Splitting methods for quadratic optimization in data analysis

Many problems arising in data analysis can be formulated as a large sparse strictly convex quadratic programming problems with equality and inequality linear constraints. In order to solve these problems, we propose an iterative scheme based on a splitting of the matrix of the objective function and called splitting algorithm (SA). This algorithm transforms the original problem into a sequence of subproblems easier to solve, for which there exists a large number of efficient methods in literature. Each subproblem can be solved as a linear complementarity problem or as a constrained least distance problem. We give conditions for SA convergence and we present an application on a large scale sparse problem arising in constrained bivariate interpolation. In this application we use a special version of SA called diagonalization algorithm (DA). An extensive experimentation on CRAY C90 permits to evaluate the DA performance

[1]  J. Pang,et al.  Iterative methods for large convex quadratic programs: a survey , 1987 .

[2]  W. J. Gordon,et al.  Smooth interpolation in triangles , 1973 .

[3]  J. Gregory,et al.  Compatable smooth interpolation in triangles , 1975 .

[4]  Olvi L. Mangasarian,et al.  Asynchronous parallel successive overrelaxation for the symmetric linear complementarity problem , 1988, Math. Program..

[5]  Alvaro R. De Pierro,et al.  Convergence Properties of Iterative Methods for Symmetric Positive Semidefinite Linear Complementarity Problems , 1993, Math. Oper. Res..

[6]  Olvi L. Mangasarian.  Convergence of Iterates of an Inexact Matrix Splitting Algorithm for the Symmetric Monotone Linear Complementarity Problem , 1991, SIAM J. Optim..

[7]  Donald Goldfarb,et al.  A numerically stable dual method for solving strictly convex quadratic programs , 1983, Math. Program..

[8]  C. Lawson.  Software for C1 Surface Interpolation , 1977 .

[9]  Hiroshi Akima,et al.  A Method of Bivariate Interpolation and Smooth Surface Fitting for Irregularly Distributed Data Points , 1978, TOMS.

[10]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[11]  O. Mangasarian,et al.  Parallel successive overrelaxation methods for symmetric linear complementarity problems and linear programs , 1987 .

[12]  G. Nielson.  A method for interpolating scattered data based upon a minimum norm network , 1983 .

[13]  C. Cryer.  The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[14]  Valeria Ruggiero,et al.  Numerical solution of equality-constrained quadratic programming problems on vector-parallel computers , 1993 .

[15]  Philip E. Gill,et al.  Numerical methods for constrained optimization , 1974 .

[16]  I. M. Klucewicz.  A piecewise c1 interpolant to arbitrarily spaced data , 1978 .

[17]  Error analysis of elimination methods for equality constrained quadratic programming problems , 1996 .

[18]  Paul Tseng,et al.  Error Bound and Convergence Analysis of Matrix Splitting Algorithms for the Affine Variational Inequality Problem , 1992, SIAM J. Optim..

[19]  P. Gill,et al.  Sparse Matrix Methods in Optimization , 1984 .

[20]  E. Loli Piccolomini,et al.  A Domain Decomposition Method for Scattered Data Approximation on a Multiprocessor System , 1993, PARCO.

[21]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[22]  A. R. De Pierro,et al.  Accelerating iterative algorithms for symmetric linear complementarity problems , 1994 .

[23]  Nira Dyn,et al.  The numerical solution of equality constrained quadratic programming problems , 1983 .

[24]  Karen T. Medhi,et al.  A two-stage successive overrelaxation algorithm for solving the symmetric linear complementarity problem , 1994, Math. Program..