The MIN PFS problem and piecewise linear model estimation

AbstractWe consider a new combinatorial optimization problem related to linear systems (MIN PFS) that consists, given an infeasible system, in finding a partition into a minimum number of feasible subsystems. MIN PFS allows formalization of the fundamental problem of piecewise linear model estimation, which is an attractive alternative when modeling a wide range of nonlinear phenomena. Since MIN PFS turns out to be NP-hard to approximate within every factor strictly smaller than 3/2 and we are mainly interested in real-time applications, we propose a greedy strategy based on randomized and thermal variants of the classical Agmon–Motzkin–Schoenberg relaxation method for solving systems of linear inequalities. Our method provides good approximate solutions in a short amount of time. The potential of our approach and the performance of our algorithm are demonstrated on two challenging problems from image and signal processing. The first one is that of detecting line segments in digital images and the second one that of modeling time-series using piecewise linear autoregressive models. In both cases the MIN PFS-based approach presents various advantages with respect to conventional alternatives, including wider range of applicability, lower computational requirements and no need for a priori assumptions regarding the underlying structure of the data

Some Structural and Algorithmic Properties of the Maximum Feasible Subsystem Problem

We consider the problem Max FS: For a given infeasible linear system, determine a largest feasible subsystem. This problem has interesting applications in linear programming as well as in fields such as machine learning and statistical discriminant analysis. Max FS is NP -hard and also difficult to approximate. In this paper we examine structural and algorithmic properties of Max FS and of irreducible infeasible subsystems (IISs), which are intrinsically related, since one must delete at least one constraint from each IIS to attain feasibility. In particular, we establish: (i) that finding a smallest cardinality IIS is NP-hard as well as very dicult to approximate; (ii) a new simplex decomposition characterization of IISs; (iii) that for a given clutter, realizability as the IIS family for an infeasible linear system subsumes the Steinitz problem for polytopes; (iv) some results on the feasible subsystem polytope whose vertices are incidence vectors of feasible subsystems of..

Irreducible Infeasible Subsystem Decomposition for Probabilistically Constrained Stochastic Integer Programs

This dissertation explores methods for finding irreducible infeasible subsystems (IISs) of systems of inequalities with binary decision variables and for solving probabilistically constrained stochastic integer programs (SIP-C). Finding IISs for binary systems is useful in decomposition methods for SIP-C. SIP-C has many important applications including modeling of strategic decision-making problems in wildfire initial response planning. New theoretical results and two new algorithms to find IISs for systems of inequalities with binary variables are developed. The first algorithm uses the new theory and the method of the alternative polyhedron within a branch-and-bound (BAB) approach. The second algorithm applies the new theory and the method of the alternative polyhedron to a system in which zero/one box constraints are appended. Decomposition schemes using IISs for binary systems can be used to solve SIP-C. SIP-C is challenging to solve due to the generally non-convex feasible region. In addition, very weak lower (upper) bounds on the objective function are obtained from the linear programming (LP) relaxation of the deterministic equivalent problem (DEP) to SIP-C. This work develops a branch-and-cut (BAC) method based on IIS inequalities to solve SIP-C with random technology matrix and random righthand- side vector. Computational results show that the LP relaxation of the DEP to SIP-C can be strengthened by the IIS inequalities. SIP-C modeling can be applied to wildfire initial response planning. A new methodology for wildfire initial response that includes a fire behavior simulation model, a wildfire risk model, and SIP-C is developed and tested. The new method- ology assumes a known standard response needed to contain a fire of given size. Likewise, this methodology is used to evaluate deployment decisions in terms of the number of firefighting resources positioned at each base, the expected number of escaped and contained fires, as well as the wildfire risk associated with fires not receiving a standard response. A study based on the Texas district 12 (TX12) that is one of the Texas A&M Forest Service (TFS) fire planning units in east Texas demonstrates the effectiveness of the new methodology towards making strategic deployment decisions for wildfire initial response planning

Preuves de non réalisabilité et filtrage de domaines pour les problèmes de satisfaction de contraintes : application à la confection d'horaires

Contexte global -- Objectifs de cette thèse -- Organisation de la thèse -- Notions préliminaires -- Problèmes de coloration de graphes -- Les problèmes de sastisfaction de contraintes -- Problème SAT de satisfaisabilité booléenne -- Programmation par contraintes -- Sous-ensembles incohérents irréductibles -- Revue de la littérature concernant l'extraction d'IIS dans les CSP, la résolution du problème SAT et l'extraction d'IIS pour le problème SAT -- Détection de sous-ensembles incohérents dans des CSP -- Le problème SAT et sa résolution -- Utilisation d'heuristiques pour trouver des sous-ensembles incohérents minimaux pour le problème SAT -- Algorithmes de détection d'IIS -- Autres procédures -- Algorithme tabou pour Max WSAT -- Détails d'implémentation -- Résultats expérimentaux -- Revue de la littérature concernant le filtrage de contraintes globales de CSP -- Algorithme de filtrage pour la contrainte AllDifferent -- Algorithme de filtrage de domaines pour la contrainte SomeDifferent -- Autres travaux concernant le filtrage de contraintes globales -- Algorithme de filtrage pour la contrainte SomeDifferent -- Description de l'algorithme de filtrage -- Résultats expérimentaux -- Revue de la littérature concernant le problème de confection d'horaires pour le personnel navigant aérien -- Les méthodes de résolution du PBS -- Détection de sous-ensembles incohérents minimaux dans le problème de confection d'horaires pour le personnel navigant aérien -- Algorithmes de détection de sous-ensembles incohérents minimaux -- Algorithme tabou -- Algorithme exact de vérification des sous-problèmes incohérents -- Résultats expérimentaux -- Méthodes de recherche locale