## Global Optimization of Mixed-Integer Nonlinear Problems (0)

Venue: | AIChE J |

Citations: | 20 - 3 self |

### BibTeX

@ARTICLE{Adjiman_globaloptimization,

author = {C. S. Adjiman and I. P. Androulakis and C. A. Floudas},

title = {Global Optimization of Mixed-Integer Nonlinear Problems},

journal = {AIChE J},

year = {},

volume = {46},

pages = {176--9}

}

### Years of Citing Articles

### OpenURL

### Abstract

Two novel deterministic global optimization algorithms for nonconvex mixed-integer problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure Mixed-Integer ffBB algorithm (SMIN-ffBB addresses problems with nonconvexities in the continuous variables and linear and mixed-bilinear participation of the binary variables. The General Structure Mixed-Integer ffBB algorithm (GMIN-ffBB), is applicable to a very general class of problems for which the continuous relaxation is twice continuously differentiable. Both algorithms are developed using the concepts of branch-and-bound, but they differ in their approach to each of the required steps. The SMIN-ffBB algorithm is based on the convex underestimation of the continuous functions while the GMIN-ffBB algorithm is centered around the convex relaxation of the entire problem. Both algorithms rely on optimization or interval based variable bound updates to enhance effici...

### Citations

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Citation Context ...e effect of such a procedure on algorithmic performance has not been previously studied and will therefore be examined in the computational section. 2.3.2 Interval-based approach Interval arithmetic (=-=Moore, 1979-=-; Neumaier, 1990) can be used to update the binary and continuous variable bounds. This approach is based on the interval evaluation of the constraints in the nonconvex MINLP as well as Eq. (5), the o... |

552 |
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Citation Context ...uch a procedure on algorithmic performance has not been previously studied and will therefore be examined in the computational section. 2.3.2 Interval-based approach Interval arithmetic (Moore, 1979; =-=Neumaier, 1990-=-) can be used to update the binary and continuous variable bounds. This approach is based on the interval evaluation of the constraints in the nonconvex MINLP as well as Eq. (5), the objective functio... |

282 |
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Citation Context ...o a lower bound on the solution of the nonconvex MINLP. In order to ensure that the GMIN-ffBB algorithm converges in a finite number of iterations, the validity of the lower bounds is not sufficient (=-=Horst and Tuy, 1996-=-) and two additional conditions must be met: 1. The global solution must be identified for nodes on the terminal level of the B&B tree, when all integer variables are fixed and the upper and lower bou... |

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Citation Context ...r and lower bounds on the mixedinteger nonconvex problem. 2.1.1 Upper bound A rigorous upper bound is obtained by solving the nonconvex MINLP (1) locally. The generalized Benders decomposition (GBD) (=-=Benders, 1962-=-; Geoffrion, 1972; Floudas et al., 1989) may be used to obtain such a solution. When there are no mixed-bilinear terms, the outer-approximation with equality relaxation (OA/ER) (Duran and Grossmann, 1... |

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Citation Context ...separated into a twice continuously differentiable part, a mixed bilinear part and linear binary part. Thus, the assumption of convexity required for commonly used MINLP algorithms such as the OA/ER (=-=Duran and Grossmann, 1986-=-; Kocis and Grossmann, 1987) and the GBD (Geoffrion, 1972) is lifted. Mathematically, the class of MINLPs for which the SMIN-ffBB algorithm is designed is given by: 3 min x;y f(x) + x T A f y + c T f ... |

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Citation Context ...linear part and linear binary part. Thus, the assumption of convexity required for commonly used MINLP algorithms such as the OA/ER (Duran and Grossmann, 1986; Kocis and Grossmann, 1987) and the GBD (=-=Geoffrion, 1972-=-) is lifted. Mathematically, the class of MINLPs for which the SMIN-ffBB algorithm is designed is given by: 3 min x;y f(x) + x T A f y + c T f y s:t: g i (x) + x T A g;i y + c T g;i ys0; i = 1; : : : ... |

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Citation Context ... (Duran and Grossmann, 1986; Kocis and Grossmann, 1987) or a local MINLP branch-and-bound algorithm (B&B) (Beale, 1977; Gupta and Ravindran, 1985; Ostrovsky et al., 1990; Borchers and Mitchell, 1991; =-=Quesada and Grossmann, 1992-=-) may also be used. 2.1.2 Lower bound In order to obtain a valid lower bound, a relaxed problem which can be solved to global optimality must be constructed from problem (1). The subclass of (1) in wh... |

