The Kaczmarz method is an iterative method for solving large systems of equations that projects iterates orthogonally onto the solution space of each equation. In contrast to direct methods such as Gaussian elimination or QR-factorization, this algorithm is efficient for problems with sparse matrices, as they appear in constraint-based user interface (UI) layout specifications. However, the Kaczmarz method as described in the literature has its limitations: it considers only equality constraints and does not support soft constraints, which makes it inapplicable to the UI layout problem. In this paper we extend the Kaczmarz method for solving specifications containing soft constraints, using the prioritized IIS detection algorithm. Furthermore, the performance and convergence of the proposed algorithms are evaluated empirically using randomly generated UI layout specifications of various sizes. The results show that these methods offer improvements in performance over standard methods like Matlab's LINPROG, a well-known efficient linear programming solver.
Algorithms and computer-based tools for analyzing infeasible linear and nonlinear programs have been developed in recent years, but few such tools exist for infeasible mixed-integer or integer linear programs. One approach that has proven especially useful for infeasible linear programs is the isolation of an Irreducible Infeasible Set of constraints (IIS), a subset of the constraints defining the overall linear program that is itself infeasible, but for which any proper subset is feasible. Isolating an IIS from the larger model speeds the diagnosis and repair of the model by focussing the analytic effort. This paper describes and tests algorithms for finding small infeasible sets in infeasible mixed-integer and integer linear programs; where possible these small sets are IISs. With ongoing advances in hardware and software, the bottleneck in linear programming is no longer a model solution, it is the correct formulation of large models in the first place. During initial formulation (or modification), a very large model may prove infeasible, but it is often difficult to determine how to correct it. We present a formulation aid which analyzes infeasible LP s and identifies minimal sets of inconsistent constraints from among the perhaps very large set of constraints defining the problem. This information helps to focus the search for a diagnosis of the problem, speeding the repair of the model. We present a series of filtering routines and a final integrated algorithm which guarantees the identification of at least one minimal set of inconsistent constraints. This guarantee is a significant advantage over previous methods. The algorithms are simple, relatively efficient, and easily incorporated into standard LP solvers. Preliminary computational results are reported. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499. This paper presents ideas from goal programming (GP) used as an accompaniment to linear programming (LP) for the analysis of LP infeasibility. A new algorithm (GPIIS) for the detection of irreducibly inconsistent systems (IIS) of constraints is presented using this approach. The structure necessary for implementing such a procedure into a commercial LP solver is outlined. Results for a selection of infeasible LP problems are given, and conclusions drawn.