Extending linear relaxation for non-square matrices and soft constraints

Noreen Jamil, Johannes Müller, M. Asif Naeem, Christof Lutteroth, Gerald Weber
2016 Journal of Computational and Applied Mathematics  
Linear relaxation is a common method for solving linear problems, as they occur in science and engineering. In contrast to direct methods such as Gausselimination or QR-factorization, linear relaxation is particularly efficient for problems with sparse matrices, as they appear in constraint-based user interface (UI) layout specifications. However, the linear relaxation method as described in the literature has its limitations: it works only with square matrices and does not support soft
more » ... nts, which makes it inapplicable to the UI layout problem. To overcome these limitations we propose two algorithms for selecting the pivot elements used during linear relaxation: random pivot assignment, and a more complex deterministic pivot assignment. Furthermore, we propose three algorithms for solving specifications containing soft constraints: prioritized IIS detection, prioritized deletion filtering and prioritized grouping constraints. With these algorithms, it is possible to prioritize constraints: if there are conflicting constraints in a specification, as it is commonly the case for UI layout, only the constraints with lower priority are violated to resolve the conflicts. 1 empirically using randomly generated UI layout specifications of various sizes. The results show that our best linear relaxation algorithm performs significantly better than Matlab's LINPROG, a well-known efficient linear programming solver.
doi:10.1016/j.cam.2016.05.006 fatcat:4k6byy6ogzghzi4u6ccxne3n3q