Stephen T. Harding

LOCATING ALL AZEOTROPES IN HOMOGENEOUS AZEOTROPIC SYSTEMS

A global optimization based approach for finding all homogeneous azeotropes in multicomponent mixtures is presented. The global optimization approach is based on a branch and bound framework in which upper and lower bounds on the solution are refined by successively partitioning the target region into small disjoint rectangles. The objective of such an approach is to locate all global minima since each global minimum corresponds to an homogeneous azeotrope. The global optimization problem is formulated from the thermodynamic criteria for azeotropy, which involve highly nonlinear and nonconvex expressions. The success of this approach depends upon constructing valid convex lower bounds for each nonconvex function in the constraints. The convex lower bounding procedure is demonstrated with the Wilson activity coefficient equation. The global optimization approach is illustrated in an example problem.

Generalized Geometric Programming Problems

In this chapter, we will discuss test problems that arise from generalized geometric programming applications. For a thorough theoretical and algorithmic exposition of global optimization approaches for generalized geometric programming problems, the reader is directed to the article of Maranas and Floudas (1997), and the book by Floudas (2000).

Quadratic Programming Problems

In this chapter nonconvex quadratic programming test problems are considered. These test problems have a quadratic objective function and linear constraints. Quadratic programming has numerous applications (Pardalos and Rosen (1987), Floudas and Visweswaran (1995)) and plays an important role in many nonlinear programming methods. Recent methods of generating challenging quadratic programming test problems and disjointly constrained bilinear programming test problems can be found in the work of Vicente et al. (1992) and Calamai et al. (1993). Furthermore, a very broad class of difficult combinatorial optimization problems such as integer programming, quadratic assignment, and the maximum clique problem can be formulated as nonconvex quadratic programming problems.

Twice Continuously Differentiable NLP Problems

Twice continuously differentiable NLPs represent a very broad class of problems with diverse applications in the fields of engineering, science, finance and economics. Specific problems include phase equilibrium characterization, minimum potential energy conformation of clusters and molecules, distillation sequencing, reactor network design, batch process design, VLSI chip design, protein folding, and portfolio optimization.

Nonlinear Systems of Equations

The solution of nonlinear systems of equations is an important problem in engineering. Applications include identifying the multiple steady states of a reactor network (Folger, 1986; Schlosser and Feinberg, 1994), predicting the azeotropes formed in a nonideal mixture (Fidkowski et al., 1993; Maranas et al., 1996), finding the steady-states of a flowsheet or part of a flowsheet (Zhang, 1987; Bullard and Biegler, 1991; Wilhelm and Swaney, 1994; Bekiaris et al., 1993), identifying equilibrium points in multiphase systems (Heidemann and Mandhane, 1973; Seader et al., 1990) .

Semidefinite Programming Problems

Semidefinite programming involves the minimization of a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Several types of problems can be transformed to this form. This constraint is in general nonlinear and nonsmooth yet convex. Semidefinite programming can be viewed as an extension of linear programming and reduces to the linear programming case when the symmetric matrices are diagonal.

Mixed-Integer Nonlinear Programming Problems (MINLPs)

Mixed-integer problems are those that involve both continuous and integer variables. The introduction of integer variables allows the modeling of complex decisions through graph theoretic representations denoted as superstructures (Floudas, 1995). This representation leads to the simultaneous determination of the optimal structure of a network and its optimum operating parameters. Thus MINLPs find applications in engineering design such as heat exchanger network synthesis, reactor-separator-recycle network synthesis or pump network synthesis (Floudas, 1995; Grossmann, 1996), in metabolic pathway engineering (Hatzimanikatis et al., 1996a,b; Dean and Dervakos, 1996), or in molecular design (Maranas, 1996; Churi and Achenie, 1996) .

Quadratically Constrained Problems

In this chapter, we discuss nonconvex quadratically constrained test problems. These include problems with separable quadratic constraints, complementarity type constraints, and integer type constraints. Notice that every simple binary constraint of the form:

Combinatorial Optimization Problems

x \in \left\{ {0,1} \right\}