Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems

In this paper, we study a class of nonlinear complementarity problems associated with circular cone (CCNCP for short), which is a type of non-symmetric cone complementarity problems. Useful properties of the circular cone are investigated, which help to reformulate equivalently CCNCP as an implicit fixed-point equation. Based on the implicit fixed-point equation and splittings of the system matrix, we establish a class of relaxation modulus-based matrix splitting iteration methods for solving such a complementarity problem. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed and the strategy choice of the parameters are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving CCNCP.

This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem with the uniqueness assumption of the multipliers replaced by a convexity assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed linear complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric variational inequality, the use of the generalized linear complementarity problem as a tool to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affine variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and various stability theorems under some weak second-order conditions.

Proposed is a time-marching al- gorithm to solve a nonlinear system of com- plementarity equations: Pi(xj) ≥ 0, Qi(xj) ≥ 0, Pi(xj)Qi(xj )= 0, i, j = 1,...,n, resulting from a discretization of nonlinear obstacle problem. We transform the above nonlinear complemen- tarity problem (NCP) into a nonlinear algebraic equations (NAEs) system: Fi(xj )= 0 with the aid of the Fischer-Burmeister NCP-function. Such NAEs are semi-smooth,highlynonlinearand usu- ally implicit, being hard to handle by the Newton- like method. Instead of, a first-order system of ODEs is derived through a fictitious time equa- tion. The time-stepping equations are obtained by applying a numerical integration on the resul- tant ODEs, which are derivative-free and do not need the inverse of any matrix. The computa- tional cost is thus greatly reduced. The numerical examples of Bratu, von Karman and other elliptic equationsareusedto demonstratethatthenewfic- titious time integration method (FTIM) is highly efficient to calculate the obstacle problems. Keyword: Nonlinear Obstacle Problem, Non- linear complementarity problem, Nonlinear alge- braic equations, Iterative method, Elliptic equa- tions, Fictitious time integration method (FTIM)

We present a new difference of convex functions algorithm (DCA) for solving linear and nonlinear mixed complementarity problems (MCPs). The approach is based on the reformulation of bilinear complementarity constraints as difference of convex (DC) functions, more specifically, the difference of scalar, convex quadratic terms. This reformulation gives rise to a DC program, which is solved via sequential linear approximations of the concave term using standard DCA techniques. The reformulation is based on a generalization of earlier results on recasting bilinearities and leads to a novel algorithmic framework for MCPs. Through extensive numerical experimentation, the proposed approach, referred to as DCA-BL, proves to be an efficient heuristic for complementarity problems. For linear complementarity problems (LCPs), we test the approach on a number of randomly generated instances by varying the size, the density, and the eigenvalue distribution of the LCP matrix, providing insights into the numerical properties of DCA-BL. In addition, we apply the framework to a market equilibrium problem and find that DCA-BL scales well on realistic LCP instances. Also, through experimentation, we find that that DCA-BL performs particularly well compared with other DC-based complementarity approaches in the literature if the LCP is highly dense, asymmetric, or indefinite. Lastly, the method is successfully applied to a set of equilibrium problems with second-order cone constraints, which give rise to nonlinear complementarity problems, with applications to stochastic equilibrium problems in water infrastructure and finance. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2113891], the Division of Electrical, Communications and Cyber Systems [Grant 2114100], the Office of Naval Research Global [Grant 00014-22-1-2649], Petrobras [Grant 4324713], the Division of Mathematical Sciences [Grant 2318519], and the Danmarks Frie Forskningsfond [Grant 0217-00009B]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0822 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0822 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush--Kuhn--Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0---1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.

The Linear Dynamic Complementarity Problem is a special case of the Extended Linear Complementarity Problem

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

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs.

