Minimal Zero Norm Solutions of Linear Complementarity Problems

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

In this article, we introduce K-type block matrices which include two new classes of block matrices, namely block triangular K-matrices and hidden block triangular K-matrices. We show that the solution of linear complementarity problem with K-type block matrices can be obtained by solving a linear programming problem. We show that block triangular K-matrices satisfy the least element property. We prove that hidden block triangular K-matrices are $$Q_0$$ and processable by Lemke’s algorithm. The purpose of this article is to study properties of K-type block matrices in the context of the solution of linear complementarity problem.

There is much discussion in the Biomagnetism Community about the differences in and usefulness of ‘dipole solutions* and ‘minimum norm solutions’. By ‘dipole solutions’ we mean the wide variety of procedures that assume the current in the brain is well modeled by a small number of current dipoles, and determine the locations and moment vectors of those dipoles. By ‘minimum norm solutions’ we mean the equally diverse group of procedures that do not assume the source is dipolar, and instead minimize the norm of the current, |J|, or a weighted version of the current. We do not wish to discuss the validity of the assumptions made in order to obtain a dipole or minimum norm Solution. Our goal is a quantitative measure for comparing dipole solutions to minimum norm solutions when the true current source is known. We hope that such an objective measure will lead to a more fruitful discussion of the usefulness of dipole solutions and minimum norm solutions, although we acknowledge it will not add anything to the discussion about the validity of the assumptions used to obtain the solutions.

In this article, we present an algorithm to compute the minimum norm solution of the positive semidefinite linear complementarity problem. We show that its solution can be obtained using the alternative theorems and a convenient characterization of the solution set of a convex quadratic programming problem. This problem reduces to an unconstrained minimization problem with once differentiable convex objective function. We propose an extension of Newton's method for solving the unconstrained optimization problem. Computational results show that convergence to high accuracy often occurs in just a few iterations.

We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.

Ben-Israel [1] proposed a method for the solution of the nonlinear least squares problem ${\min _{x \in D}}{\left \| {F(x)} \right \|_2}$ where $F:D \subset {R^n} \to {R^m}$. This procedure takes the form ${x_{k + 1}} = {x_k} - F’{({x_k})^ + }F({x_k})$ where $F’{({x_k})^ + }$ denotes the Moore-Penrose generalized inverse of the Fréchet derivative of F. We give a general convergence theorem for the method based on Lyapunov stability theory for ordinary difference equations. In the case where there is a connected set of solution points, it is often of interest to determine the minimum norm least squares solution. We show that the Ben-Israel iteration has no predisposition toward the minimum norm solution, but that any limit point of the sequence generated by the Ben-Israel iteration is a least squares solution.

We show that the alternating direction implicit algorithm of Peaceman—Rachford can be adapted to solve linear complementarity problems arising from free boundary problems. The alternating direction implicit algorithm is significantly faster than modified SOR algorithms. Some numerical results are given.

Previous article Next article Complementarity Theorems for Linear ProgrammingA. C. WilliamsA. C. Williamshttps://doi.org/10.1137/1012015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Frank Eugene Clark, Mathematical Notes: Remark on the Constraint Sets in Linear Programming, Amer. Math. Monthly, 68 (1961), 351–352 MR1531192 0109.38204 CrossrefISIGoogle Scholar[2] A. J. Goldman, Resolution and separation theorems for polyhedral convex setsLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, 41–51, in [4] MR0089113 0072.37505 Google Scholar[3] A. J. Goldman and , A. W. Tucker, Theory of linear programmingLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, 53–97, in [4] MR0101826 0072.37601 Google Scholar[4] H. W. Kuhn and , A. W. Tucker, Linear Inequalities and Related Systems, Princeton University Press, Princeton, 1956 0072.37502 Google Scholar[5] A. C. Williams, Boundedness relations for linear constraint sets, Linear Algebra and Appl., 3 (1970), 129–141 10.1016/0024-3795(70)90009-1 MR0266612 0201.22003 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Sparse solutions to an underdetermined system of linear equations via penalized Huber loss6 November 2020 | Optimization and Engineering, Vol. 22, No. 3 Cross Ref Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming26 February 2019 | Journal of Optimization Theory and Applications, Vol. 181, No. 3 Cross Ref Necessary and Sufficient Conditions for Noiseless Sparse Recovery via Convex Quadratic SplinesMustafa Ç Pinar12 February 2019 | SIAM Journal on Matrix Analysis and Applications, Vol. 40, No. 1AbstractPDF (416 KB)A labeling algorithm for the sensitivity ranges of the assignment problemApplied Mathematical Modelling, Vol. 35, No. 10 Cross Ref Bibliography15 August 2011 Cross Ref Sensitivity Analysis and Dynamic Programming15 February 2011 Cross Ref Bibliography Cross Ref Condition measures and properties of the central trajectory of a linear programMathematical Programming, Vol. 83, No. 1-3 Cross Ref New characterizations of ℓ1 solutions to overdetermined systems of linear equationsOperations Research Letters, Vol. 16, No. 3 Cross Ref Limiting behavior of weighted central paths in linear programmingMathematical Programming, Vol. 65, No. 1-3 Cross Ref Stability of linearly constrained convex quadratic programsJournal of Optimization Theory and Applications, Vol. 64, No. 1 Cross Ref Marginal values in mixed integer linear programmingMathematical Programming, Vol. 44, No. 1-3 Cross Ref A theory of linear inequality systemsLinear Algebra and its Applications, Vol. 106 Cross Ref Boundedness relations in linear semi-infinite programmingAdvances in Applied Mathematics, Vol. 8, No. 1 Cross Ref A Variable-Complexity Norm Maximization ProblemO. L. Mangasarian and T. -H. Shiau17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 3AbstractPDF (698 KB)Simple computable bounds for solutions of linear complementarity problems and linear programs26 February 2009 Cross Ref On polyhedral extension of some LP theoremsMathematical Programming, Vol. 30, No. 1 Cross Ref Polyhedral extensions of some theorems of linear programmingMathematical Programming, Vol. 24, No. 1 Cross Ref Optimal simplex tableau characterization of unique and bounded solutions of linear programsJournal of Optimization Theory and Applications, Vol. 35, No. 1 Cross Ref Projection and Restriction Methods in Geometric Programming and Related Problems Cross Ref Representation of Convex Sets Cross Ref Projection and restriction methods in geometric programming and related problemsJournal of Optimization Theory and Applications, Vol. 26, No. 1 Cross Ref The complementary unboundedness of dual feasible solution sets in convex programmingMathematical Programming, Vol. 12, No. 1 Cross Ref Theorems on the dimension of convex setsLinear Algebra and its Applications, Vol. 12, No. 1 Cross Ref On the primal and dual constraint sets in geometric programmingJournal of Mathematical Analysis and Applications, Vol. 32, No. 3 Cross Ref Volume 12, Issue 1| 1970SIAM Review History Submitted:23 December 1968Published online:18 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1012015Article page range:pp. 135-137ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Missing data estimation for run-to-run EWMA-controlled processes

