An optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications

• Published in: Mathematical programming Vol. 135; no. 1-2; pp. 195 - 220
• Main Authors: Dostál, Zdeněk, Kozubek, Tomáš
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2012; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Calculus of variations and optimal control; Calculus of Variations and Optimal Control; Optimization; Cantilever beams; Combinatorics; Conjugate gradients; Eigenvalues; Exact sciences and technology; Friction; Full Length Paper; Innovations; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Minimization; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Quadratic programming; Sciences and techniques of general use; Theoretical; 90C20; 90C25; 65K10; 90C90; QPQC with separable constraints; Rate of convergence; Spherical constraints; Error estimation; Active set method; Modeling; Convex programming; Interval arithmetic; Cantilever beam; Bounded solution; Separation method; Convergence rate; Variational calculus; Convex function; Eigenvalue problem; Hessian matrices; Mathematical programming; Conjugate gradient method; Optimal algorithm; Minimization; Quadratic programming; Friction; Particular solution; Equality constraint; Quadratic function
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-011-0454-2

We propose a modification of our MPGP algorithm for the solution of bound constrained quadratic programming problems so that it can be used for minimization of a strictly convex quadratic function subject to separable convex constraints. Our active set based algorithm explores the faces by conjugate gradients and changes the active sets and active variables by gradient projections, possibly with the superrelaxation steplength. The error estimate in terms of extreme eigenvalues guarantees that if a class of minimization problems has the spectrum of the Hessian matrix in a given positive interval, then the algorithm can find and recognize an approximate solution of any particular problem in a number of iterations that is uniformly bounded. We also show how to use the algorithm for the solution of separable and equality constraints. The power of our algorithm and its optimality are demonstrated on the solution of a problem of two cantilever beams in mutual contact with Tresca friction discretized by almost twelve millions nodal variables.