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Citation Context ...ne algorithm of Westerlund et al. (1998), an alternative to branch-and-bound techniques, tackles problems involving pseudo-convex functions and is an extension of the ECP algorithm for convex MINLPs (=-=Westerlund and Pettersson, 1995-=-). A review of these algorithms is presented in Adjiman et al. (1999). More specialized algorithms have also been proposed for certain classes of applications, as in the 2 work of Zamora and Grossmann... |

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Citation Context ...d Stephanopoulos (1975). While such techniques have long been used to solve convex mixedinteger problems (see, for instance, Beale, 1977; Ostrovsky et al., 1990), and certain types of nonconvex NLPs (=-=Falk and Soland, 1969-=-; McCormick, 1976), they have only recently started to be applied to the global optimization of nonconvex MINLPs. This introduction provides an overview of work in this specific area, and the reader i... |

47 |
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Citation Context ...ce, approaches based on nonlinear programming (NLP), such as those described in Horst and Tuy (1996), have been used to determine optimum equipment sizes and operating conditions for a given process (=-=Floudas, 1995-=-; Grossmann, 1996). The economic benefits to be derived from identifying the global solution 1 Current address: Centre for Process Systems Engineering, Imperial College of Science, Technology and Medi... |

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Citation Context ... twice continuously differentiable, the General Structure Mixed-Integer ffBB algorithm (GMIN-ffBB) is then introduced. While based on a classical branch-and-bound approach for mixed-integer problems (=-=Gupta and Ravindran, 1985-=-), it circumvents the problems such algorithms encounter when dealing with nonconvex problems through efficient and rigorous strategies for bounding, branching and variable bound tightening. The algor... |

37 | An improved branch and bound algorithm for mixed integer nonlinear programs
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Citation Context ...h equality relaxation (OA/ER) (Duran and Grossmann, 1986; Kocis and Grossmann, 1987) or a local MINLP branch-and-bound algorithm (B&B) (Beale, 1977; Gupta and Ravindran, 1985; Ostrovsky et al., 1990; =-=Borchers and Mitchell, 1991-=-; Quesada and Grossmann, 1992) may also be used. 2.1.2 Lower bound In order to obtain a valid lower bound, a relaxed problem which can be solved to global optimality must be constructed from problem (... |

32 | Finding all solutions of nonlinearly constrained systems of equations - Maranas, Floudas - 1995 |

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(Show Context)
Citation Context ...inuously differentiable part, a mixed bilinear part and linear binary part. Thus, the assumption of convexity required for commonly used MINLP algorithms such as the OA/ER (Duran and Grossmann, 1986; =-=Kocis and Grossmann, 1987-=-) and the GBD (Geoffrion, 1972) is lifted. Mathematically, the class of MINLPs for which the SMIN-ffBB algorithm is designed is given by: 3 min x;y f(x) + x T A f y + c T f y s:t: g i (x) + x T A g;i ... |

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21 | Optimization models for reliability of modular software systems - Berman, Ashrafi - 1993 |

20 |
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(Show Context)
Citation Context ...icipation of the binary variables, is discussed. It is based on the ffBB global optimization algorithm for twice continuously differentiable NLPs (Androulakis et al., 1995; Adjiman and Floudas, 1996; =-=Adjiman et al., 1998-=-a). The performance of the algorithm is studied on a number of small literature problems and some medium-size heat exchanger network problems. Particular attention is paid to the selection of options ... |

20 |
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Citation Context ...y the main driver. The use of specialized algorithms to solve subproblems is a common feature of branch-and-bound and decomposition algorithms, as exemplified by the OA/ER (Duran and Grossmann, 1986; =-=Kocis and Grossmann, 1989-=-) or the GBD (Geoffrion, 1972). In general, the interaction between the higher level driver and the algorithms it uses is limited to the exchange of subproblem formulation and solution. This level of ... |

17 |
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Citation Context ...the SMIN-ffBB algorithm. All runs are performed on an HP-C160 with a relative tolerance of 10 \Gamma3 . 3.1 Small examples The examples used here have all appeared in the literature as test examples (=-=Kocis and Grossmann, 1988-=-; Floudas et al., 1989; Ryoo and Sahinidis, 1995; Cardoso et al., 1997). The results are summarized in Table 2. Example 3.1.1 This example, which was first proposed by Kocis and Grossmann (1988), invo... |

15 | An extended cutting planemethod for a class of non-convexMINLP problems - Westerlund, Skrifvars, et al. - 1998 |

