A hierarchy of relaxations and convex hull characterizations for mixed-integer zero—one programming problems

Sherali, Hanif D.; Adams, Warren P.

doi:10.1016/0166-218x(92)00190-w

Cited by 258 publications

(134 citation statements)

References 8 publications

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“…A different lift-and-project method was suggested by Sherali and Adams [39]. Chronologically, this came before Lovász and Schrijver, but for the purposes of exposition it makes more sense to reverse the order.…”

Section: Connection With Lift-and-project Methodsmentioning

confidence: 99%

“…Both SA and LS achieve this by letting P k be the polytope defined by the inequalities that have an SA k or LS k -proof from the inequalities that define the initial polytope. Moreover, and this is the main point of the methods, in both cases there is an algorithm running in time n O(k) to optimize any given linear objective function over the polytope P k (see [39], [33]). …”

Section: Degree-related Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

Atserias

2013

Theory and Applications of Satisfiability Testing – SAT 2013

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Abstract. In recent years there has been some progress in our understanding of the proof-search problem for very low-depth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNF-formulas (i.e. R(k) systems), or polynomial inequalities (i.e. semi-algebraic proof systems). In this talk I will overview this progress. I will start with bounded-width resolution, whose specialized proof-search algorithm is as easy as uninteresting, but whose proof-search problem is unintentionally solved by certain versions of conflict-driven clause-learning algorithms with restarts. I will continue with R(k) systems, whose proof-search problem turned out to hide the complexity of certain two-player games of interest in the area of systems synthesis and verification. And I will close with bounded-degree semialgebraic proof systems, whose proof-search problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems.

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Section: Connection With Lift-and-project Methodsmentioning

confidence: 99%

Section: Degree-related Algorithmsmentioning

confidence: 99%

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

Atserias

2013

Theory and Applications of Satisfiability Testing – SAT 2013

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“…Next, we use conic duality to construct the set of valid linear inequalities for the tighter relaxationC and show how to interpret these inequalities as linear and convex quadratic valid inequalities for C • . The tighter relaxations are generated by suitably extending the sequential convexification approaches proposed by Balas, Ceria and Cornuéjols [6]; Sherali and Adams [40,41]; Lovász and Schrijver [31]; and Lasserre [25] (see also [26]). While some of these hierarchies were proposed for pure 0-1 LPs, they can all be extended to mixed 0-1 LPs.…”

Section: Lemmamentioning

confidence: 99%

“…The intent of this paper is to show that cut generation strategies for general mixed 0-1 LPs, such as the Gomory cuts [36] and cuts from tighter convex relaxations [6,31,41,25], extend in a very natural manner to general MCPs. While many of our results follow from those in the literature, these extensions have never been carefully explored.…”

Section: Introductionmentioning

confidence: 99%

Cuts for mixed 0-1 conic programming

2005

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In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

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“…In this work the nonconvex MINLP will be transformed into an MILP as will be shown in the next paragraphs. Theoretically (Sherali andAdams, 1989 , Lovacz andShrijver , 1989), it is possible to transform the MILP model to an LP. This step however may require an exponential number of steps and therefore is not useful for practical purposes.…”

Section: Minlp Modelmentioning

confidence: 99%

Optimal synthesis of multiproduct batch plants with cyclic scheduling and inventory considerations

Voudouris¹,

Grossmann

1993

Ind. Eng. Chem. Res.

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This paper addresses the problem of determining the optimal configuration and cyclic operation of batch plants in which all the products require the same processing sequence. In particular, the problem can be stated as follows. Given are demands of a number of products, as well as technical information on the processing tasks (size factors, processing times, clean-up times) which are not restricted to a zero-wait policy. Given are also cost data for investment and product inventories, a list of candidate equipment and a list of candidate storage vessels with standard sizes. The problem then consists in determining the following items: number, type and size of equipment, as well as their allocation to one or multiple tasks and possible parallel operation; location and size of intermediate storage vessels; the length of the production cycle including the sequence of production of the products; levels of product inventories. The objective is to maximize the net present value. The major complication of this design problem lies in the many trade-offs that are involved, as for instance the merging of tasks versus its impact on the schedule, and length of production cycle versus inventory levels. By using a novel representation for cyclic schedules and exact linearization schemes, it is shown that this problem can for formulated as a mixed-integer linear programming problem, and solved rigorously to global optimality. An efficient computational scheme is proposed for this purpose. Compared to the previous work by Birewar and Grossmann (1990), the proposed model provides a significant extension of the scope of the operational problem, while at the same time yielding an optimization problem that does not involve nonlinearities. Several example problems are presented to illustrate the capability of this method.Introduction.

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A hierarchy of relaxations and convex hull characterizations for mixed-integer zero—one programming problems

Cited by 258 publications

References 8 publications

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

Cuts for mixed 0-1 conic programming

Optimal synthesis of multiproduct batch plants with cyclic scheduling and inventory considerations

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