Convex generalized disjunctive programming (GDP)

Edited By: Nicholas Schafhauser, Blerand Qeriqi, Ryan Cuppernull

Introduction

[ Insert picture from google doc of GDP branching to Logic Based Methods and Reformulation MI(N)LP ]

Methodology

The two most common ways of reformulating a GDP problem into an MINLP are through Big-M (BM) and Hull Reformulation (HR). BM is the simpler of the two, while HR results in tighter relaxation (smaller feasible region) and faster solution times. (https://kilthub.cmu.edu/articles/A_hierarchy_of_relaxations_for_nonlinear_convex_generalized_disjunctive_programming/6466535)

Below is an example of the reformulation of the GDP problem from the Theory section reformulated into an MINLP by using the Big-M method.

${\begin{aligned}\min z=f(x)\\s.t.g(x)\leq 0\\r_{ki}(x)\leq M^{ki}(1-y_{ki})\quad k\in K,i\in D_{k}\\\sum _{i\in D_{k}}y_{ki}=1\quad k\in K\\Hy\geq h\\x^{lo}\leq x\leq x^{up}\\x\in \Re ^{n}\\y_{ki}\in {0,1}\quad k\in K,i\in D_{k}\end{aligned}}$

Notice that the boolean term from the original GDP has been converted into a numerical {0,1}. The logic relations have also been converted into linear integer constraints (Hy).

(https://kilthub.cmu.edu/articles/journal_contribution/Improved_Big-M_Reformulation_for_Generalized_Disjunctive_Programs/6467063)

This MINLP reformulation can now be used in well-known solvers (list them here) to calculate a solution.

Numerical Example

The following example was taken from: http://egon.cheme.cmu.edu/Papers/IMAGrossmannRuiz.pdf