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A Global Optimization Method, αBB, for General TwiceDifferentiable Constrained NLPs: I  Theoretical Advances
, 1997
"... In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the constru ..."
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Cited by 52 (3 self)
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In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...
Global Optimization of MINLP Problems in Process Synthesis and Design
 Computers & Chemical Engineering
, 1997
"... : Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMINffBB, and the General structure Mixed Integer Nonlinear ffBB, GMINffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of ..."
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Cited by 16 (6 self)
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: Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMINffBB, and the General structure Mixed Integer Nonlinear ffBB, GMINffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of MINLPs involving twicedifferentiable nonconvex functions in the continuous variables can be identified. The conditions imposed on the functionality of the binary variables differ for each method : linear and mixed bilinear terms can be treated with the SMINffBB; mixed nonlinear terms whose continuous relaxation is twicedifferentiable are handled by the GMINffBB. While both algorithms use the concept of a branch & bound tree, they rely on fundamentally different bounding and branching strategies. In the GMINffBB algorithm, lower (upper) bounds at each node result from the solution of convex (nonconvex) MINLPs derived from the original problem. The construction of convex lower bound...
Global Optimization of MixedInteger Nonlinear Problems
 AIChE J
"... Two novel deterministic global optimization algorithms for nonconvex mixedinteger problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure MixedInteger ffBB algorithm (SMINffBB addresses problems with nonconvexities ..."
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Cited by 14 (2 self)
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Two novel deterministic global optimization algorithms for nonconvex mixedinteger problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure MixedInteger ffBB algorithm (SMINffBB addresses problems with nonconvexities in the continuous variables and linear and mixedbilinear participation of the binary variables. The General Structure MixedInteger ffBB algorithm (GMINffBB), 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 branchandbound, but they differ in their approach to each of the required steps. The SMINffBB algorithm is based on the convex underestimation of the continuous functions while the GMINffBB 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...
Deterministic Global Optimization In Design, Control, And Computational Chemistry
 IMA Volumes in Mathematics and its Applications : Large Scale Optimization with Applications, Part II
, 1997
"... . This paper presents an overview of the deterministic global optimization approaches and their applications in the areas of Process Design, Control, and Computational Chemistry. The focus is on (i) decompositionbased primal dual methods, (ii) methods for generalized geometric programming problems, ..."
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Cited by 10 (7 self)
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. This paper presents an overview of the deterministic global optimization approaches and their applications in the areas of Process Design, Control, and Computational Chemistry. The focus is on (i) decompositionbased primal dual methods, (ii) methods for generalized geometric programming problems, and (iii) global optimization methods for general nonlinear programming problems. The classes of mathematical problems that are addressed range from indefinite quadratic programming to concave programs, to quadratically constrained problems, to polynomials, to general twice continuously differentiable nonlinear optimization problems. For the majority of the presented methods nondistributed global optimization approaches are discussed with the exception of decompositionbased methods where a distributed global optimization approach is presented. 1. Background. A significant effort has been expended in the last five decades toward theoretical and algorithmic studies of applications that arise...
A Global Optimization Method, αBB, for Process Design
 COMPUT. CHEM. ENG
, 1996
"... A global optimization algorithm, αBB, for twicedifferentiable NLPs is presented. It operates within a branchandbound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twicedifferentiable functions ..."
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Cited by 7 (1 self)
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A global optimization algorithm, αBB, for twicedifferentiable NLPs is presented. It operates within a branchandbound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twicedifferentiable functions is described. The αBB has been applied to a variety of problems and a summary of the results obtained is provided.
Distributed Branch And Bound Algorithms For Global Optimization
 IMA Volumes in Mathematics and its Applications, Vol. 106. Parallel Processing of Discrete Problems
, 1998
"... . This paper presents computational results of the parallelized version of the ffBB global optimization algorithm. Important algorithmic and implementational issues are discussed and their impact on the design of parallel branch and bound methods is analyzed. These issues include selection of the ap ..."
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Cited by 5 (0 self)
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. This paper presents computational results of the parallelized version of the ffBB global optimization algorithm. Important algorithmic and implementational issues are discussed and their impact on the design of parallel branch and bound methods is analyzed. These issues include selection of the appropriate architecture, communication patterns, frequency of communication, and termination detection. The approach is demonstrated with a variety of computational studies aiming at revealing the various types of behavior of the distributed branch and bound global optimization algorithm can exhibit. These include ideal behavior, speedup, detrimental, and deceleration anomalies. Key words. Global optimization, parallel computing, branch and bound. 1. Introduction. A wide range of optimal selection problems, in diverse scientific areas, can be formulated as non linear constrained optimization problems. One of the most common characteristics of these problems is the presence of nonconvexitie...