15 |
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Citation Context ...LP where the objective is to minimize x i (\Gammax i ) subject to the convexified constraints from the original problem. In addition, a so-called "objective function cut" can be added to the=-= problem (Zamora and Grossmann, 1998-=-a) as a constraint: f(x; w) + x T A f y + c T f ysf ; (5) where thessuperscript denotes the convex underestimator of the specified function valid for the current domain [x L ; x U ], f denotes the bes... |

13 |
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(Show Context)
Citation Context ...ernative structures within a single problem. This is achieved through the introduction of integer variables which leads to the formulation of a mixed-integer nonlinear problem (MINLP) (Floudas, 1995; =-=Floudas and Grossmann, 1995-=-; Grossmann, 1996). Such an approach has already been used for a wide array of applications including process synthesis. The solution of many MINLPs relevant to chemical engineering is made challengin... |

13 | Global optimization in generalized geometric programming - Maranas, Floudas - 1997 |

11 |
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Citation Context ...LP where the objective is to minimize x i (\Gammax i ) subject to the convexified constraints from the original problem. In addition, a so-called "objective function cut" can be added to the=-= problem (Zamora and Grossmann, 1998-=-a) as a constraint: f(x; w) + x T A f y + c T f ysf ; (5) where thessuperscript denotes the convex underestimator of the specified function valid for the current domain [x L ; x U ], f denotes the bes... |

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(Show Context)
Citation Context ...omplete implementation of the algorithm has been developed, with links to the Outer Approximation and the Generalized Benders Decomposition algorithms that are provided as part of the MINOPT package (=-=Schweiger et al., 1997-=-). This section is dedicated to the investigation of the performance of the proposed SMIN-ffBB algorithm on a series of small test examples from the literature and three heat exchanger network synthes... |

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Citation Context ... costs, the fixed charges for the required heat-exchangers and an area-based cost for each heat-exchanger. Since the Chen approximation for the logarithmic mean of the temperature difference is used (=-=Chen, 1987-=-), the area is a highly nonlinear function of the heat duty and the temperature differences at both ends of the heat exchanger. The binary variables, which represent the existence of a given heat-exch... |

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Citation Context ...nteger nonconvex problem. 2.1.1 Upper bound A rigorous upper bound is obtained by solving the nonconvex MINLP (1) locally. The generalized Benders decomposition (GBD) (Benders, 1962; Geoffrion, 1972; =-=Floudas et al., 1989-=-) may be used to obtain such a solution. When there are no mixed-bilinear terms, the outer-approximation with equality relaxation (OA/ER) (Duran and Grossmann, 1986; Kocis and Grossmann, 1987) or a lo... |

2 |
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Citation Context ...icipation of the binary variables, is discussed. It is based on the ffBB global optimization algorithm for twice continuously differentiable NLPs (Androulakis et al., 1995; Adjiman and Floudas, 1996; =-=Adjiman et al., 1998-=-a). The performance of the algorithm is studied on a number of small literature problems and some medium-size heat exchanger network problems. Particular attention is paid to the selection of options ... |

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Citation Context ...ities in the continuous variables and restricted participation of the binary variables, is discussed. It is based on the ffBB global optimization algorithm for twice continuously differentiable NLPs (=-=Androulakis et al., 1995-=-; Adjiman and Floudas, 1996; Adjiman et al., 1998a). The performance of the algorithm is studied on a number of small literature problems and some medium-size heat exchanger network problems. Particul... |

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Citation Context ...e its introduction in the literature by Lee et al. (1970) and Westerberg and Stephanopoulos (1975). While such techniques have long been used to solve convex mixedinteger problems (see, for instance, =-=Beale, 1977-=-; Ostrovsky et al., 1990), and certain types of nonconvex NLPs (Falk and Soland, 1969; McCormick, 1976), they have only recently started to be applied to the global optimization of nonconvex MINLPs. T... |

1 | Application of MINLP Methods on a Scheduling Problem - Harjunkoski - 1997 |

1 |
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Citation Context ...ction in the literature by Lee et al. (1970) and Westerberg and Stephanopoulos (1975). While such techniques have long been used to solve convex mixedinteger problems (see, for instance, Beale, 1977; =-=Ostrovsky et al., 1990-=-), and certain types of nonconvex NLPs (Falk and Soland, 1969; McCormick, 1976), they have only recently started to be applied to the global optimization of nonconvex MINLPs. This introduction provide... |

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1 | Sahinidis, "A Finite Algorithm for Global Minimization of Separable Concave Programs - Shectman, V - 1998 |

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