Ignacio E. Grossmann

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.

Simultaneous optimization models for heat integration. II, Heat exchanger network synthesis

Abstract In this paper, a mixed integer nonlinear programming (MINLP) model is presented which can generate networks where utility cost, exchanger areas and selection of matches are optimized simultaneously. The proposed model does not rely on the assumption of fixed temperature approaches (HRAT or EMAT), nor on the prediction of the pinch point for the partitioning into subnetworks. The model is based on the stage-wise representation introduced in Part I of this series of papers, where within each stage, potential exchanges between each hot and cold stream can occur. The simplifying assumption on isothermal mixing to calculate heat transfer area for stream splits allows the feasible space to be defined by a set of linear constraints. As a result, the model is robust and can be solved with relative ease. Constraints on the network design that simplify its structure, e.g. no stream splits, forbidden matches, required and restricted matches as well as the handling of multiple utilities can be easily included in the model. In addition, the model can consider matches between pairs of hot streams or pairs of cold streams, as well as variable inlet and outlet temperatures. Several examples are presented to illustrate the capabilities of the proposed simultaneous synthesis model. The results show that in many cases, heuristic rules such as subnetwork partitioning, no placement of exchangers across the pinch, number of units, fail to hold when the optimization is performed simultaneously.

A combined penalty function and outer-approximation method for MINLP optimization

Abstract An improved outer-approximation algorithm for MINLP optimization is proposed in this paper which is aimed at the solution of problems where convexity conditions may not hold. The proposed algorithm starts by solving the NLP relaxation. If an integer solution is not found, a sequence of iterations consisting of NLP subproblems and MILP master problems is solved. The proposed MILP master problem is based on the outer-approximation/equality-relaxation algorithm and features an exact penalty function that allows violations of linearizations of nonconvex constraints. The search proceeds until no improvement is found in the NLP subproblems. Computational experience is presented on a set of 20 test problems. Included are problems for optimum feed tray location and number of plates for distillation columns which are described in detail. The results show that although no theoretical guarantee can be given, the proposed method has a high degree of reliability for finding the global optimum in nonconvex problems.

State-of-the-art review of optimization methods for short-term scheduling of batch processes

There has been significant progress in the area of short-term scheduling of batch processes, including the solution of industrial-sized problems, in the last 20 years. The main goal of this paper is to provide an up-to-date review of the state-of-the-art in this challenging area. Main features, strengths and limitations of existing modeling and optimization techniques as well as other available major solution methods are examined through this paper. We first present a general classification for scheduling problems of batch processes as well as for the corresponding optimization models. Subsequently, the modeling of representative optimization approaches for the different problem types are introduced in detail, focusing on both discrete and continuous time models. A comparison of effectiveness and efficiency of these models is given for two benchmarking examples from the literature. We also discuss two real-world applications of scheduling problems that cannot be readily accommodated using existing methods. For the sake of completeness, other alternative solution methods applied in the field of scheduling are also reviewed, followed by a discussion related to solving large-scale problems through rigorous optimization approaches. Finally, we list available academic and commercial software, and briefly address the issue of rescheduling capabilities of the various optimization approaches as well as important extensions that go beyond short-term batch scheduling.

An algorithmic framework for convex mixed integer nonlinear programs

This paper is motivated by the fact that mixed integer nonlinear programming is an important and difficult area for which there is a need for developing new methods and software for solving large-scale problems. Moreover, both fundamental building blocks, namely mixed integer linear programming and nonlinear programming, have seen considerable and steady progress in recent years. Wishing to exploit expertise in these areas as well as on previous work in mixed integer nonlinear programming, this work represents the first step in an ongoing and ambitious project within an open-source environment. COIN-OR is our chosen environment for the development of the optimization software. A class of hybrid algorithms, of which branch-and-bound and polyhedral outer approximation are the two extreme cases, are proposed and implemented. Computational results that demonstrate the effectiveness of this framework are reported. Both the library of mixed integer nonlinear problems that exhibit convex continuous relaxations, on which the experiments are carried out, and a version of the software used are publicly available.

A structural optimization approach in process synthesis—II: Heat recovery networks

Abstract Several formulations of the transshipment model from Operations Research are proposed for the optimal synthesis of heat exchanger networks. The linear programming versions are used for predicting the minimum utility cost, and can handle restricted matches and multiple utilities. The mixed-integer programming version yields minimum utility cost networks in which the number of units is minimized, while allowing stream splitting and selection of most preferred matches. It is shown that the transshipment models can also be incorporated easily within a mixed-integer programming approach for synthesizing chemical processing systems. Several numerical examples are presented which show that the proposed models are computationally very efficient.

Retrospective on optimization

In this paper, we provide a general classification of mathematical optimization problems, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering. We then provide a review of solution methods of the major types of optimization problems for continuous and discrete variable optimization, particularly nonlinear and mixed-integer nonlinear programming (MINLP). We also review their extensions to dynamic optimization and optimization under uncertainty. While these areas are still subject to significant research efforts, the emphasis in this paper is on major developments that have taken place over the last 25 years.

A structural optimization approach in process synthesis—I: Utility systems

Abstract A mixed-integer linear programming approach is presented for performing structural and parameter optimization in the synthesis of processing systems. This approach is applied to the synthesis of utility systems that have to provide fixed demands of electricity, power for drivers and steam at various pressure levels. A superstructure that has embedded many potential configurations of utility systems is proposed, as well as its corresponding mixed-integer programming model. The application of the model is illustrated with a large example problem.

Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques

This paper has as a major objective to present a unified overview and derivation of mixed-integer nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.

Modelling and computational techniques for logic based integer programming

This paper presents a modelling framework for discrete optimization problems that relies on a logic representation in which mixed-integer logic is represented through disjunctions, and integer logic through propositions. It is shown that transformation of the logic formulation into the equation form is not always desirable, and that therefore there is a need to address the solution of mixed-integer programming problems where some of the mixed-integer relationships are expressed in disjunctions while others are expressed as algebraic constraints. A theoretical characterization of disjunctive constraints is proposed which can serve as a criterion for deciding whether a disjunction should be transformed into equation form. A solution algorithm that generalizes the method of Raman and Grossmann (Computers & Chemical Engineering, 17, 909, 1993) for handling mixed-integer disjunctions symbolically is also proposed. Several examples are presented to illustrate the proposed modelling framework and the potential of the solution method.