Results 1  10
of
31
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 ..."
Abstract

Cited by 52 (3 self)
 Add to MetaCart
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...
Molecular Modeling Of Proteins And Mathematical Prediction Of Protein Structure
 SIAM Review
, 1997
"... . This paper discusses the mathematical formulation of and solution attempts for the socalled protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein, given its sequence of amino acids. The dynamic aspect asks about the possib ..."
Abstract

Cited by 47 (4 self)
 Add to MetaCart
. This paper discusses the mathematical formulation of and solution attempts for the socalled protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein, given its sequence of amino acids. The dynamic aspect asks about the possible pathways to folding and unfolding, including the stability of the folded protein. From a mathematical point of view, there are several main sides to the static problem:  the selection of an appropriate potential energy function;  the parameter identification by fitting to experimental data; and  the global optimization of the potential. The dynamic problem entails, in addition, the solution of (because of multiple time scales very stiff) ordinary or stochastic differential equations (molecular dynamics simulation), or (in case of constrained molecular dynamics) of differentialalgebraic equations. A theme connecting the static and dynamic aspect is the determination and formation of...
Rigorous Convex Underestimators for General TwiceDifferentiable Problems
 Journal of Global Optimization
, 1996
"... . In order to generate valid convex lower bounding problems for nonconvex twicedifferentiable optimization problems, a method that is based on second order information of general twicedifferentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues ..."
Abstract

Cited by 35 (15 self)
 Add to MetaCart
. In order to generate valid convex lower bounding problems for nonconvex twicedifferentiable optimization problems, a method that is based on second order information of general twicedifferentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues of such functions are obtained and used in constructing convex underestimators. By solving several nonlinear example problems, it is shown that the lower bounds are sufficiently tight to ensure satisfactory convergence of the ffBB, a branch and bound algorithm which relies on this underestimation procedure [3]. Key words: convex underestimators; twicedifferentiable; interval anlysis; eigenvalues 1. Introduction The mathematical description of many physical phenomena, such as phase equilibrium, or of chemical processes generally requires the introduction of nonconvex functions. As the number of local solutions to a nonconvex optimization problem cannot be predicted a priori, the identifi...
Finding All Solutions of Nonlinearly Constrained Systems of Equations
 Journal of Global Optimization
, 1995
"... . A new approach is proposed for finding all fflfeasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem(P) whose multiple global minimum solutionswith a zero object ..."
Abstract

Cited by 30 (14 self)
 Add to MetaCart
. A new approach is proposed for finding all fflfeasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem(P) whose multiple global minimum solutionswith a zero objectivevalue (if any)correspond to all solutionsof the initial constrainedsystem of equalities. All fflglobally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite fflconvergence to each of the multiple global minima of (P) through the successive refinement of a convexrelaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products o...
Global Optimization in Parameter Estimation of Nonlinear Algebraic Models via the ErrorInVariables Approach
 Ind. Eng. Chem. Res
, 1998
"... The estimation of parameters in nonlinear algebraic models through the errorinvariables method has been widely studied from a computational standpoint. The method involves the minimization of a weighted sum of squared errors subject to the model equations. Due to the nonlinear nature of the models ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
The estimation of parameters in nonlinear algebraic models through the errorinvariables method has been widely studied from a computational standpoint. The method involves the minimization of a weighted sum of squared errors subject to the model equations. Due to the nonlinear nature of the models used, the resulting formulation is nonconvex, and may contain several local minima in the region of interest. Current methods tailored for this formulation, although computationally efficient, can only attain convergence to a local solution. In this paper, a global optimization approach based on a branch and bound framework and convexification techniques for general twice differentiable nonlinear optimization problems is proposed for the parameter estimation of nonlinear algebraic models. The proposed convexification techniques exploit the mathematical properties of the formulation. Classical nonlinear estimation problems were solved and will be used to illustrate the various theoretical an...
GLOPT  A Program for Constrained Global Optimization
 Developments in Global Optimization
, 1996
"... . GLOPT is a Fortran77 program for global minimization of a blockseparable objective function subject to bound constraints and blockseparable constraints. It finds a nearly globally optimal point that is near a true local minimizer. Unless there are several local minimizers that are nearly global, ..."
Abstract

