R.G. Jeroslow, Logic Based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics 40 (1989).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as lift and project, with an emphasis on current research on the subject. The foundations of this approach were laid in a July 1974 technical report, published 24 years later as an invited paper [1] with a foreword. For additional work on disjunctive programming in the seventies and eighties see [2, 3, 8, 9, 12, 13, 17, 18, 19, 22]. In particular, 2] contains a detailed account of the origins of the disjunctive approach and the relationship of disjunctive cuts to Gomory s mixed integer cut, intersection cuts and others. Disjunctive programming received a new impetus in the early nineties from the work on matrix cones by ....

R.G. Jeroslow, \Logic Based Decision Support: Mixed Integer Model Formulation." Annals of Discrete Mathematics, 40, 1989.

....approach is to try to generate all complete instantiations of a first order formula and to use a fast satisfiability algorithm to find a truth valuation that satisfies the resulting ground level formulas. But the number of instantiations can be astronomically large or even infinite. R. Jeroslow [19, 20] addressed this problem with a partial instantiation (PI) approach. It solves a series of propositional satisfiability problems, each obtained by instantiating one or more of the variables in the last. With luck, the first order satisfiability question is resolved Graduate School of Industrial ....

Jeroslow, R. G. Logic-based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics 40, North-Holland (Amsterdam, 1989).

....the form (4) From this he derived that a subset of continuous space is the feasible set of some mixed integer model if and only if it is the union of finitely many polyhedra, all of which have the same set of recession directions. He proved a similar result for mixed integer nonlinear programming [97, 98]. His analysis provides a general tool for obtaining continuous relaxations for nonconvex regions of continuous space, which again may or may not be practical in a given case. 2.3 Links between Logic and Mathematical Programming Williams was among the first to point out parallels between logic ....

Jeroslow, R. E., Logic-Based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics 40. North-Holland (Amsterdam, 1989).

....problem is (P7) Min T 16 subject to P m i=1 x ij = 1 j = 1; 2; n P n j=1 p ij x ij Gamma T 0 i = 1; 2; m x ij = 0 or 1 8 i j Note that if all p ij are integers, the optimal solution will be such that T is an integer. 3.2 Jeroslow s Representability Theorem R. Jeroslow [70], building on joint work with J. K. Lowe [71] characterized subsets of n space that can be represented as the feasible region of a mixed integer (Boolean) program. They proved that a set is the feasible region of some mixed integer linear programming problem (MILP) if and only if it is the union ....

....We say that a set S ae n is represented by (3) if, x 2 S if and only if (x; y; satisfies (3) for some y; If f is a linear transformation, so that (3) is a MILP constraint set, we will say that S is MILP representable. The main result can now be stated. Theorem 3. 1 (Jeroslow, Lowe [71,70]) A set in n space is MILP representable if and only if it is the union of finitely many polyhedra having the same set of recession directions. 17 3.3 Benders Representation Any mixed integer linear program (MILP) can be reformulated so that there is only one continuous variable. This ....

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R.G.Jeroslow, Logic-Based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics, Volume 40, North Holland, 1989.

....convex hull of the 0 1 points satisfying them. Rather, the difficulty is that there are typically a large number of clauses in CNF. This results in a loose representation of the formula as a whole, even though each individual clause is tightly represented. R. Jeroslow discusses this principle in [10]. Thus when logical constraints occur or can be expressed in the form of elementary cardinality rules, it is far better to give each a convex hull representation directly than to convert it to CNF first. The resulting description may be no shorter than the CNF equivalent, because it may contain ....

Jeroslow, R. S., Logic-Based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics 40, North-Holland (Amsterdam, 1989).

....approach is to try to generate all complete instantiations of a first order formula and to use a fast satisfiability algorithm to find a truth valuation that satisfies the resulting ground level formulas. But the number of instantiations can be astronomically large or even infinite. R. Jeroslow [35, 36] addressed this problem with a partial instantiation (PI) approach. It solves a series of propositional satisfiability problems, each obtained by instantiating one or more of the variables in the last. With luck, the first order satisfiability question is resolved when only a few of the possible ....

Jeroslow, R. G. Logic-based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics 40, North-Holland (Amsterdam, 1989).

....in the MIPLIB library. 1 Introduction Disjunctive programming is optimization over a finite union of convex sets. Its foundations were developed, and the term itself coined in the early seventies by Balas [4, 5] since then it attracted the attention of numerous researchers, including Jeroslow [18, 19], Blair [12] Williams [25] Hooker [15] Beaumont [11] Sherali and Shetty [23] Meyer [21] Besides having an elegant theory, disjunctive programming provides a way to formulate a wide variety of optimization problems, such as mixed integer programs, linear complementarity, job shop scheduling, ....

R. Jeroslow, Logic based decision support: mixed-integer model formulation, Annals of Discrete Mathematics 40, (1989) North Holland, Amsterdam.

....operations research was discussed as early as 1968 in Hammer and Rudeanu s treatise on boolean methods [26] Granot and Hammer [24] suggested in 1971 the possibility of using boolean methods for integer programming. The MLLP approach described here was perhaps first clearly articulated by Jeroslow [43, 44], who was primarily interested in issues of representability. He viewed discrete variables as artifices for representing a feasible subset of continuous space, which in the case of an MLLP or MILP model is a union of finitely many polyhedra. From this it follows that MLLP and MILP models are ....

Jeroslow, R. E., Logic-Based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics 40. North-Holland (Amsterdam 1989).

....till no further changes occur. It is easy to see that both these Simplifications preserve the set of minimal models of P . An additional optimization that we are currently working on is the idea of simultaneously grounding and compacting a program. We are investigating how to use Jeroslow s idea [21] of partial instantiation to devise a row generation algorithm (where rows correspond to ground clauses) 5 Computing Stable Models In this section, we present three alternative approaches for computing stable models of logic programs. We discuss the advantages and disadvantages of the different ....

R. E. Jeroslow. (1989) Logic-based Decision Support: Mixed Integer Model Formulation, North Holland.

....mixed integer programs. 1 Introduction Disjunctive programming is optimization over a finite union of convex sets. Its foundations were developed, and the term itself coined in the early seventies by Balas [4, 5] since then it attracted the attention of numerous researchers, including Jeroslow [17, 18], Blair [11] Williams [25] Hooker [14] Beaumont [10] Sherali and Shetty [23] Meyer [20] Besides having an elegant theory, disjunctive programming provides a natural way to formulate a wide variety of optimization problems, such as (the arguably most important class) mixed 0 1 programs , ....

R. Jeroslow, Logic based decision support: mixed-integer model formulation, Annals of Discrete Mathematics 40, (1989) North Holland, Amsterdam.

.... 1. Here, the component f e of f gives the number of times the weight of hyperarc e has been counted in the computation of v(x; Formulation (P d ) meets a negative property, since it is not an ILP, but rather a Disjunctive Programming problem. However, a bounded MIP representation [9] of (P d ) can be given [6] Denote by (P c ) P d ) and (P v ) the linear relaxations obtained by replacing the constraints x 2 f0; 1g m by x 2 [0; 1] m in the formulations (P c ) P d ) and (P v ) respectively. In the following, S( and z( denote the set of feasible solutions and the ....

R. G. Jeroslow, Logic-based Decision Support - Mixed Integer Model formulation Annals of Discrete Mathematics, 40 (1989).

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R.G. Jeroslow, Logic Based Decision Support: Mixed Integer Model Formulation, Annals of Discrete Mathematics 40 (1989).

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Jeroslow, R.G., Logic-Based Decision Support: Mixed Integer Model Formulation. North-holland. New York (1989).

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