# Sensitivity analysis of QP

Given a quadratic program $$f^* \equiv x^\top Q x + b^\top x \\ x \geq 0 \\ A^\top x = d \\ x \in \mathbb{R}^n$$ I would like to analyze the sensitivity of the solution $$x^*$$ to perturbations in $$Q$$ and $$b$$. Could anything be said regarding variations of $$x^*$$ depending on for instance variations in $$Q$$ and $$b$$ with respect to a matrix norm? One special case that could be interesting is if the equality constraint is replaced by $$1^\top x = 1$$, i.e., the QP is solved on the simplex.

One special case analysis I found is for the linear least squares problem, see [On the Perturbation of Pseudo-Inverses, Projections and Linear Least Squares Problems, SIAM Review Vol. 19, No. 4 (Oct., 1977), pp. 634-662]. However, this does not generalize to the quadratic program in a straightforward manner.

The short version is that you describe the variability of your optimization problem as a function of certain parameters, typically denotes as $$\theta$$. Then, you calculate the optimization problem as a function of $$\theta$$, i.e. $$x^*(\theta)$$. For QPs, the best way to do this is by solving the parametric version of the KKT conditions.
The solution of a parametric optimization problem is a series of so-called "critial regions", i.e. areas where an certain active set continues to be optimal despite a variation of the parameters. For each critical region you can calculate the analytical expression $$x^*(\theta)$$ (also for the Lagrangian multipliers).