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...

: Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMIN--ffBB, and the General structure Mixed Integer Nonlinear ffBB, GMIN--ffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of MINLPs involving twice--differentiable 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 SMIN--ffBB; mixed nonlinear terms whose continuous relaxation is twice--differentiable are handled by the GMIN--ffBB. While both algorithms use the concept of a branch & bound tree, they rely on fundamentally different bounding and branching strategies. In the GMIN--ffBB 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...

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...

. 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) decomposition-based 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 decomposition-based 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...

Part I of this paper (Adjiman et al., 1997) described the theoretical foundations of a global optimization algorithm, the ffBB algorithm, which can be used to solve problems belonging to the broad class of twice-differentiable NPLs. For any such problem, the ability to automatically generate progressively tighter convex lower bounding problems at each iteration guarantees the convergence of the branchand -bound ffBB algorithm to within ffl of the global optimum solution. Several methods were presented for the construction of convex valid underestimators for general nonconvex functions. In this second part, the performance of the proposed algorithm and its alternative underestimators is studied through their application to a variety of problems. An implementation of the ffBB is described and a number of rules for branching variable selection and variable bound updates are shown to enhance convergence rates. A user-friendly parser facilitates problem input and provides flexibility in the...

A global optimization algorithm, αBB, for twice-differentiable NLPs is presented. It operates within a branch-and-bound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twice-differentiable functions is described. The αBB has been applied to a variety of problems and a summary of the results obtained is provided.

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 PNS-problems as set-covering or set-partitioning problems. These problems are well-known to be NPcomplete, thus, a PNS-problem is NP-hard. 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...

. 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 non--convexitie...

: 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...

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 NP-complete. In this work, we present such a subclass of PNS problems which is well-solvable. 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...