#### Date of Award

Fall 9-2015

#### Embargo Period

6-11-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Chemical Engineering

#### Advisor(s)

Ignacio E. Grossmann

#### Abstract

Many optimization problems require the modelling of discrete and continuous variables, giving rise to mixed-integer linear and mixed-integer nonlinear programming (MILP / MINLP). An alternative representation of MINLP is Generalized Disjunctive Programming (GDP)1. GDP models are represented through continuous and Boolean variables, and involve algebraic equations, disjunctions, and logic propositions. This higher level representation facilitates the modelling process while keeping the logic structure of the problem. GDP models are typically reformulated as MINLP problems to exploit the developments in these solvers. The two traditional GDP-to-MINLP reformulations are the Big-M (BM) and Hull-reformulation (HR). Alternatively to direct MINLP reformulations, special techniques can help to improve the performance in solving GDP problems. There are two main contributions in this thesis. The first contribution involves the development of reformulations and methods that generate improved MINLP models form GDP problems. This development is achieved by exploiting the logic-nature of GDP, as well as alternative GDP-to-MINLP reformulations, to obtain relatively small MINLP models with tight continuous relaxations. The second contribution of this thesis is the improvement of existing GDP solution methods by the use of novel concepts. In particular, we improve the linear disjunctive branch and bound through the use of a Lagrangean relaxation of the HR. Also, we extend the logic-based outer-approximation to nonconvex problems, and develop a novel method to obtain cutting planes that improves the linear relaxation of the nonconvex problem. In the thesis, we first present a new Big-M reformulation of GDPs. Unlike the traditional iii Big-M reformulation that uses one M-parameter for each constraint, the new approach uses multiple M-parameters for each constraint. The multiple-parameter Big-M (MBM) reformulation is at least as tight as the traditional BM. Furthermore, it does not require additional variables or constraints. We present the new MBM and analyze the strength in its continuous relaxation compared to that of the traditional Big-M. We then present two algorithmic approaches to improve mixed-integer models that are originally formulated as convex GDPs. The algorithms seek to obtain an improved continuous relaxation of the MINLP reformulation of the GDP, while limiting the growth in the problem size. Both algorithms make use of the logic operation called basic step. This operation allows the derivation of formulations with continuous relaxations that are stronger than the direct BM and HR reformulations. The two algorithms differ in the method to exploit the advantages of the small problem size of the BM, and the tight continuous relaxation of the HR after the application of basic steps. The first algorithm uses a hybrid reformulation of GDP that seeks to exploit both advantages of the BM and HR. The second algorithm uses the strong formulation to derive cuts for the BM, generating a stronger formulation with small growth in problem size. In terms of GDP solution methods, we first present an enhancement to the disjunctive branch and bound for linear GDPs. In particular, we present a Lagrangean relaxation of the HR. The proposed Lagrangean relaxation can be applied to any linear GDP, and it always assigns 0-1 values to the binary variables of the HR. Furthermore, this relaxation is much simpler to solve than the continuous relaxation of the HR. The Lagrangean relaxation can be used in different manners to improve GDP solution methods. In this thesis, we explore the use of the Lagrangean relaxation as a primal heuristic to find feasible solutions in a disjunctive branch and bound. We note that the proposed Lagrangean relaxation, and its use in the disjunctive branch and bound, can be extended to nonlinear convex problems. We then extend the logic-based outer-approximation to the global solution of non-convex GDPs. The general idea of the algorithm is to have a linear master GDP that overestimates the feasible region of the GDP. This master problem provides a valid lower bound (in a minimization problem), and the selection of only one disjunctive term in each of the disjunctions. With the alternative provided by the master problem, an NLP subprobiv lem is solved to global optimality. This NLP subproblem is smaller and simpler than the continuous relaxation of the MINLP reformulation of the original GDP. After solving the subproblem, infeasibility or optimality integer cuts can be added to the master problem. This basic algorithm has the advantage of solving only small NLP problems to global optimality, instead of solving a larger MINLP to global optimality from the beginning. Furthermore, by using GDP as framework the NLP subproblem is smaller and simpler than an equivalent method directly applied to the MINLP reformulation. In order to further improve the performance of this logic-based outer approximation, two main features were implemented: derivation of additional cuts and partition of the algorithm in two stages. Finally, we apply a modified version of the global logic-based outer-approximation to the multiperiod blending problem. In addition to the proposed solution method, we present an improved problem formulation that makes use of redundant constraints. In order to generate such constraints, an alternative formulation was derived. The main idea in the new formulation is to track sources or commodities in the system, instead of tracking compositions. The main advantage is that it is possible to create redundant constraints in which the sum of individual source flows adds up to the total flow. Similarly, the sum of individual source inventories adds up to the total inventory. These redundant constraints considerably improve the relaxation of the model when linear approximations are used for the bilinear terms. Furthermore, the additional constraints can be included in the original model, strengthening its linear relaxation. This thesis makes several important contributions. From an aggregated perspective, our most significant contribution is the use of GDP and its logic structure to obtain improved models and develop solution methods. In this thesis we show that GDP is not only an intuitive and structured modeling framework, but it also opens a set of tools that are not accessible when modeling problems using mixed-integer programming. The tools we have developed can help to solve some problems in Process Systems Engineering (PSE). Furthermore, we hope that the advantages of formulating some problems using GDP become apparent. As the PSE community continues to increasingly use GDP as modeling framework, we hope it brings greater attention to the OR community.

#### Recommended Citation

Trespalacios, Francisco, "Improved Formulations and Computational Strategies for the Solution and Nonconvex Generalized Disjunctive Programs" (2015). *Dissertations*. 651.

http://repository.cmu.edu/dissertations/651