## An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms (2005)

Venue: | Journal of Global Optimization |

Citations: | 21 - 9 self |

### BibTeX

@ARTICLE{Liberti05anexact,

author = {Leo Liberti and Constantinos C. Pantelides},

title = {An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms},

journal = {Journal of Global Optimization},

year = {2005},

volume = {36},

pages = {161--189}

}

### OpenURL

### Abstract

Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of Reformulation-Linearization Technique constraints) which do not a#ect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an e#cient graph-theoretical algorithm for e#ecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branchand Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.

### Citations

1125 |
Graph theory
- Harary
- 1969
(Show Context)
Citation Context ...he identification of linear constraint subsets L that satisfy (27) for a given multiplier variable zl. We start by constructing a bipartite graph (Korte and Vygen, 2000; Goodaire and Parmenter, 1998; =-=Harary, 1971-=-) Gl where the set of nodes is partitioned into two disjoint subsets N L and N V l . The former correspond to the set of linear constraints in [P] while the latter correspond to those problem variable... |

544 |
An n5/2 algorithm for maximum matchings in bipartite graphs
- Hopcroft, Karp
- 1973
(Show Context)
Citation Context ...hose edges alternate between not belonging and belonging to µ. If such a path can be found, then a matching of cardinality |µ| + 1 can be obtained from µ by replacing all edges in µ ∩ p by edges p\µ (=-=Hopcroft and Karp, 1973-=-; Korte and Vygen, 2000). A recursive procedure for the identification of an augmenting path emanating from a node λ is described in (Duff, 1981). The theoretical complexity of the algorithm is at mos... |

186 | Combinatorial Optimization: Theory and Algorithms. Algorithms and Combinatorics - Korte, Vygen - 2005 |

107 |
Computability of global solutions to factorable nonconvex programs: Part I - convex underestimating problems
- McCormick
- 1976
(Show Context)
Citation Context ... can be obtained in a simple and completely automatic fashion. Tight convex relaxations for the most common nonconvex terms are available in the literature. One of the best known is that proposed by (=-=McCormick, 1976-=-) (see Example 1.1) for the relaxation of bilinear terms. Linear fractional terms like those appearing in constraint (4) are reformulated to zizk = zj and replaced by the McCormick convex relaxation. ... |

94 |
A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Nonconvex Optimiza- Optimization of Mixed-Integer Quadratically-Constrained Quadratic Programs 25 tion and Its Applications
- Sherali, Adams
(Show Context)
Citation Context ...ing a proper subset of the set of all bilinear constraints and a particular set of RLT constraints implies that the rest of the bilinear constraints are also satisfied. Propositions 8.11 and 8.12 in (=-=Sherali and Adams, 1999-=-) show a similar result to our Theorem 3.1 but with stronger assumptions. In particular, they show that the bilinear terms are automatically implied by the RLT relaxation provided a certain subset of ... |

85 |
Jointly constrained biconvex programming
- Al-Khayyal, Falk
- 1983
(Show Context)
Citation Context ...¯ Cl but w ′ [i] �∈ ¯ Rl, which implies that ¯ Rl is a proper subset of ¯ Cl. ✷ The above theorem makes use of the fact that bilinear envelopes are loose everywhere other than at the variable bounds (=-=Al-Khayyal and Falk, 1983-=-) to show that valid RRLT constraint sets are always useful, since they tighten the convexification of the original problem. However, despite the apparent simplicity of criterion (27), applying it in ... |

52 |
On algorithms for obtaining a maximum transversal
- Duff
- 1981
(Show Context)
Citation Context ...replacing all edges in µ ∩ p by edges p\µ (Hopcroft and Karp, 1973; Korte and Vygen, 2000). A recursive procedure for the identification of an augmenting path emanating from a node λ is described in (=-=Duff, 1981-=-). The theoretical complexity of the algorithm is at most of the order (number of constraint nodes × number of edges). However, as observed by (Duff, 1981), the computational complexity of such algori... |

