Discrete-continuous optimization problems in process systems engineering are commonly modeled in algebraic form as mixed-integer linear or nonlinear programming models. Since these models can often be formulated in different ways, there is a need for a systematic modeling framework that provides a fundamental understanding on the nature of these models, particularly their continuous relaxations. This paper describes a modeling framework, Generalized Disjunctive Programming (GDP), which represents problems in terms of Boolean and continuous variables, allowing the representation of constraints as algebraic equations, disjunctions and logic propositions. We provide an overview of major research results that have emerged in this area. Basic concepts are emphasized as well as major classes of formulations that can be derived. These are illustrated with a number of examples in the area of process systems engineering. As will be shown, GDP provides a structured way for systematically deriving mixed-integer optimization models that exhibit strong continuous relaxations. IntroductionMixed-integer optimization provides a powerful framework for the mathematical modeling of many optimization problems that involve discrete and continuous variables. Over the last few years there has been a pronounced increase in the development of Mixed-integer Linear/ Nonlinear Programming (MILP/MINLP) models in Process Systems Engineering [10][12] [21][23] [29] . Synthesis models can be formulated as MILP for high level targeting, and as MINLP for more detailed superstructure optimization [11] . For planning and scheduling, the majority tends to be MILP models [6][9] [24][28] although gradually there is also an increasing trend to MINLP, especially as process models are incorporated [17] . MILP/MINLP models, however, are based on algebraic formulations which are not unique. Although there has been significant progress in software for solving mixed-integer problems, especially MILP, how one formulates a model can have a major impact in the performance and capability to find a solution. Therefore, deriving "good" formulations or finding potential improvements in existing models is commonly regarded as an art and strongly depends on the modeler skills.

This work presents a review of the applications of mixed-integer nonlinear programming (MINLP) in process systems engineering (PSE). A review on the main deterministic MINLP solution methods is presented, including an overview of the main MINLP solvers. Generalized disjunctive programming (GDP) is an alternative higher-level representation of MINLP problems. This work reviews some methods for solving GDP models, and techniques for improving MINLP methods through GDP. The paper also provides a high-level review of the applications of MINLP in PSE, particularly in process synthesis, planning and scheduling, process control and molecular computing.

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In this work, we present a new Big-M reformulation for Generalized Disjunctive Programs. The proposed MINLP reformulation is stronger than the traditional Big-M, and it does not require additional variables or constraints. We present the newBig-M, and analyze the strength in its continuous relaxation compared to that of the traditional Big-M. The new formulation is tested by solving several instances of process networks and muli-product batch plant problems with an NLP-based branch and bound method. The results show that, in most cases, the new reformulation requires fewer nodes and less time to find the optimal solution.

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This paper introduces the concept of optimization equilibrium as an equivalently versatile definition of a generalized Nash equilibrium for multi-agent non-cooperative games. Through this modified definition of equilibrium, we draw precise connections between generalized Nash equilibria, feasibility for bilevel programming, the Nikaido-Isoda function, and classic arguments involving Lagrangian duality and social welfare maximization. Significantly, this is all in a general setting without the assumption of convexity. Along the way, we introduce the idea of minimum disequilibrium as a solution concept that reduces to traditional equilibrium when equilibrium exists. The connections with bilevel programming and related semi-infinite programming permit us to adapt global optimization methods for those classes of problems, such as constraint generation or cutting plane methods, to the problem of finding a minimum disequilibrium solution. We show that this method works, both theoretically and with a numerical example, even when the agents are modeled by mixed-integer programs.

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