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

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group(X¯1,X¯2,…,X¯q)in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

Iterative methods applied to the normal equationsA T Ax=A T b are sometimes used for solving large sparse linear least squares problems. However, when the matrix is rank-deficient many methods, although convergent, fail to produce the unique solution of minimal Euclidean norm. Examples of such methods are the Jacobi and SOR methods as well as the preconditioned conjugate gradient algorithm. We analyze here an iterative scheme that overcomes this difficulty for the case of stationary iterative methods. The scheme combines two stationary iterative methods. The first method produces any least squares solution whereas the second produces the minimum norm solution to a consistent system.

Distributed linear solutions are widely used in source localization to solve the ill-posed EEG/MEG inverse problem. In the classical approach based on dipole sources, these methods estimate the current densities at a great number of brain sites, typically at the nodes of a 3-D grid which discretizes the chosen solution space. The estimated current density distributions are displayed as brain electromagnetic tomography (BET) images. We have tested well known minimum norm solutions (MN, WMN, LORETA) and other linear inverse solutions [WROP, sLORETA, interference uniform, gain uniform, weight vector normalized (WVN), and a new solution named SLF (Standardized Lead Field)], using a MEG configuration (BTi Magnes 2500 WH with 148 axial magnetometers) and a realistic head model using BEM (Boundary Element Method). The solutions were compared in a noise-free condition and in the presence of noise using the classical dipole localization errors (DLE) together with a new figure of merit that we called max gain uniformity, which measures the capability of an inverse linear solution to show spots of activity with similar amplitudes on the brain electromagnetic tomographies when multiple dipole sources with similar moments are simultaneously active. Whereas some solutions (sLORETA, interference uniform and SLF) were capable of zero dipole localization errors in the noise-free case, none of them reached 100% of correct dipole localizations in the presence of a high level of Gaussian noise. The SLF solution, which has the advantage to be independent from any regularization parameter, presented the best results with the lowest max gain uniformities, with almost 100% of correct dipole localizations with 10% of noise and more than 90% of correct localizations with 30% of noise added to the data. Nevertheless, no solution was able to combine at the same time a correct localization of single sources and the capability to visualize multiple sources with comparable amplitudes on the brain electromagnetic tomographies.

In this paper, we first study a hierarchical problem of Baillon’s type, and we study a strong convergence theorem of this problem. For the special case of this convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon’s type. Our result of the ergodic theorem of Baillon’s type improves and generalizes many existence theorems of this type of problem. Two numerical examples are given to demonstrate our results. As applications of our convergence theorem of the hierarchical problem, we study the unique solution for the following problems: mathematical programming with multiply sets split variational inclusion and fixed point set constraints; mathematical programming with multiple sets split variational inequalities and fixed point set constraints; the variational inequality problem with a system of mixed type equilibria and fixed point set constraints; the variational inequality problem with multiple sets split system of mixed type equilibria and fixed point set constraints; mathematical programming with a system of mixed type equilibria and fixed point set constraints. We give iteration processes for these types of problems and establish the strong convergence for the unique solution of these problems. For our special case, our results can be reduced to the following problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibria problem; the minimum norm solution of the system of mixed type equilibria problem. Our results will have many applications in diverse fields of science.

Inverse electromagnetic scattering problems related to objects having inhomogeneous isotropic impedance boundaries are addressed by considering small impedance variations between a measured object and a reference one. For many non-destructive electromagnetic testing, the structure geometries are known and only coated materials properties variations from reference numerical model are of interest. Consequently, the scattered field and equivalent surface current of the reference object can be precisely evaluated. We use the Chu-Stratton integral equation to express the differences of scattered far field between the measured and reference configurations in terms of surface current variation. Physical Optics approximation is used to derive a model of these surface current variations and we keep the first order term to solve a linear inverse problem. Consequently, the surface impedance can be reconstructed by a minimum norm solution in the least squares sense. Some promising numerical results on spherical geometries are presented with fixed incidence and frequency, in bistatic configuration. © Institution of Engineering and Technology.All Rights Reserved.