48 |
A new reformulationlinearization technique for bilinear programming problems
- Alameddine, Sherali
- 1992
(Show Context)
Citation Context ...LT). The RLT uses linear constraints built in this way to provide a lower bound to 0-1 mixed-integer linear and bilinear programming problems. In the case of continuous bilinear programming problems (=-=Sherali and Alameddine, 1992-=-), the RLT generates all possible new linear constraints that can be obtained by multiplying together bound factors (zj − z L j ≥ 0, zU j − zj ≥ 0 for all j ≤ p) and constraint factors ( � j aijzj − b... |

42 |
Convex Analysis and Global Optimization
- Tuy
- 1998
(Show Context)
Citation Context ...99; Smith, 1996). Therefore, theoretical results, manipulations and solution algorithms derived on the basis of this standard form are generally applicable. Spatial Branch-and-Bound (sBB) algorithms (=-=Tuy, 1998-=-) are among the most effective methods currently available for the global solution of nonconvex NLPs. An important requirement for any sBB algorithm is to be able to construct a tight convex underesti... |

41 |
C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs
- Smith, Pantelides
- 1999
(Show Context)
Citation Context ...denominator does not include zero. In any case, the above standard form is practically important as it can be shown that all NLPs can automatically be reformulated to it using symbolic manipulations (=-=Smith and Pantelides, 1999-=-; Smith, 1996). Therefore, theoretical results, manipulations and solution algorithms derived on the basis of this standard form are generally applicable. Spatial Branch-and-Bound (sBB) algorithms (Tu... |

41 | Convexification and global optimization in continuous and mixed-integer nonlinear programming - Tawarmalani, Sahinidis - 2002 |

21 | New properties and computational improvement of the GOP algorithm for problems with quadratic objective function and constraints
- Floudas, Visweswaran
- 1993
(Show Context)
Citation Context ...rochemical industry and are well known to exhibit multiple local minima. There is a wealth of publications on the pooling problem in the literature (Foulds et al., 1992; Sherali and Alameddine, 1992; =-=Visweswaran and Floudas, 1993-=-; Ben-Tal et al., 1994; Quesada and Grossmann, 1995; Visweswaran and Floudas, 1996; Adhya et al., 1999; Tawarmalani and Sahinidis, 2004; Audet et al., 2004; Tawarmalani and Sahinidis, 2002a; Sahinidis... |

18 |
Global optimization of bilinear process networks with multicomponents flows. Comput
- Grossmann, Quesada
- 1995
(Show Context)
Citation Context ...tiple local minima. There is a wealth of publications on the pooling problem in the literature (Foulds et al., 1992; Sherali and Alameddine, 1992; Visweswaran and Floudas, 1993; Ben-Tal et al., 1994; =-=Quesada and Grossmann, 1995-=-; Visweswaran and Floudas, 1996; Adhya et al., 1999; Tawarmalani and Sahinidis, 2004; Audet et al., 2004; Tawarmalani and Sahinidis, 2002a; Sahinidis and Tawarmalani, 2005). Application of RLT-type co... |

17 |
Discrete Mathematics with Graph Theory, Third Edition
- Goodaire, Parmenter
- 2006
(Show Context)
Citation Context ...ph-theoretical algorithm for the identification of linear constraint subsets L that satisfy (27) for a given multiplier variable zl. We start by constructing a bipartite graph (Korte and Vygen, 2000; =-=Goodaire and Parmenter, 1998-=-; Harary, 1971) Gl where the set of nodes is partitioned into two disjoint subsets N L and N V l . The former correspond to the set of linear constraints in [P] while the latter correspond to those pr... |

