From Exact to Approximate Solutions

Abstract

The exact constraint Ax = b is often relaxed, with an approximated equality measured using the quadratic penalty function $$Q\left(\mathbf{X}\right)=\parallel \mathbf{A}\mathbf{x}-\mathbf{b}{\parallel^{2}_{2}}.$$Such relaxation allows us to (i) define a quasi-solution in case no exact solution exists (even in cases where A has more rows than columns); (ii) exploit ideas from optimization theory; (iii) measure the quality of a candidate solution; and more.