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Abstract: Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering. [Copyright &y& Elsevier]

Abstract: This paper presents a systematic method for optimal lumping of a large number of components in order to minimize the loss of information. In principle, a rigorous composition-based model is preferable to describe a system accurately. However, computational intensity and numerical issues restrict such applications in process modeling, simulation and design. A pseudo-component approach that lumps a large number of components in a system into a much smaller number of hypothetical groups reduces the dimensionality at the cost of losing information. Moreover, empirical and heuristic approaches are commonly used to determine the lumping scheme. Given an objective function defined with a linear weighting rule, an optimal lumping problem is formulated as a mixed integer nonlinear programming (MINLP) problem both in discrete and in continuous settings. A reformulation of the original problem is also presented, which significantly reduces the number of independent variables. The application to a system with 144 components demonstrates that the optimal lumping problem can be efficiently solved with a stochastic optimization method, Tabu Search (TS) algorithm. The case study also reveals that the discrete formulation is preferable due to the reduced search space compared to a continuous model formulation. [Copyright &y& Elsevier]

This paper presents a technique for simultaneous targeting and design in cooling water systems comprising of at least two cooling towers and several cooling water using operations. The presented technique is based on a superstructure from which a mathematical formulation is derived using system specific variables and parameters. It is demonstrated that in a system like this, true optimality can only be realized by a holistic consideration of the entire cooling water system. Consideration of individual subsets of cooling towers with their dedicated cooling water operations yields suboptimal results. Four operational cases are considered and structural considerations of corresponding mathematical formulations presented. The first case results in a linear programming (LP) formulation, the second case yields a mixed integer linear programming (MILP) formulation whilst the other two cases yield mixed integer nonlinear programming (MINLP) formulations which cannot be exactly linearized. However, in all cases significant improvements in excess of 40% were realized in targeting, without compromising the heat duty of the cooling water using operations. The main objective of this investigation is to debottleneck the overall cooling water supply for the cooling water network. [Copyright &y& Elsevier]

This paper treats the solution of nonlinear optimization problems involving discrete decision variables, also known as generalized disjunctive programming (GDP) or mixed-integer nonlinear programming (MINLP) problems, that arise in process engineering. The key idea is to eliminate the discrete decision variables by adding a set of continuous variables and constraints that represent the discrete decision space of the optimization problem. With such a reformulation, we are able to apply solution algorithms for purely continuous nonlinear optimization problems to efficiently calculate local minima of GDP or MINLP problems. In this contribution, we propose different alternatives to reformulate GDP/MINLP problems as continuous optimization problems. We furthermore investigate theoretical properties of the different reformulations with regard to their numerical solution. The proposed formulations are illustrated and analyzed on the basis of optimization problems dealing with process engineering applications involving stationary as well as dynamic process models. [Copyright &y& Elsevier]