15 |
A Lagrangian approach to the pooling problem
- Adhya, Sahinidis, et al.
- 1999
(Show Context)
Citation Context ... pooling problem in the literature (Foulds et al., 1992; Sherali and Alameddine, 1992; Visweswaran and Floudas, 1993; Ben-Tal et al., 1994; Quesada and Grossmann, 1995; Visweswaran and Floudas, 1996; =-=Adhya et al., 1999-=-; Tawarmalani and Sahinidis, 2004; Audet et al., 2004; Tawarmalani and Sahinidis, 2002a; Sahinidis and Tawarmalani, 2005). Application of RLT-type constraints to the problem formulation was carried ou... |

14 |
Global minimization by reducing the duality gap
- Ben-Tal, Eiger, et al.
- 1994
(Show Context)
Citation Context ...l known to exhibit multiple local minima. There is a wealth of publications on the pooling problem in the literature (Foulds et al., 1992; Sherali and Alameddine, 1992; Visweswaran and Floudas, 1993; =-=Ben-Tal et al., 1994-=-; Quesada and Grossmann, 1995; Visweswaran and Floudas, 1996; Adhya et al., 1999; Tawarmalani and Sahinidis, 2004; Audet et al., 2004; Tawarmalani and Sahinidis, 2002a; Sahinidis and Tawarmalani, 2005... |

12 |
New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems
- Sherali, Tuncbilek
- 1997
(Show Context)
Citation Context ...hout the RLT constraints and only generates and includes those RLT constraints which are valid cuts with respect to the relaxed optimum. A more advanced constraint filtering technique is proposed in (=-=Sherali and Tuncbilek, 1997-=-), based on the sign of the coefficient of the variable zi in the original constraint and in the RLT constraint being generated. The approach adopted in this paper is fundamentally different from thos... |

11 |
A tight linearization and an algorithm for 0-1 quadratic programming problems
- Adams, Sherali
- 1986
(Show Context)
Citation Context ...ced computational effort, sometimes even by several orders of magnitude. The creation of new linear constraints via multiplication of existing linear constraints by problem variables was proposed in (=-=Sherali and Adams, 1986-=-; Sherali, 2002) under the name “ReformulationLinearization Technique” (RLT). The RLT uses linear constraints built in this way to provide a lower bound to 0-1 mixed-integer linear and bilinear progra... |

11 |
EMB: On the Optimal Design of Continuous Processes
- Smith
- 1996
(Show Context)
Citation Context ... zero. In any case, the above standard form is practically important as it can be shown that all NLPs can automatically be reformulated to it using symbolic manipulations (Smith and Pantelides, 1999; =-=Smith, 1996-=-). Therefore, theoretical results, manipulations and solution algorithms derived on the basis of this standard form are generally applicable. Spatial Branch-and-Bound (sBB) algorithms (Tuy, 1998) are ... |

6 |
A bilinear approach to the pooling problem
- Foulds, Haugland, et al.
- 1992
(Show Context)
Citation Context ...operties. Such problems occur frequently in the petrochemical industry and are well known to exhibit multiple local minima. There is a wealth of publications on the pooling problem in the literature (=-=Foulds et al., 1992-=-; Sherali and Alameddine, 1992; Visweswaran and Floudas, 1993; Ben-Tal et al., 1994; Quesada and Grossmann, 1995; Visweswaran and Floudas, 1996; Adhya et al., 1999; Tawarmalani and Sahinidis, 2004; Au... |

5 | Tight relaxations for nonconvex optimization problems using the reformulationlinearization/convexification technique - Sherali |

1 | Pantelides: 2003, ‘Convex Envelopes of Monomials of Odd Degree - Liberti, C |

1 | Tawarmalani: 2005, ‘Accelerating Branch-and-Bound through a Modeling Language Construct for Relaxation-Specific Constraints - Sahinidis, M |

1 | Sahinidis: 2002b, ‘Exact Algorithms for Global Optimization of Mixed-Integer Nonlinear Programs - Tawarmalani, N |

1 | Sahinidis: 2004, ‘Global Optimization of Mixed Integer Nonlinear Programs: A - Tawarmalani, N |