An interior point method and Sherman-Morrison formula for solving large scale convex quadratic problems with diagonal Hessians

Abstract

We develop an approach for solving large scale convex quadratic problems with quadratic matrices subject to linear equalities and box-constraints. These problems appear in real-life applications. At first glance, this is a simple convex optimisation problem. However, the size of this problem (109 variables and 106 constraints for some applications) makes it very challenging to apply traditional convex optimisation techniques. Therefore, one needs to develop a specific algorithm for solving such kind of problems. We apply a combination of the Interior Point method and Sherman--Morrison formula to solve this problem. We test our approach on smaller size datasets (1000 variables and 100 constraints). Our numerical experiments show that this combination is efficient, fast and computationally stable. This approach is suitable for large scale convex quadratic optimisation problems