Optimal Quadratic and Qcqp Algorithms with Applications

This book presents cutting-edge algorithms for solving large-scale quadratic programming (QP) and/or QCQP. While applying these algorithms to the class of QP problems with the spectrum confined to a positive interval, the theory guarantees finding the prescribed precision solution through a uniformly bounded number of simple iterations, like matrix-vector multiplications.

Key concepts explored include the active set strategy, spectral gradients, and augmented Lagrangian methods. The book provides a comprehensive quantitative convergence theory, avoiding unspecified constants. Through detailed numerical experiments, the author demonstrates the algorithms’ superior performance compared to traditional methods, especially in handling large problems with sparse Hessian. The performance of the algorithms is shown on large-scale (billions of variables) problems of mechanics, optimal control, and support vector machines.

Ideal for researchers and practitioners in optimization and computational mathematics, this volume is also an introductory text and a reference for advanced studies in nonlinear programming. Whether you’re a scholar in applied mathematics or an engineer tackling complex optimization challenges, this book offers valuable insights and practical tools for your work.

Zdeněk Dostál is a professor at the Department of Applied Mathematics and Senior Researcher at IT4Innovations National Supercomputing Center, VSB-Technical University of Ostrava. Zdeněk works in Numerical Linear Algebra, Optimization, and Computational Mechanics. He published his results in more than 120 papers (Scopus). He is an author of the book ’Optimal Quadratic Programming Algorithms’ (Springer 2009) and coauthor of ’Scalable Algorithms for Contact Problems’ (Springer 2017) on massively parallel algorithms with theoretically supported linear (optimal) complexity. His current research concerns QP, QCQP, and generalization of the above results to H-TFETI and H-TBETI.