Cited by 15 (7 self)
 Add to MetaCart
. GLOPT is a Fortran77 program for global minimization of a blockseparable objective function subject to bound constraints and blockseparable constraints. It finds a nearly globally optimal point that is near a true local minimizer. Unless there are several local minimizers that are nearly global, we thus find a good approximation to the global minimizer. GLOPT uses a branch and bound technique to split the problem recursively into subproblems that are either eliminated or reduced in their size. This is done by an extensive use of the block separable structure of the optimization problem. In this paper we discuss a new reduction technique for boxes and new ways for generating feasible points of constrained nonlinear programs. These are implemented as the first stage of our GLOPT project. The current implementation of GLOPT uses neither derivatives nor simultaneous information about several constraints. Numerical results are already encouraging. Work on an extension using curvature inf...
Floudas. ASTROFOLD: A Combinatorial and Global Optimization Framework for Ab Initio Prediction of ThreeDimensional Structures of Proteins from the Amino Acid Sequence
 Biophysical Journal
, 2003
"... ABSTRACT The field of computational biology has been revolutionized by recent advances in genomics. The completion of a number of genome projects, including that of the human genome, has paved the way toward a variety of challenges and opportunities in bioinformatics and biological systems engineeri ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
ABSTRACT The field of computational biology has been revolutionized by recent advances in genomics. The completion of a number of genome projects, including that of the human genome, has paved the way toward a variety of challenges and opportunities in bioinformatics and biological systems engineering. One of the first challenges has been the determination of the structures of proteins encoded by the individual genes. This problem, which represents the progression from sequence to structure (genomics to structural genomics), has been widely known as the structurepredictioninproteinfolding problem. We present the development and application of ASTROFOLD, a novel and complete approach for the ab initio prediction of protein structures given only the amino acid sequences of the proteins. The approach exhibits many novel components and the merits of its application are examined for a suite of protein systems, including a number of targets from several criticalassessmentofstructureprediction experiments.
Global Optimization in Generalized Geometric Programming
 Engng
, 1997
"... A deterministic global optimization algorithm is proposed for locating the global minimum of generalized geometric (signomial) problems (GGP). By utilizing an exponential variable transformation the initial nonconvex problem (GGP) is reduced to a (DC) programming problem where both the constraints ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
A deterministic global optimization algorithm is proposed for locating the global minimum of generalized geometric (signomial) problems (GGP). By utilizing an exponential variable transformation the initial nonconvex problem (GGP) is reduced to a (DC) programming problem where both the constraints and the objective are decomposed into the difference of two convex functions. A convex relaxation of problem (DC) is then obtained based on the linear lower bounding of the concave parts of the objective function and constraints inside some box region. The proposed branch and bound type algorithm attains finite fflconvergence to the global minimum through the successive refinement of a convex relaxation of the feasible region and/or of the objective function and the subsequent solution of a series of nonlinear convex optimization problems. The efficiency of the proposed approach is enhanced by eliminating variables through monotonicity analysis, by maintaining tightly bound variables thro...
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, ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
. 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...
Global Optimization Approaches in Protein Folding and Peptide Docking
, 1999
"... . The recent advances in genetic engineering, high powered computing and global optimization continue to stimulate interest in the area of molecular modeling and protein structure prediction. The goal of these efforts is the ability to correctly predict native protein conformations and the binding i ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
. The recent advances in genetic engineering, high powered computing and global optimization continue to stimulate interest in the area of molecular modeling and protein structure prediction. The goal of these efforts is the ability to correctly predict native protein conformations and the binding interactions of macromolecules. These two problems currently dominate the field of computational chemistry and, through the use of detailed molecular models, they have also greatly influenced research in the area of global optimization. This article examines some aspects related to conformational energy modeling and reviews a variety of global optimization approaches developed for the protein folding and peptide docking problems. 1. Introduction Proteins are undoubtedly the most complex and vital molecules in nature. This complexity arises from an intricate balance of intra and intermolecular interactions which define the native threedimensional structure of the system, and subsequently i...