# Metric selection and scale-based preconditioning in quadratic optimization problem

I'm going to use scale-based preconditioning in a quadratic optimization problem: minimize $x^T Q x + p^T x$ such that $A x + b = 0$ and $D x + E \leq 0$, I want to speed up finding the optimal $x$ (note that $x$ is a vector with size $n$)by scaling the coefficient matrices ,I have studied the article "The ADMM algorithm for distributed quadratic problems: parameter selection and constraint preconditioning" and "Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems" and so many other articles on this issue, but all of them recommend an algorithm that costs time and computation , so it is not applicable in my case,I simply need to scale coefficient matrices to decrease the number of iterations needed to reach optimal point,Can you suggest an article or some thing that gives the best scaling rules?Or some hints on how should I choose the scaling matrix ?

I have tried $Q$ diagonalization and normalization ,both mehtods caused a little speed up.(I guess it can get better by using more sophisticated scaling).

• A simple scaling is generally not terribly effective as a preconditioner. What do you know about $Q,A,D$? Where do they come from, and what structural properties do you have for them? – Wolfgang Bangerth Jul 14 '18 at 3:25
• @WolfgangBangerth Q is a symmetric positive semi definite matrix($n \times n$), A is $m \times n$ matrix in which m is the number of equalities(so is D) .So what preconditioning that doesn't consume too much time, do you suggest ? – MAh2014 Jul 14 '18 at 5:29
• It all depends on where these matrices come from. Are they discretizations of differential equations, for example? – Wolfgang Bangerth Jul 15 '18 at 9:19
• @WolfgangBangerth actually there is no restriction on these matrices, it depends on the problem that user is trying to solve,and may be some problem that has been converted to a quadratic problem by means of changing variables. – MAh2014 Jul 15 '18 at 9:51
• Then there is no good preconditioner. Good preconditioners use the structure of matrices. If you don't know anything about the matrix, you will not be able to devise a good preconditioner. – Wolfgang Bangerth Jul 20 '18 at 8:47