critical. This study presents a multi-stage stochastic Mixed Complementarity Problem for the energy sector, considering renewables variability and water scarcity’s impact on natural gas extraction. The power and natural gas sectors were modeled with hourly and daily resolution, respectively. Results demonstrate that stochastic planning enables smoother natural gas production and storage strategies, ensuring consistent operations with a narrower production range. Water taxation affects gas production and the generation mix, leading to electricity price changes. Gas-based electricity generation varies up to 16.2% based on water taxation and wind availability. Furthermore, water taxation of up to 20% can raise gas prices by up to 21.1%. This study underscores the need to integrate water scarcity considerations into energy planning, offering insights into sustainable decision-making, water resource management, reliable electricity generation, and price stability amidst evolving renewable portfolios.

A nonsmooth dynamics framework for simulating frictionless spatial joints with clearances

AbstractReal-world multibody systems do not have ideal joints; most joints have some clearance. The clearance allows the connected bodies to undergo a misalignment, and the resulting dynamics is governed by the contacts thus formed. Two approaches are typically taken to deal with contacts: the commonly used continuous dynamics approaches assume the Hertzian nature of the contact modeled by nonlinear unilateral spring-damper elements; while the nonsmooth dynamics approach results in a complementarity problem. This paper employs a nonsmooth dynamics approach to develop a coherent framework for the simulation of multibody systems having frictionless joints with clearances. Because clearances are of small magnitude relative to the dimensions of the mechanical components, the nature of the contact in the joints is assumed to be inelastic. Using this assumption and the general nonsmooth dynamics framework, the parametric formulations for cylindrical, prismatic, and revolute joints with clearances are derived. The equations of motion are formulated, and their time-discretized counterparts are cast as a nonlinear programming problem. The proposed scheme also enforces normalization constraint on Euler parameters in contrast to state-of-the-art methods that is conducive to stability of the solution for a suitable range of step sizes. In addition, a variable time-stepping scheme that includes the step size as an extra variable in the optimization is introduced and its stability properties are discussed. The versatility of the proposed framework is demonstrated through numerical experiments.

On the number of pivots of Dantzig's simplex methods for linear and convex quadratic programs

Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegatively constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency of simplex-type methods for solving these problems. Two applications of the quadratic programming results are presented.