In this paper, the deterministic global optimization algorithm, αBB, (α-based 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 twice-differentiable 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...

. In order to generate valid convex lower bounding problems for nonconvex twice--differentiable optimization problems, a method that is based on second-- order information of general twice--differentiable 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; twice--differentiable; 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...

. Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, the quadratic problem is known to be NP-hard, which makes this one of the most interesting and challenging class of optimization problems. In this chapter, we review various properties of the quadratic problem, and discuss different techniques for solving various classes of quadratic problems. Some of the more successful algorithms for solving the special cases of bound constrained and large scale quadratic problems are considered. Examples of various applications of quadratic programming are presented. A summary of the available computational results for the algorithms to solve the various classes of problems is presented. Key words: Quadratic optimization, bilinear programming, concave pro...

Calculation of phase and chemical equilibrium represents a crucial phase in the modeling of many separation processes. For conditions of constant temperature and pressure, a necessary and sufficient condition for the true equilibrium solution is that (i) the total Gibbs free energy of the system be at its global minimum, or (ii) the minimum of the tangent plane distance function be nonnegative for all phase models used to represent the system. In this work, the goal is to obtain equilibrium solutions corresponding to a global minimum of the Gibbs free energy as efficiently as possible, for cases where the liquid phase or phases can be modeled by the NRTL, UNIQUAC, UNIFAC, Wilson, modified Wilson and ASOG equations. Vapor phases whose behavior can be described as ideal can also be handled. In achieving this goal, there are two distinct problems of relevance: (i) the minimization of the Gibbs free energy, denoted (G), and (ii) the minimization of the tangent plane distance fun...

The Gibbs tangent plane criterion has become an important tool in determining the quality of obtained solutions to the phase and chemical equilibrium problem. The ability to determine if a postulated solution is thermodynamically stable with respect to perturbations in any or all of the phases is very useful in the search for the true equilibrium solution. Previous approaches have concentrated on finding the stationary points of the tangent plane distance function. However, no guarantee of obtaining all stationary points can be provided. These difficulties arise due to the complex and nonlinear nature of the models used to predict equilibrium. In this work, simpler formulations for the stability problem are presented for the special class of problems where nonideal liquid phases can be adequately modeled using the NRTL and UNIQUAC activity coefficient equations. It is shown how the global minimum of the tangent plane distance function can be obtained for this class of problems. The adv...

In Floudas and Visweswaran (1990, 1992), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series of primal and relaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an ffl-global optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and in...

Several approaches have been proposed for the computation of solutions to the phase and chemical equilibrium problem when the problem is posed as the minimization of the Gibbs free energy function. None of them can guarantee convergence to the true optimal solution, and are highly dependent on the supplied initial point. Convergence to local solutions often occurs, yielding incorrect phase and component distributions. This work examines the problem when the liquid phase is adequately modeled by the Non-Random Two Liquid (NRTL) activity coefficient expression and the vapor phase is assumed to be ideal. The contribution of the proposed approach is twofold. Firstly, a novel and important property of the Gibbs free energy expression involving the NRTL equation is provided. It is subsequently shown that by introducing new variables, the problem can then be transformed into one where a biconvex objective function is minimized over a set of bilinear constraints. Secondly, the Global OPtimizat...

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εk-global minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

The Wilson equation for the excess Gibbs energy has found wide use in successfully representing the behavior of polar and nonpolar multicomponent mixtures with only binary parameters, but was incapable of predicting more than one liquid phase. The UNIFAC and ASOG group contribution methods do not have this limitation and can predict the presence of multiple liquid phases. The most important area of application of all these equations is in the prediction of phase equilibrium. The calculation of phase equilibrium involves two important problems: (i) the minimization of the Gibbs free energy, and (ii) the tangent plane stability criterion. Problem (ii), which can be implemented as the minimization of the tangent plane distance function, has found wide application in aiding the search for the global minimum of the Gibbs free energy. However, a drawback of all previous approaches is that they could not provide theoretical guarantees that the true equilibrium solution will be obtained. The g...

An increasingly popular approach when solving the phase and chemical equilibrium problem is to pose it as an optimization problem. However, difficulties are encountered due to the highly nonlinear nature of the models used to represent the behavior of the fluids, and because of the existence of multiple local solutions. This work shows how it is possible to guarantee ffl-global solutions for a certain important class of the phase and chemical equilibrium problem, namely when the liquid phase can be modeled using either the Non-Random Two-Liquid (NRTL) equation, or the UNIversal QUAsi Chemical (UNIQUAC) equation. Ideal vapor phases are easily incorporated into the global optimization framework. A number of interesting properties are described which drastically alter the structure of the respective problems. For the NRTL equation, it is shown that the formulation can be converted into a biconvex optimization problem. The GOP algorithm of Floudas and Visweswaran [8, 9] can then be used to...