J. Gleeson and J. Ryan. Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 3:61--63, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of many overlapping IISs, this does not provide enough information to repair the original system. To achieve feasibility, one must delete at least one inequality from each IIS. If all IISs were known, the complementary version of MAX FS could be formulated as the following covering problem [26]. MIN IIS COVER: Given an infeasible system # : and the set of all its IISs, minimize y i subject to i#C # C, 1 , 1 m. Note that can grow exponentially with m and n [17] An exact algorithm based on a partial cover formulation is proposed in [38, 39] and ....

....m n 1. Now let # : b be an infeasible system which is not necessarily an IIS. The following result relates the IISs of # to the vertices of a given alternative polyhedron. Recall that the support of a vector is the set of indices of its nonzero components. Theorem 2 (Gleeson and Ryan [26]) Let # : b be an infeasible system with A, b as above. Then the IISs of # are in one to one correspondence with the vertices of the polyhedron P : yA = 0, yb = 0 . In particular, the nonzero components of any vertex of P index an IIS. See [39] for this statement that slightly ....

J. GLEESON AND J. RYAN, Identifying minimally infeasible subsystems of inequalities, ORSA J. Comput. 2, no. 1 (1990), pp. 61--63.

....and the (dual) system of linear inequalities # # # b 0 (2) 333 A variant of the well known Farkas lemma (see for instance [2] for systems of linear inequalities asserts that exactly one of the following holds: 1) is feasible, 2) is feasible. Basing on this, Gleeson and Ryan [4] showed that an irreducible infeasible subsystem (IIS, see e.g. 1] of a given system of linear inequalities can be selected by solving the new (dual) system # # # b # 1 (3) If (3) is infeasible, 1) is feasible. On the contrary, if (3) is feasible, 1) is infeasible and each IIS ....

J. Gleeson and J. Ryan. Identifying Minimally Infeasible Subsystems of Inequalities. ORSA Journal on Computing Vol.2 N.1 61-63, 1990.

....when there are many overlapping IISs, this does not provide enough information to repair the original system. To achieve feasibility, one must delete an inequality from each IIS. If all IISs were known, the complementary version of Max FS could be formulated as the following covering problem [17]. Min IIS Cover: Given an infeasible system : fAx bg with A 2 IR m n and b 2 IR m and the set C of all its IISs, minimize P m i=1 y i subject to P i2C y i 1 8C 2 C, y i 2 f0; 1g, 1 i m. Note that jCj can grow exponentially with m and n [10] An exact algorithm based on a ....

....m n 1. Now let : fAx bg be an infeasible system which is not necessarily an IIS. The following result relates the IISs of to the vertices of a given alternative polyhedron. Recall that the support of a vector is the set of indices of its nonzero components. Theorem 2 (Gleeson and Ryan [17]) Let : fAx bg be an infeasible system with A; b as above. Then the IISs of are in one to one correspondence with the supports of the vertices of the polyhedron P : fy 2 IR m j yA = 0; yb 1; y 0g : The inequality in the alternative system can obviously be replaced by the equation ....

J. Gleeson and J. Ryan, Identifying minimally infeasible subsystems of inequalities, ORSA Journal on Computing, 2 (1990), pp. 61-63.

....inspection. With the new software and hardware advances, dealing with infeasible programs is becoming a major bottleneck in linear programming. Several methods have been proposed in order to try to locate the source of infeasibility. While the first ones looked for minimal infeasible subsystems [28, 15], the later ones aim at removing as few constraints as possible to achieve feasibility [29, 53, 51, 50, 14] In fact, the more practical approach in which the modeler is allowed to weight the constraints according to their importance and flexibility leads to weighted versions of Min ULR [51, 50] ....

J. Gleeson and J. Ryan. Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 3:61--63, 1990.

....mentioned below) on the diagnosis of infeasibilities in mathematical programming models. In general, infeasibility diagnosis is a non trivial task, and automated assistance for it is certainly desirable in a computer aided modeling system. In the case of linear programming models, Gleeson and Ryan [17] describe an efficient method for identifying a minimally infeasible subsystem (also called irreducibly inconsistent system or IIS [28] of constraints, a subsystem of Ax b that is infeasible, but which could be made feasible by dropping any equality from it. It is useful to identify such ....

Gleeson, John, and Jennifer Ryan, "Identifying Minimally Infeasible Subsystems of Inequalities," ORSA Journal on Computing 2:1, pp. 61-- 63, 1991.

....with respect to s 1 , is an I IS. Van Loon also notes that by pivoting through other bases of the Phase 1 LP other I ISs can be identified. By using I ISs as a tool in LP infeasibility analysis, Van Loon provides the foundation of much of the more recent work in the area. Gleeson and Ryan [19] derive a geometric approach to I IS identification that is based upon Farkas Theorem of the Alternative and basic polyhedral theory. They identify an alternative system whose extreme vertices are in one to one correspondence with the I ISs of the original infeasible LP. In the absence of ....

....are in one to one correspondence with the I ISs of the original infeasible LP. In the absence of degeneracy of the alternative system, this improves upon the work of Van Loon in that every pivot in the alternative system identifies a unique I IS in the original system. Theorem 2. 6 (Gleeson Ryan [19]) Let Ax b denote an inconsistent set of inequalities. Then the I ISs are in 1 1 correspondence with the extreme points of the polyhedron P = fy 2 m j y T A = 0; y T b = Gamma1; y 0g: In particular, the nonzero components of any extreme point of P index an I IS. Once a basic feasible ....

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Gleeson J., and Ryan J., Identifying Minimally Infeasible Subsystems of Inequalities, ORSA Journal on Computing, Vol. 2 (1990), pp. 6163.

....of constraints frequently occur and cannot be repaired by simple inspection. With the new software and hardware advances, dealing with infeasible programs is becoming a major bottleneck in linear programming. Several methods have been proposed in order to try to locate the source of infeasibility [20, 11] and to remove as few constraints as possible to achieve feasibility [21, 37] There have recently been new substantial progresses in the study of the approximability of NP hard optimization problems. Various classes have been defined and different reductions preserving approximability have been ....

J. Gleeson and J. Ryan. Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 3:61--63, 1990.

....new constraint indicate the constraints that contradict it. When the simplified constraint contains only slack variables, it is added into the slack equations; then the Simplex is activated to solve the system. If this system is infeasible, one can apply the technique proposed by Gleeson and Ryan [8] to identify the minimally infeasible subsystems and hence decide which constraints should be removed to obtain feasibility. It follows from the construction of the normal form that the number of slack equations is less than or equal to the number of inequalities in the original system. This is ....

J. Gleeson and J. Ryan. Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 2(1):61--63, Winter 1990.

....errors and to guarantee feasibility. Infeasible programs with thousands of constraints frequently occur and cannot be repaired by simple inspection. Several methods have been proposed in order to try to locate the source of infeasibility. While earlier ones look for minimal infeasible subsystems [28, 15], the later ones aim at removing as few constraints as possible to achieve feasibility [60, 57, 58, 13, 29] The more practical approach in which the modeler is allowed to weight the constraints according to their importance and flexibility leads to weighted versions of Min ULR [58, 57] The ....

J. Gleeson and J. Ryan. Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 3:61--63, 1990.

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J. Gleeson and J. Ryan. Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 3:61--63, 1990.

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