Abstract: Parameter estimation is a key problem in the development of process models, both steady- and unsteadystate, and thus is an important issue in both process design and control. The error-in-variable approach differs distinctly from the standard approach in that measurement errors in both dependent and independent system variables are taken into account when formulating the objective function in the parameter estimation problem. It is not uncommon for the objective function in nonlinear parameter estimation problems to have multiple local optima. However, the usual methods used to solve these problems are local methods that offer no guarantee that the global optimum, and thus the best set of model parameters, has been found. We demonstrate here a technique, based on interval analysis, that can solve the error-in-variable parameter estimation problem with complete reliability, providing a mathematical and computational guarantee that the global optimum is found. As examples, we consider the estimation of parameters in both steady and unsteady-state models, including a vapor-liquid equilibrium (VLE) model, a CSTR model, and a reaction kinetics model.

The use of interval methods, in particular interval-Newton/generalized-bisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inverse-midpoint method, is presented, as is a new scheme for selection of the real point used in formulating the interval-Newton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the intervalNewton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.

A strategy is described for using linear programming (LP) to bound the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method for deterministic global optimization. An implementation of this technique is described in detail, and several important issues are considered. These include selection of the interval corner required by the LP strategy, and determination of rigorous bounds on the solutions of the LP problems. The impact of using a local minimizer for updating the upper bound on the global minimum in this context is also considered. The procedure based on these techniques, LISS LP, is demonstrated using several global optimization problems, with focus on problems arising in chemical engineering. Problems with a very large number of local optima can be effectively solved, as well as problems with a relatively large number of variables.

In recent years, it has been shown that strategies based on an interval-Newton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior. These strategies provide a mathematical and computational guarantee either that all solutions have been located in an equation solving problem or that the global optimum has been found in an optimization problem. The primary drawback to this approach is the potentially high computational cost. In this paper, we consider strategies for bounding the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method. Recent preconditioning techniques for this purpose are reviewed, and a new bounding approach based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to determine the desired bounds exactly (within round out), leading to significant overall improvements in computational efficiency. These techniques will be demonstrated using several global optimization problems, with focus on problems arising in chemical engineering, including parameter estimation and molecular modeling. These problems range in size from under ten variables to over two hundred, and are solved deterministically using the interval methodology.

Branch-and-prune (BP) and branch-and-bound (BB) techniques are commonly used for intelligent search in finding all solutions, or the optimal solution, within a space of interest. The corresponding binary tree structure provides a natural parallelism allowing concurrent evaluation of subproblems using parallel computing technology. Of special interest here are techniques derived from interval analysis, in particular an interval-Newton/generalized-bisection procedure. In this context, we discuss issues of load balancing and work scheduling that arise in the implementation of parallel interval-Newton on a cluster of workstations using message passing, and describe and analyze techniques for this purpose. Results using an asynchronous diffusive load balancing strategy show that a consistently high efficiency can be achieved in solving nonlinear equations, providing excellent scalability, especially with the use of a two-dimensional torus virtual network. The effectiveness of the approach used, especially in connection with a novel stack management scheme, is also demonstrated in the consistent superlinear speedups observed in performing global optimization.

Abstract—This paper presents a sophisticated and efficient parallel scheme for the DIRECT global optimization algorithm of Jones et al. (1993). Although several sequential implementations for this algorithm have been successfully applied to large scale MDO problems, few parallel versions of the DIRECT algorithm have addressed well algorithm characteristics such as a single starting point, an unpredictable workload, and a strong data dependency. These challenges engender many interesting design issues including domain decomposition, data access and management, and workload balancing. In the present work, a hierarchical parallel scheme has been developed to address these challenges at three levels. Each level is supported by parallel and distributed data structures to access shared data sets, distribute workload, or exchange messages. Parameter estimation problems in systems biology provide an ideal application context for the present work. Global nonlinear parameter estimation results obtained on a 200 node Linux cluster are given for a cell cycle model for frog eggs. Index Terms—DIRECT (DIviding RECTangles) algorithm, global optimization, GPSHMEM (generalized portable shared memory), load balancing strategy, multidisciplinary design optimization, parallel and distributed data structures, parameter estimation, systems biology 1.

Recent advances in global optimization for process synthesis, design and control are discussed. After a review of the chemical engineering contributions, we focus on the enclosure of all solutions of nonlinear constrained systems of equations. Important theoretical results are presented accompanied with computational studies on the enclosure of multiple steady states and all homogeneous azeotropes. 1 Introduction and Review A significant effort has been expended in the last four decades toward theoretical and algorithmic studies of applications that arise in Chemical Engineering Process Design, Process Synthesis, Process Control, as well as in Computational Chemistry and Molecular Biology. In the last decade we have experienced a dramatic growth of interest in Chemical Engineering for new methods of global optimization and their application to important engineering, as well as computational chemistry and molecular biology problems. Contributions from the chemical engineering communit...

Models in computational biology, such as those used in binding, docking and folding, are often empirical and have adjustable parameters. Since few of these models are yet fully predictive, the problem may be nonoptimal choices of parameters. We describe an algorithm called ENPOP (ENergy function Parameter OPtimization) that improves, and sometimes optimizes, the parameters for any given model and for any given search strategy that identies the stable state of that model. ENPOP iteratively adjusts the parameters simultaneously to move the model global minimum energy conformation for each of m dierent molecules as close as possible to the true native conformations, based on some appropriate measure of structural error. A proof of principle is given for two very dierent test problems. The rst involves 3 dierent 2-dimensional model protein molecules having 12 or 13 monomers, and 4 parameters in common. The parameters converge to the values used to design the model native structures.

: 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) twice-differentiable constrained nonlinear optimization problems, (b) mixed-integer 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; Mixed-Integer 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...

Abstract- In this paper we report the application and evaluation of the simulated annealing (SA) optimization method in parameter estimation for vapor-liquid equilibrium (VLE) modeling. We tested this optimization method using the classical least squares and error-in-variable approaches. The reliability and efficiency of the data-fitting procedure are also considered using different values for algorithm parameters of the SA method. Our results indicate that this method, when properly implemented, is a robust procedure for nonlinear parameter estimation in thermodynamic models. However, in difficult problems it still can converge to local optimums of the objective function. Keywords: Vapor-liquid equilibrium; Simulated annealing; Nonlinear parameter estimation; Error-in-variable method; Global optimization.