Locating Minimal Infeasible Constraint Sets in Linear Programs

Abstract

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 LPs 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.

On Computing Minimal Equivalent Subformulas

“…MUS extraction algorithms can be broadly characterized as deletion-based [8,3] or insertion-based. Moreover, insertion-based algorithms can be characterized as linear search [11] or dichotomic search [23,21].…”

“…The remainder of this section illustrates how this can be achieved. Let F be a CNF formula partitioned as follows, F = E ∪ R ∪ S. A subformula R is redundant in F iff E ∪S R. Given an under-approximation E of an MES of F, and a working subformula S F, the objective is to decide whether E and S entail all the other clauses of F. MUS extraction algorithms can be organized as (see Section 2.1): (i) deletion-based [8,3], (ii) insertion-based [11,38], (iii) insertion-based with dichotomic search [23,21], and (iv) insertion-based with relaxation variables [28].…”

Towards a Notion of Unsatisfiable Cores for LTL

“…Using UCs to help a user debugging by pointing out a subset of the input as part of some problem is stated explicitly as motivation in many works on cores, e.g., [10,4,9,48].…”

Restoring Satisfiability or Maintaining Unsatisfiability by finding small Unsatisfiable Subformulae

“…Moreover, finding a MUS typically requires much more time than just solving the SAT problem, just like finding an IIS requires much more time than just solving the feasibility of a system of linear inequalities [6]. We therefore introduce the concept of approximation for an unsatisfiable subformula.…”