Complementarity Problem Research Articles

The double-relaxation modulus-based matrix splitting iteration method for linear complementarity problems

For the large sparse linear complementarity problems (LCP), by adopting the relaxation and relaxation two-sweep techniques for the modulus-based matrix splitting (MMS) iteration method, we establish a double-relaxation two-sweep modulus-based matrix splitting (DRTMMS) iteration method. This new method contains some known ones developed recently. Some sufficient conditions for guaranteeing the convergence of the DRTMMS method are presented when the system matrices are positive definite matrices and H+-matrices, which generalize some existing results. And the parameter selection strategy of the DRTMMS iteration method is given. Besides, the convergence analyses of the DRTM accelerated overrelaxation (DRTMAOR) method are given. Finally, numerical experiments illustrate that the DRTMMS method is efficient and has better performance than several existing MMS-like methods, and outperforms the projected splitting methods in some cases.

Modulus-based matrix splitting methods for complex linear complementarity problem

At present, the numerical solution of the complex linear complementarity problem (CLCP) by iteration methods is scanty. To fill in this gap, the goal of this paper is to establish a class of modulus-based matrix splitting methods for solving the CLCP by expressing the real form of the CLCP as the equivalent absolute value equations. Some sufficient conditions to guarantee the convergence of the proposed methods are provided. Numerical examples are offered to confirm the feasibility and efficiency of the proposed methods.

Existence of the least element solution of the vertical block Z-tensor complementarity problem

Existence of the least element solution of the vertical block Z-tensor complementarity problem

A Fixed Point Iterative Method for Third-order Tensor Linear Complementarity Problems

A Fixed Point Iterative Method for Third-order Tensor Linear Complementarity Problems

A Finsler Geometrical Programming Approach to the Nonlinear Complementarity Problem of Traffic Equilibrium

We consider a geometrical approach to the optimisation problems motivated by transportation system management. Here, we provide a comprehensive account of geometric programming based on the elementary Finsler geometry in {mathbb {R}}^n. Then, we present a Finslerian dynamical model for the nonlinear complementarity problem of traffic equilibrium that can be applied to a variety of equilibrium problems.

On the new modulus-based matrix splitting method for linear complementarity problem of $$H_{+}$$-matrix

On the new modulus-based matrix splitting method for linear complementarity problem of $$H_{+}$$-matrix

Effects of Human Activities on Evapotranspiration and Its Components in Arid Areas

With the increasing impact of human activities on the environment, evapotranspiration (ET) has changed in arid areas, which further affects the water resources availability in the region. Therefore, understanding the impact of human activities on ET and its components is helpful to the management of water resources in arid areas. This study verified the accuracy of Fisher's model (PT-JPL model) for ET estimation in southern Xinjiang, China by using the evaporation complementarity theory dataset (AET dataset). The ET and the evapotranspiration components (T:E) of six land-use types were estimated in southern Xinjiang from 1982 to 2015, and the impact of human activities on ET was analyzed. In addition, the impact of four environmental factors (temperature (Temp), net radiation (Rn), relative humidity (RH), and NDVI) on ET were evaluated. The results showed that the calculated ET values of the PT-JPL model were close to the ET values of the AET dataset. The correlation coefficient (R2) was more than 0.8, and the NSE was close to 1. In grassland, water area, urban industrial and mining land, forest land, and cultivated land, the ET values were high, and in unused land types, the ET values were the lowest. The T:E values varied greatly in urban industrial and mining land, forest land, and cultivated land, which was due to the intensification of human activities, and the values were close to 1 in summer in recent years. Among the four environmental factors, temperature largely influenced the monthly ET. These findings suggest that human activities have significantly reduced soil evaporation and improved water use efficiency. The impact of human activities on environmental factors has caused changes in ET and its components, and appropriate oasis expansion is more conducive to regional sustainable development.

English

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A globally convergent smoothing Newton method for the second order cone complementarity approach of elastoplasticity problems

A new algorithm for solving small-deformation elastoplasticity problems as second order cone complementarity problems (SOCCPs) is presented. The SOCCP is derived from the Hellinger–Reissner variational principle, and then solved with a smoothing Newton method that is modified from the globally convergent Qi-Sun-Zhou method. The method is compared with two well-established methods: the return-mapping method and the conic programming (CP) approach with primal–dual interior-point method. Numerical results indicate that the proposed method shares common characteristics of CP class methods and offers better convergence efficiency in some cases. Moreover, it is shown that the proposed method can be imbedded in the framework of conventional finite element codes with only a few modifications, which makes it easy-to-implement and adaptive to large-scale parallel computing.

Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions

This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by “standard” inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given.

Asymptotic Analysis for One-Stage Stochastic Linear Complementarity Problems and Applications

One-stage stochastic linear complementarity problem (SLCP) is a special case of a multi-stage stochastic linear complementarity problem, which has important applications in economic engineering and operations management. In this paper, we establish asymptotic analysis results of a sample-average approximation (SAA) estimator for the SLCP. The asymptotic normality analysis results for the stochastic-constrained optimization problem are extended to the SLCP model and then the conditions, which ensure the convergence in distribution of the sample-average approximation estimator for the SLCP to multivariate normal with zero mean vector and a covariance matrix, are obtained. The results obtained are finally applied for estimating the confidence region of a solution for the SLCP.

Interior point methods for solving Pareto eigenvalue complementarity problems

In this paper, we propose to solve Pareto eigenvalue complementarity problems by using interior-point methods. Precisely, we focus the study on an adaptation of the Mehrotra Predictor Corrector Method (MPCM) and a Non-Parametric Interior Point Method (NPIPM). We compare these two methods with two alternative methods, namely the Lattice Projection Method (LPM) and the Soft Max Method (SM). On a set of data generated from the Matrix Market, the performance profiles highlight the efficiency of MPCM and NPIPM for solving eigenvalue complementarity problems. We also consider an application to a concrete and large size situation corresponding to a geomechanical fracture problem. Finally, we discuss the extension of MPCM and NPIPM methods to solve quadratic pencil eigenvalue problems under conic constraints.

A Class of Smoothing Modulus-Based Iterative Methods for Solving the Stochastic Mixed Complementarity Problems

In this paper, we present a smoothing modulus-based iterative method for solving the stochastic mixed complementarity problems (SMCP). The main idea is that we firstly transform the expected value model of SMCP into an equivalent nonsmooth system of equations, then obtain an approximation smooth system of equations by using a smoothing function, and finally solve it by the Newton method. We give the convergence analysis, and the numerical results show the effectiveness of the new method for solving the SMCP with symmetry coefficient matrices.

AN INTERIOR-POINT ALGORITHM FOR SIMPLICIAL CONE CONSTRAINED CONVEX QUADRATIC OPTIMIZATION

In this paper, we are concerned with the numerical solution of simplicial cone constrained convex quadratic optimization (SCQO) problems. A reformulation of the K.K.T optimality conditions of SCQOs as an equivalent linear complementarity problem with $\mathcal{P}$-matrix ($\mathcal{P}$-LCP) is considered. Then, a feasible full-Newton step interior-point algorithm (IPA) is applied for solving SCQO via $\mathcal{P}$-LCP. For the completeness of the study, we prove that the proposed algorithm is well-defined and converges locally quadratic to an optimal of SCQOs. Moreover, we obtain the currently best well-known iteration bound for the algorithm with short-update method, namely,$ \mathcal{O}(\sqrt{n}\log\frac{n}{\epsilon })$. Finally, we present a various set of numerical results to show its efficiency.

A new search direction of IPM for horizontal linear complementarity problems

This study presents a new search direction for the horizontal linear complementarity problem. A vector-valued function is applied to the system of xy=μe, which defines the central path. Usually, the way to get the equivalent form of the central path is using the square root function. However, in our study, we substitute a new search function formed by a different identity map, which obtains the equivalent shape of the central path using the square root function. We get the new search directions from Newton’s Method. Given this framework, we prove polynomial complexity for the Newton directions. We show that the algorithm’s complexity is O(nlognϵ), which is the same as the best-given algorithms for the horizontal linear complementarity problem.

