28-11-2012, 02:59 PM
Quadratic Optimization Models and Algorithms. One Algorithm for Linear Bound Constraints based on Neural Networks
Abstract:
The topic begins with several quadratic programming (QP) models (1. unconstrained model; 2. QP with linear and
symmetric bound constraints; 3. QP with linear bound constraints; 4. QP with one quadratic constraint; 5. QP in
standard form). The solving of QP models 2, 3 and 4 is associated with a neural network frame. For QP models 2
and 3 a preconditioning technique is developed. This technique reduces the susceptibility of the system to round off
errors. Two algorithms of preconditioning are presented: the preconditioning algorithm 1 is based on one associated
matrix and the preconditioning algorithm 2 is based on two associated matrices. Both algorithms are used in several
applications. Each application ends by a test of correctitude of computations, which validates the theory. The solving
of models 2 and 3 is done by a general neural network algorithm. For model 5 a dual quadratic problem (DQP) is
associated. The DQP is studied in two cases: for invertible matrix and for non-invertible matrix. In the first case an
iterative algorithm is developed ( based on Hildreth and D’ Esopo ideas). Numerical examples illustrate the theory.