An algorithme for quadratic optimization with one quadratic constraint and bounds on the variables

This paper presents an efficient algorithm to solve a constrained optimisation problem with a quadratic object function, one quadratic constraint and (positivity) bounds on the variables. Against little computational cost, the algorithm allows for the inclusion of positivity of the solution as prior knowledge. This is very useful for the solution of those (linear) inverse problems where negative solutions are unphysical. The algorithm rewrites the solution as a function of the Lagrange multipliers, which is achieved with the help of the generalised eigenvectors, or equivalently, the generalised singular value decomposition. The next step is to find the Lagrange multipliers. The multiplier corresponding to the quadratic constraint, which is known to be active, is easy to find. The Lagrange multipliers corresponding to the positivity constraints are found with an iterative method that can be likened to the activeset methods from quadratic programming