The Linear Complementarity Problem in Static and Dynamic Games

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

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/ .

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.

In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.

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.

The Linear Complementarity Problem LCP(M,q) is to find a vector x in IRn satisfying x0, Mx+q0 and xT(Mx+q)=0, where M as a matrix and q as a vector, are given data. In this paper we show that the linear complementarity problem is completely equivalent to finding the fixed point of the map x = max (0, (I-M)x-q); to find an approximation solution to the second problem, we propose an algorithm starting from any interval vector X(0) and generating a sequence of the interval vector (X(k))k=1 which converges to the exact solution of our linear complementarity problem. We close our paper with some examples which illustrate our theoretical results.

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.

It is shown that solutions of linear inequalities, linear programs and certain linear complementarity problems (e.g. those with P-matrices or Z-matrices but not semidefinite matrices) are Lipschitz continuous with respect to changes in the right-hand side data of the problem. Solutions of linear programs are not Lipschitz continuous with respect to the coefficients of the objective function. The Lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right-hand side perturbation.

A number of equivalent characterizations for the existence and boundedness of solutions of the linear complementarity problem: Mx+q≥0, x≥0, x T(Mx+q)=0 where M is an n×n real matrix and q is an n-vector, are given for the case when M is copositive plus. The special case when M is skew-symmetric covers the linear programming case. One useful characterization of existence and boundedness of solutions is given by solving a simple linear program. Other important characterizations are the Slater constraint qualification and the stability condition that for all arbitrary but sufficiently small perturbations of the data M and q which maintain copositivity plus, the perturbed linear complementarity problem is solvable and its solutions are uniformly bounded. An interesting sufficient condition for boundedness of solutions is that the linear complementarity problem have a nondegenerate vertex solution. Another result is that the subclass ™ of copositive plus matrices for which the linear complementarity problem has a solution for each q in R n, that is ™⊂Q, coincides with the subclass of copositive plus matrices for which the linear complementarity problem has a nonempty bounded solution set for each q in R n.Key wordsLinear Complementarity ProblemSolution SetBoundednessCopositive-plus Matrices

Linear complementarity problems provide a powerful framework to model nonsmooth phenomena in dynamical control systems. Mimicking the general strategy that led to the foundation of bifurcation theory in smooth maps, we introduce a novel notion of equivalence between linear complementarity problems that sets the basis for a theory of bifurcations in a large class of nonsmooth maps, including steady-state bifurcations in linear complementarity systems. Our definition leads to constructive algebraic conditions for identifying and classifying the nonsmooth singularities associated with nonsmooth bifurcations. We thoroughly illustrate our theory on an extended applied example and on the identification and classification of all possible equivalence classes in two-dimensional linear complementarity problems.

For a given function f: ℝ k → ℝ k we mean by the complementarity problem, corresponding to f, the problem CP: Find x ∈ ℝ k , such that x ≥ 0, f(x) ≥ 0, xT f(x) = 0. If f is an affine function, say f(x) = q + Mx for all x ∈ ℝ n for some n × n-matrix M and q ∈ ℝ n , then we call the problem a linear complementarity problem (LCP). Let us denote the solution set of CP by O(f) and the solution set of LCP, corresponding to (q, M), by O(q, M). Hence, \(O(f) = \{ \hat x \in {\mathbb{R}^k}|\hat x \geqslant 0,f(\hat x) \geqslant 0,{\hat x^ \top }f(\hat x) = 0\}\) etc. Many problems in Operations Research, mathematical economics and game theory can be translated into a complementarity problem. We give some examples in the following exercises.

It is shown that the linear complementarity problem of finding an n-by-1 vector x such that Mx+q≧0, x≧0, and x T(Mx+q)=0, where M is a given n-by-n real matrix and q is a given n-by-1 vector, is solvable if and only if the linear program: minimize p T x subject to Mx+q≧0, x≧0, is solvable, where p is an n-by-1 vector which satisfies certain conditions. Furthermore each solution of the linear program, solves the linear complementarity problem. For a number of special cases including those when M has nonpositive off-diagonal elements, or when M is strictly or irreducibly diagonally dominant, or when M is a positive matrix with a dominant diagonal columnwise, p is very easily determined and the linear complementarity problem can be solved as an ordinary linear program. Examples of linear complementarity problems are given which can be solved as linear programs, but not by Lemke's method or the principal pivoting method.Key wordsLinear Complementarity ProblemLinear Programming

New characterizations of row sufficient matrices

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 linear complementarity problem is a mathematical problem with important theoretical value and wide practical applications. Research on linear complementarity problems mainly focuses on algorithm improvement and innovation; Combining with fields such as machine learning and artificial intelligence to expand the application scope of linear complementarity problems; Further research on the properties of linear complementarity problems, the existence and uniqueness of solutions, and other theoretical issues. In this paper, by virtue of the properties of DZ-matrices and DZ-B matrices, error bound estimation formulas for the extended vertical linear complementarity problem are given respectively for block matrices composed of DZ-matrices and DZ-B matrices under the condition of satisfying the row W-property. At the same time, the feasibility and superiority of the results are illustrated by numerical examples.