## Reformulation in mathematical programming: an application to quantum chemistry (2007)

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Venue: | DISCRETE APPLIED MATHEMATICS, ACCEPTED FOR PUBLICATION |

Citations: | 6 - 6 self |

### BibTeX

@MISC{Liberti07reformulationin,

author = {Leo Liberti and Carlile Lavor and Nelson Maculan and Marco Antonio Chaer Nascimento},

title = {Reformulation in mathematical programming: an application to quantum chemistry},

year = {2007}

}

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### Abstract

### Citations

772 |
Data Structures and Algorithms
- AHO, HO’CROFT, et al.
- 1983
(Show Context)
Citation Context ...ly employed deterministic GO method is an extension to continuous spaces, called spatial Branch-and-Bound (sBB) [1,22,13], of the well-known Branch-and-Bound algorithm for implicit binary enumeration =-=[2]-=-. In sBB, branching occurs by partitioning the continuous variable range in two or more sub-ranges. The recursive application of branching gives rise to a search tree. A node is fathomed (i.e. no furt... |

227 | Global Optimization
- Törn, Z̆ilinskas
- 1989
(Show Context)
Citation Context ...finite time (for some problems exactness can also be proved, i.e. ε = 0), or they employ random search techniques (which are usually associated to a convergence result in probability in infinite time =-=[23]-=-, Ch. 4). The most commonly employed deterministic GO method is an extension to continuous spaces, called spatial Branch-and-Bound (sBB) [1,22,13], of the well-known Branch-and-Bound algorithm for imp... |

119 |
Computability of global solutions to factorable nonconvex programs. Part I—Nonconvex underestimating problems
- McCormick
- 1976
(Show Context)
Citation Context ...elaxation of (16) by replacing (14)-(15) by their convex envelopes. For simple constraints of the form z = xt where x L ≤ x ≤ x U and t L ≤ t ≤ t U (such as (14)-(15)) the convex envelope is given by =-=[18,3]-=-: z ≤ min{x U t + t L x − x U t L ,x L t + t U x − x L t U } (17) z ≥ max{x L t + t L x − x L t L ,x U t + t U x − x U t U }, (18) which defines an enveloping tetrahedron around the points (x L ,t L )... |

101 |
A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems
- Sherali, Adams
- 1999
(Show Context)
Citation Context ...sed to obtain the lower bound at each sBB node. Section 4 explains how to tighten the lower bound: Section 4.1 provides a brief introduction to reduced Reformulation-Linearization Technique (RLT, see =-=[20]-=-) constraints [16] used to tighten the bound; Section 4.2 shows a method to choose the best reduced RLT constraint system; Section 4.3 discusses the application to the HFP in 1 This formulation recent... |

91 |
Jointly constrained biconvex programming
- Al-Khayyal, Falk
- 1983
(Show Context)
Citation Context ...elaxation of (16) by replacing (14)-(15) by their convex envelopes. For simple constraints of the form z = xt where x L ≤ x ≤ x U and t L ≤ t ≤ t U (such as (14)-(15)) the convex envelope is given by =-=[18,3]-=-: z ≤ min{x U t + t L x − x U t L ,x L t + t U x − x L t U } (17) z ≥ max{x L t + t L x − x L t L ,x U t + t U x − x U t U }, (18) which defines an enveloping tetrahedron around the points (x L ,t L )... |

56 | A global optimization method, αBB, for general twice-differentiable constrained NLPs: I. Theoretical advances
- Adjiman, Dallwig, et al.
- 1998
(Show Context)
Citation Context ...ed to a convergence result in probability in infinite time [23], Ch. 4). The most commonly employed deterministic GO method is an extension to continuous spaces, called spatial Branch-and-Bound (sBB) =-=[1,22,13]-=-, of the well-known Branch-and-Bound algorithm for implicit binary enumeration [2]. In sBB, branching occurs by partitioning the continuous variable range in two or more sub-ranges. The recursive appl... |

50 |
A new reformulation—linearization technique for bilinear programming problems
- Sherali, Alameddine
- 1992
(Show Context)
Citation Context ...p (a) reformulates the problem to a standard form [22] consisting of a linear objective function subject to linear constraints and a set of quadratic constraints. By introducing new variable sets y,w =-=[21,20]-=- with the following constraints: y ij rs =cricsj ∀r,s ≤ b, i,j ≤ n (14) w ij rstu =y ii rsy jj tu ∀r,s,t,u ≤ b, i,j ≤ n (15) we can replace all the quartic and quadratic terms in the problem, obtainin... |

44 |
A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs
- Smith, Pantelides
- 1999
(Show Context)
Citation Context ...ed to a convergence result in probability in infinite time [23], Ch. 4). The most commonly employed deterministic GO method is an extension to continuous spaces, called spatial Branch-and-Bound (sBB) =-=[1,22,13]-=-, of the well-known Branch-and-Bound algorithm for implicit binary enumeration [2]. In sBB, branching occurs by partitioning the continuous variable range in two or more sub-ranges. The recursive appl... |

30 |
Writing global optimization software
- Liberti
(Show Context)
Citation Context ...ed to a convergence result in probability in infinite time [23], Ch. 4). The most commonly employed deterministic GO method is an extension to continuous spaces, called spatial Branch-and-Bound (sBB) =-=[1,22,13]-=-, of the well-known Branch-and-Bound algorithm for implicit binary enumeration [2]. In sBB, branching occurs by partitioning the continuous variable range in two or more sub-ranges. The recursive appl... |

