Cones of Matrices and Set-Functions and 0–1 Optimization

Lovász, László; Schrijver, Alexander

doi:10.1137/0801013

Cited by 849 publications

(794 citation statements)

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“…(by (12)- (14)), and therefore (1 − γ) n j=1 w j x j ≤ W , by (10) and (11), thereby complementing the bound (4). We will apply this technique in our construction.…”

Section: The Disjunctive Proceduresmentioning

confidence: 86%

“…An interesting open question is whether the lift-and-project operators in [10,13,12] can be used to obtain a result similar to Theorem 1.1. However, these operators do not create disjunctions based on the structure of the constraints of an integer program, in particular, the numerical value of coefficients.…”

Section: Resultsmentioning

confidence: 99%

“…Given a 0 − 1 integer program, such operators create "lifted" formulations that are polynomially larger, both in terms of constraints and variables, but which are provably strong. In particular, [10] shows how, in the case of the clique problem, the resulting formulation provably satisfies strong inequalities. Also see [4,5,3,6,7].…”

Section: Motivationmentioning

confidence: 99%

“…Or, using the equivalence between separation and optimization, we can always work in R n but separate vectors from Q, thus obtaining a cutting-plane algorithm. This is the genesis of so-called lift-and-project reformulation operators, see [10,13,2,12,11,4]. Given a 0 − 1 integer program, such operators create "lifted" formulations that are polynomially larger, both in terms of constraints and variables, but which are provably strong.…”

Section: Motivationmentioning

confidence: 99%

See 3 more Smart Citations

Approximate formulations for 0-1 knapsack sets

Bienstock

2008

Operations Research Letters

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“…(by (12)- (14)), and therefore (1 − γ) n j=1 w j x j ≤ W , by (10) and (11), thereby complementing the bound (4). We will apply this technique in our construction.…”

Section: The Disjunctive Proceduresmentioning

confidence: 86%

Section: Resultsmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

See 2 more Smart Citations

Approximate formulations for 0-1 knapsack sets

Bienstock

2008

Operations Research Letters

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“…Of course, the domain {0, 1} n can be enforced by adding the quadratic constraints x 2 i − x i ≥ 0 and x i − x 2 i ≥ 0, but now, if we want to make use of these constraints, we are forced to go beyond positive linear combinations and use some multiplications or squares. The lift-and-project method of Lovász and Schrijver [33] allows these rules but only in the following limited forms:…”

Section: Connection With Lift-and-project Methodsmentioning

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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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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Advances in Logic‐Based Optimization Approaches to Process Integration and Supply Chain Management

Grossmann¹

2005

Chemical Engineering

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Optimization as an enabling technology has been one of the big success stories in process systems engineering. In this paper we present a review on recent research work in the area of logic-based discrete/continuous optimization. In particular, recent advances are presented in the modeling and solution of nonlinear mixed-integer and generalized disjunctive programming, global optimization and constraint programming. The impact of these techniques is illustrated with several examples in the areas of process integration and supply chain management.. where f(x, y) is the objective function (e.g. cost), h(x, y) = 0 are the equations that describe the performance of the system (material balances, production rates), and g(x,y) ≤ 0 are inequalities that define the specifications or constraints for feasible plans and schedules. The variables x are continuous and generally correspond to state variables, while y are the discrete variables, which generally are restricted to take 0-1 values to define for instance the assignments of equipment and sequencing of tasks. Problem (MIP) corresponds to a mixed-integer nonlinear program (MINLP) when any of the functions involved are nonlinear. If all functions are linear it corresponds to a mixed-integer linear program (MILP). If there are no 0-1 variables, the problem (MIP) reduces to a nonlinear program (NLP) or linear program (LP) depending on whether or not the functions are linear.It should be noted that (MIP) problems, and their special cases, may be regarded as steady-state models. Hence, one important extension is the case of dynamic models, which in the case of discrete time models gives rise to multiperiod optimization problems, while for the case of continuous time it gives rise to optimal control problems that contain differential-algebraic equation (DAE) models.Mathematical programming, and optimization in general, have found extensive use in process systems engineering. A major reason for this is that in these problems there are often many alternative solutions, and hence, it is often not easy to find the optimal solution. Furthermore, in many cases the economics is such that finding the optimum solution translates into large savings. Therefore, there might be a large economic penalty to just sticking to suboptimal solutions. In summary, optimization has become a major technology that helps companies to remain competitive.Applications in Process Integration (Process Design and Synthesis) have been dominated by NLP and MINLP models due to the need for the explicit handling of performance equations, although simpler targeting models in process synthesis can give rise to LP and MILP problems. An extensive review of optimization models for process integration can be found in Grossmann et al. (1999). In contrast, Supply Chain Management problems tend to be dominated by linear models, LP and MILP, for planning and scheduling (see Grossmann et al. 2002 for a review). Finally, global optimization has concentrated more on

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Cones of Matrices and Set-Functions and 0–1 Optimization

Cited by 849 publications

References 20 publications

Approximate formulations for 0-1 knapsack sets

Approximate formulations for 0-1 knapsack sets

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

Advances in Logic‐Based Optimization Approaches to Process Integration and Supply Chain Management

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