Stochastic ridesharing equilibrium problem with compensation optimization

In the urban traffic system with ridesharing programs, we develop a generalized stochastic user equilibrium model to formulate travelers’ mode and route choice behavior. To suit more general scenarios, the proposed model takes into consideration travelers’ heterogeneity in terms of car ownership and value of time, and travelers’ limited perceived information based on the stochastic user equilibrium principle instead of Wardrop’s user equilibrium principle. The proposed model is formulated as variational inequalities and an equivalent nonlinear mixed complementarity problem due to the inseparable and asymmetric travel cost functions. Furthermore, we address the decision-making problem of ridesharing compensation from the perspective of traffic managers and policy-makers who want to minimize the total travel cost and vehicular air pollution emissions simultaneously. A bi-objective optimization model and two single-objective optimization models are proposed to formulate this decision-making problem, in which travelers’ mode and route choice behavior has been respected. As a mathematical problem with complementarity constraints, the bi-objective optimization model is solved by an improved Non-Dominated Sorting Genetic Algorithm II to generate a set of Pareto-optimal solutions for policy-makers and allow them to choose desired solutions. Finally, several numerical experiments based on two different scales of networks are conducted to demonstrate the properties of the problem and the performance of the proposed model and algorithm. The results show that rational pricing of ridesharing compensation can indeed mitigate urban traffic congestion and pollution emissions simultaneously. Moreover, by integrating travelers’ choice behavior based on the stochastic user equilibrium principle instead of the user equilibrium principle in the ridesharing compensation optimization model, this study derives a series of more effective decision-making strategies for ridesharing compensation.

Polyhedral complementarity problem with quasimonotone decreasing mappings

The fixed point problem of piecewise constant mappings in Rn is investigated. This is a polyhedral complementarity problem, which is a generalization of the linear complementarity problem. Such mappings arose in the author?s research on the problem of economic equilibrium in exchange models, where mappings were considered on the price simplex. The author proposed an original approach of polyhedral complementarity, which made it possible to obtain simple algorithms for solving the problem. The present study is a generalization of linear complementarity methods to related problems of a more general nature and reveals a close relationship between linear complementarity and polyhedral complementarity. The investigated method is an analogue of the well-known Lemke method for linear complementarity problems. A class of mappings is described for which the process is monotone, as it is for the linear complementarity problems with positive principal minors of the constraint matrix (class P). It is shown that such a mapping has always unique fixed point.

The new full-Newton step interior-point algorithm for the Fisher market equilibrium problems based on a kernel function

In this paper, we propose a full-Newton step interior-point method for the weighted linear complementarity problem (wLCP) model of the Fisher market equilibrium problem. The weighted complementarity problem (wCP) is a generalization of the complementarity problem (CP) with the nonnegative weight vector, where the zero on the right-hand side is replaced by the nonnegative weight vector. The importance of a wCP lies in the fact that many equilibrium problems in science, engineering, and economics can be reformulated as wCPs, which lead to the development of highly efficient algorithms. We extend a full-Newton step interior-point method (IPM) to the wLCP model of the Fisher problem. New search directions are obtained by using a kernel function in the scaled Newton system. The algorithm takes only full-Newton steps, thus, avoiding the calculation of the step size which is computationally advantageous. Under the suitable assumptions, the algorithm is shown to have global convergence and polynomial complexity. Some numerical results indicate the efficiency of the algorithm. To the best of our knowledge, this is the first full-Newton step IPM for Fisher problems, which uses the kernel function to obtain new search directions.

A numerical method for parabolic complementarity problem

<abstract><p>In this paper, we study the numerical solution of a parabolic complementarity problem which is a widely used model in many fields, such as option pricing, risk measures, etc. Using a power penalty method we represent the complementarity problem as a nonlinear parabolic partial differential equation (PDE). Then, we use the trapezoidal rule as the time discretization, for which we have to solve a nonlinear equation at each time step. We solve such a nonlinear equation by the fixed-point iteration and in this methodology solving a tridiagonal linear system is the major computation. We present an efficient backward substitution algorithm to handle this linear system. Numerical results are given to illustrate the advantage of the proposed algorithm (compared to the built-in command backslash in Matlab) in terms of CPU time.</p></abstract>

A New Dynamic State Estimation Approach Including Hard Limits on Control Devices

This paper puts forward a comprehensive formulation for the wide area dynamic state estimation of power systems. This is a problem of great current interest in the power systems community, where significant progress has been achieved in recent years. This proposal builds on the existing work and formulates a novel approach to estimate dynamic and algebraic state variables of the entire power system in a simultaneous solution method. In contrast to existing formulations, the proposed approach simultaneously co-estimates the transient evolution of the transmission grid's nodal voltages and the dynamic states associated with synchronous machines and their associated controllers. Since controllers with hard limits are always present in a power system, the proposal also estimates when and how hard limits constrain the time evolution of the power system operating state. In this case, constrained dynamic variables associated with non-windup and windup limiters are automatically handled in the estimation process by transforming the inequality constraints of hard limits into equality constraints using the complementarity theory and the Fischer-Burmeister merit function. The feasibility of the proposed method is demonstrated on the WSCC 9-bus power system and the Mexican 190-bus power system.