A note on connection between PNS and set covering problems
 Acta Cybernetica
, 1996
"... Process network synthesis (PNS) has enormous practical impact; however, its mixed integer programming model is tedious to solve because it usually involves a large number of binary variables. Using a combinatorial approach, a structural model of PNS can be given, and a branchand bound technique ..."
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Cited by 3 (3 self)
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Process network synthesis (PNS) has enormous practical impact; however, its mixed integer programming model is tedious to solve because it usually involves a large number of binary variables. Using a combinatorial approach, a structural model of PNS can be given, and a branchand bound technique can be applied for searching an optimal solution. In some realistic examples of PNS, this method is ecient. Nevertheless, ecient methods are unavailable for solving these models generally. In this note, we describe a special class of PNSproblems as setcovering or setpartitioning problems. These problems are wellknown to be NPcomplete, thus, a PNSproblem is NPhard. 1 Introduction In a manufacturing system, materials of dierent properties are consumed through various mechanical, physical and chemical transformation to yield desired products. Devices in which these transformations are carried out are called operating units, e.g., a lathe or a chemical reactor. Thus, a manufactur...
Global Optimization In Design And Control Of Chemical Process Systems
 J. of Proc. Control
, 2001
"... : This paper presents an overview of the recent advances in deterministic global optimization approaches and their applications in the areas of Process Design and Control. The focus is on global optimization methods for (a) twicedifferentiable constrained nonlinear optimization problems, (b) mixed ..."
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Cited by 3 (0 self)
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: This paper presents an overview of the recent advances in deterministic global optimization approaches and their applications in the areas of Process Design and Control. The focus is on global optimization methods for (a) twicedifferentiable constrained nonlinear optimization problems, (b) mixedinteger nonlinear optimization problems, and (c) locating all solutions of nonlinear systems of equations. Theoretical advances and computational studies on process design, batch design under uncertainty, phase equilibrium, location of azeotropes, stability margin, process synthesis, and parameter estimation problems are discussed. Keywords: Global Optimization; Twice Differentiable NLPs; MixedInteger Nonlinear Optimization; Locating All Solutions; ffBB approach, Design and Control 1. INTRODUCTION A significant effort has been expended in the last five decades toward theoretical and algorithmic studies of applications that arise in Process Design and Control. In the last decade we have expe...
On a wellsolvable class of the PNS problem
, 2000
"... A manufacturing system consists of operating units converting materials of dierent properties into further materials. In a design problem, we are to nd a suitable network of operating units which produces the desired products from the given raw materials. If we consider this network design from ..."
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Cited by 1 (1 self)
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A manufacturing system consists of operating units converting materials of dierent properties into further materials. In a design problem, we are to nd a suitable network of operating units which produces the desired products from the given raw materials. If we consider this network design from structural point of view, then we obtain a combinatorial optimization problem called Process Network Synthesis or (PNS) problem. It is known that the PNS problem is NPcomplete. In this work, we present such a subclass of PNS problems which is wellsolvable. 1 Introduction In a manufacturing system, materials of dierent properties are consumed through various mechanical, physical and chemical transformations to result in desired products. Devices in which these transformations are carried out are called operating units, e.g., a lathe or a chemical reactor. Hence, a manufacturing system can be considered as a network of operating units which is called process network. A process desig...
On DecisionMappings Related to Process Network Synthesis Problem
, 1998
"... . Process network synthesis (PNS) has enormous practical impact and a structural model can be given for it on the basis of a combinatorial approach. An important tool of this approach is the notion of the decisionmapping. In the present work, the number of the consistent decisionmappings is co ..."
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. Process network synthesis (PNS) has enormous practical impact and a structural model can be given for it on the basis of a combinatorial approach. An important tool of this approach is the notion of the decisionmapping. In the present work, the number of the consistent decisionmappings is counted and an upper bound is presented for the number of the feasible solutions of a PNS problem. Introduction In a manufacturing system, materials of dierent properties are converted into desired products through various physical, chemical, and biological transformations. Devices in which these transformations are carried out are called operating units and a manufacturing system can be considered as a network of operating units, i.e., process network. Naturally, minimizing the cost of a process network is indeed essential. For this purpose, several papers have appeared for solving PNS problems by global optimization methods (see, e.g., [2] and [8]) and by combinatorial approach based on ...