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Mathematics | Optimal Quadratic Programming Algorithms - With Applications to Variational Inequalities

Optimal Quadratic Programming Algorithms

With Applications to Variational Inequalities

Dostál, Zdenek

2009

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  • The first monograph to present the solution to quadratic programming problems, a topic usually addressed only in journal publications
  • Offers theoretical and practical results in the field of bound-constrained and equality-constrained optimization
  • Provides algorithms with the rate of convergence independent of constraints
  • Develops theoretically supported scalable algorithms for variational inequalities
  • Comprehensive presentation of working set methods and inexact augmented Lagrangians

Solving optimization problems in complex systems often requires the implementation of advanced mathematical techniques. Quadratic programming (QP) is one technique that allows for the optimization of a quadratic function in several variables in the presence of linear constraints. QP problems arise in fields as diverse as electrical engineering, agricultural planning, and optics. Given its broad applicability, a comprehensive understanding of quadratic programming is a valuable resource in nearly every scientific field.

Optimal Quadratic Programming Algorithms presents recently developed algorithms for solving large QP problems. The presentation focuses on algorithms which are, in a sense optimal, i.e., they can solve important classes of problems at a cost proportional to the number of unknowns. For each algorithm presented, the book details its classical predecessor, describes its drawbacks, introduces modifications that improve its performance, and demonstrates these improvements through numerical experiments.

This self-contained monograph can serve as an introductory text on quadratic programming for graduate students and researchers. Additionally, since the solution of many nonlinear problems can be reduced to the solution of a sequence of QP problems, it can also be used as a convenient introduction to nonlinear programming. The reader is required to have a basic knowledge of calculus in several variables and linear algebra.

Content Level » Research

Keywords » Algebra - SOIA - algorithm - algorithms - calculus - complex systems - electrical engineering - linear algebra - linear optimization - nonlinear optimization - optimization - programming - quadratic programming

Related subjects » Applications - Computational Intelligence and Complexity - Computational Science & Engineering - Mathematics

Table of contents 

I Background.- Linear Algebra.- Optimization.- II Algorithms.- Conjugate Gradients for Unconstrained Minimization.- Equality Constrained Minimization.- Bound Constrained Minimization.- Bound and Equality Constrained Minimization.- III Applications to Variational Inequalities.- Solution of a Coercive Variational Inequality by FETI—DP Method.- Solution of a Semicoercive Variational Inequality by TFETI Method.

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