Reducing horizontal linear complementarity problems

A popular approach for addressing uncertainty in variational inequality problems is by solving the expected residual minimization (ERM) problem. This avenue necessitates distributional information associated with the uncertainty and requires minimizing nonconvex expectation-valued functions. We consider a distinctly different approach in the context of uncertain linear complementarity problems with a view towards obtaining robust solutions. Specifically, we define a robust solution to a complementarity problem as one that minimizes the worst-case of the gap function. In what we believe is amongst the first efforts to comprehensively address such problems in a distribution-free environment, we show that under specified assumptions on the uncertainty sets, the robust solutions to uncertain monotone linear complementarity problem can be tractably obtained through the solution of a single convex program. We also define uncertainty sets that ensure that robust solutions to non-monotone generalizations can also be obtained by solving convex programs. More generally, robust counterparts of uncertain non-monotone LCPs are proven to be low-dimensional nonconvex quadratically constrained quadratic programs. We show that these problems may be globally resolved by customizing an existing branching scheme. We further extend the tractability results to include uncertain affine variational inequality problems defined over uncertain polyhedral sets as well as to hierarchical regimes captured by mathematical programs with uncertain complementarity constraints. Preliminary numerics on uncertain linear complementarity and traffic equilibrium problems suggest that the presented avenues hold promise.

In this paper, we propose two splitting methods for solving horizontal linear complementarity problems characterized by matrices with positive diagonal elements. The proposed procedures are based on the Jacobi and on the Gauss–Seidel iterations and differ from existing techniques in that they act directly and simultaneously on both matrices of the problem. We prove the convergence of the methods under some assumptions on the diagonal dominance of the matrices of the problem. Several numerical experiments, including large-scale problems of practical interest, demonstrate the capabilities of the proposed methods in various situations.

Given a linear transformation L on a finite dimensional real inner product space V to itself and an element q ∈ V we consider the general linear complementarity problem LCP (L, K, q) on a proper cone K ⊆ V. We observe that the iterates generated by any closed algorithmic map will converge to a solution for LCP (L, K, q), whenever L is strongly monotone. Lipschitz constants of L is vital in establishing the above said convergence. Hence we compute the Lipschitz constants for certain classes of Lyapunov, Stein and double-sided multiplicative transformations in the setting of semidefinite linear complementarity problems. We give a numerical illustration of a closed algorithmic map in the setting of a standard linear complementarity problem. On account of the difficulties in numerically implementing such algorithms for general linear complementarity problems, we give an alternative algorithm for computing the solution for a special class of strongly monotone semidefinite linear complementarity problems along with a numerical example.

The extended linear complementarity problem (denoted by XLCP), of which the linear and horizontal linear complementarity problems are two special cases, can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed smoothing Fischer-Burmeister function, the XLCP is approximated by a family of parameterized smoothness optimization problems. Asmoothing damped Gauss-Newton method is designed for solving the XLCP. The proposed algorithm is proved to be convergent globally under suitable assumptions. Some numerical results are reported in the paper

Given a certain monotone horizontal linear complementarity problem (HLCP), we can naturally construct a family of systems of nonlinear equations parametrized by a parameter t ∈ (0, 1] with the property that, as t tends to 0, the corresponding system “converges” to the HLCP. Under reasonable conditions, it has been shown that each system of the family has a unique solution and that, as t tends to 0, these solutions converge to a specific solution of the HLCP. The main purpose of this paper is to study the asymptotic behavior of the derivative of the trajectory of solutions and therefore obtain information on the way the trajectory approaches the solution set of the HLCP. We show that the trajectory of solutions converges to the solution set along a unique and well-characterized direction. Moreover, if the HLCP has a solution satisfying strict complementarity then the direction forms a definite angle with any face of the feasible region which contains the limit point; otherwise, the direction is tangent to some face of the feasible region.

In this paper, we deal with unique solvability and numerical solution ofabsolute value equations (AVE),Ax−B|x|=b, (A,B∈Rn×n,b∈Rn).Under some weaker conditions, a simple proof is given for unique solvabilityof AVE. Furthermore, we demonstrate with an example that these results arereliable to detect unique solvability of AVE. These results are also extendedto unique solvability of standard and horizontal linear complementarity prob-lems. Finally, we suggest a Picard iterative method to compute an approx-imated solution of some uniquely solvable AVE problems where its globallylinear convergence is guaranteed via one of our weaker sufficient condition.