21 | Global Optimization: From Theory to Implementation (Nonconvex Optimization and Its Applications - Liberti, Maculan - 2006 |

19 | An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms
- Liberti, Pantelides
(Show Context)
Citation Context ...lower bound at each sBB node. Section 4 explains how to tighten the lower bound: Section 4.1 provides a brief introduction to reduced Reformulation-Linearization Technique (RLT, see [20]) constraints =-=[16]-=- used to tighten the bound; Section 4.2 shows a method to choose the best reduced RLT constraint system; Section 4.3 discusses the application to the HFP in 1 This formulation recently appeared in Eur... |

18 |
Solving a spreadspectrum radar polyphase code design problem by tabu search and variable neighbourhood search
- Mladenović, Petrović, et al.
(Show Context)
Citation Context ...ic approach) solving the original instances. The last two columns refer to heuristic methods. The fourth column contains results obtained with an implementation of Variable Neighbourhood Search (VNS) =-=[19,14]-=-. The fifth column contains results obtained with a variant of the Multi Level Single Linkage (MLSL) algorithm called SobolOpt [7], which uses deterministic low-discrepancy Sobol’ sequences 15to gene... |

14 |
Application of deterministic low-discrepancy sequences in global optimization
- Sytsko
(Show Context)
Citation Context ...ned with an implementation of Variable Neighbourhood Search (VNS) [19,14]. The fifth column contains results obtained with a variant of the Multi Level Single Linkage (MLSL) algorithm called SobolOpt =-=[7]-=-, which uses deterministic low-discrepancy Sobol’ sequences 15to generate a uniform sampling of starting points. All computational results have been obtained by running global solvers within the ooOP... |

14 |
Reduction constraints for the global optimization of NLPs
- Liberti
(Show Context)
Citation Context ...e original surface (14)-(15). We try to improve this situation by adding some valid inequalities to the convexification of (16) obtained in Section 3. For this task we turn to reduced RLT constraints =-=[11,12,16]-=-, which form a subclass of the RLT constraints described in [21]. 4.1 Reduced RLT constraints primer In this section only, we attach a different meaning to the symbols n and b. Assume the feasible reg... |

14 |
Linearity embedded in nonconvex programs
- Liberti
(Show Context)
Citation Context ...e original surface (14)-(15). We try to improve this situation by adding some valid inequalities to the convexification of (16) obtained in Section 3. For this task we turn to reduced RLT constraints =-=[11,12,16]-=-, which form a subclass of the RLT constraints described in [21]. 4.1 Reduced RLT constraints primer In this section only, we attach a different meaning to the symbols n and b. Assume the feasible reg... |

11 |
User’s Guide for SNOPT 5.3. Systems Optimization
- Gill
- 1999
(Show Context)
Citation Context ...al problem (13) to the node’s variable ranges. The upper bound is provided by locally solving the original nonconvex problem with restricted bound by a general purpose local NLP solver (such as SNOPT =-=[5]-=-). The lower bound is given by the solution of a linear relaxation of the restricted problem using an LP solver (such as CPLEX [6]). The linear relaxation is built in two steps: (a) reformulation and ... |

5 |
Basis sets for ab initio molecular orbital calculations and intermolecular interactions
- Feller, Davidson
- 1990
(Show Context)
Citation Context ...he several types of basis sets available in the literature, the Gaussian-type basis is the one most frequently used because it represents the best compromise between accuracy and computational effort =-=[4]-=-. The choice of the size of the basis set (parameter b) is dictated by the degree of accuracy expected from the calculations. In the standard HF method, as the basis set increases, the energy converge... |

4 |
Quantum Chemistry. Prentice-Hall, Upper Saddle River
- Levine
- 2000
(Show Context)
Citation Context ... the approximation) is given by the energy function E associated with the approximating set {¯ϕi}, which is guaranteed to be an upper bound to the energy function associated with the spatial orbitals =-=[9]-=-. The set {¯ϕi} is required to be an orthonormal for the reasons already mentioned. Furthermore, once each member of the set {¯ϕi} is expanded in a complete orthonormal basis {χs}, a normalization con... |

3 |
Solving hartree-fock systems with global optimization metohds
- Lavor, Liberti, et al.
(Show Context)
Citation Context ...the bound; Section 4.2 shows a method to choose the best reduced RLT constraint system; Section 4.3 discusses the application to the HFP in 1 This formulation recently appeared in Europhysics Letters =-=[8]-=- too: this is a journal targeted at physics researchers and not generally read by the OR community. 4general and the He and Be instances in particular. Section 5 discusses the computational experienc... |

3 |
Automatic reformulation of bilinear MINLPs
- Liberti
- 2004
(Show Context)
Citation Context ...remain in the problem formulation) have the least total convexity gap. Equivalently, we need to choose a set of basic variables with the largest total convexity gap ∑ all (i,k) V i k. It can be shown =-=[10]-=- that V i k depends on the widths of the variable ranges of xk,xi: the larger the variable ranges, the larger the convexity gap. We therefore choose the basic variables of the companion system to incl... |