William R. Esposito

Deterministic Global Optimization in Nonlinear Optimal Control Problems

The accurate solution of optimal control problems is crucial in many areas of engineering and applied science. For systems which are described by a nonlinear set of differential-algebraic equations, these problems have been shown to often contain multiple local minima. Methods exist which attempt to determine the global solution of these formulations. These algorithms are stochastic in nature and can still get trapped in local minima. There is currently no deterministic method which can solve, to global optimality, the nonlinear optimal control problem. In this paper a deterministic global optimization approach based on a branch and bound framework is introduced to address the nonlinear optimal control problem to global optimality. Only mild conditions on the differentiability of the dynamic system are required. The implementa-tion of the approach is discussed and computational studies are presented for four control problems which exhibit multiple local minima.

Deterministic global optimization in isothermal reactor network synthesis

The reactor network synthesis problem involves the simultaneous determination of the structure and operating conditions of a reactor system to optimize a given performance measure. This performance measure may be the yield of a given product, the selectivity between products, or the overall profitability of the process. The problem is formulated as a nonlinear program (NLP) using a superstructure based method in which plug flow reactors (PFRs) in the structure are modeled using differential-algebraic equations (DAEs). This formulation exhibits multiple local minima. To overcome this, a novel deterministic global optimization method tailored to the special structure and characteristics of this problem will be presented. Examples of isothermal networks will be discussed to show the nature of the local minima and illustrate various components of the proposed approach.

Parameter estimation in nonlinear algebraic models via global optimization

The estimation of parameters in semi-empirical nonlinear models through the error-in-variables method has been widely studied from a computational standpoint. This method involves the minimization of a quadratic objective function subject to the model equations being satisfied. Due to the nonlinear nature of these models, the resulting formulation is nonconvex in nature. The approaches to solve this problem presented so far in the literature, although computationally efficient, only offer convergence to a local solution of a model which may contain multiple minima. In this paper a global optimization approach based on a branch-and-bound framework will be presented to solve the error-in-variables formulation. Various estimation problems were solved and will be presented to illustrate the theoretical and computational aspects of the proposed method.

Global optimization of nonconvex problems with differential-algebraic constraints

Differential-algebraic systems of constraints, in particular, initial value ordinary differential equations, appear in numerous optimization problems in the chemical engineering field. A difficulty in the solution of this formulation which has not been throughly addressed, is the problem of multiple local minima. In this paper, a novel deterministic global optimization method using a sequential approach will be presented.

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) .