Logic-based solution methods for GDP

GDP problems can be reformulated as MINLP, or solved with specialized algorithms16. In particular, the disjunctive branch and bound7,17and logic-based outer-approximation18 have proven to be successful in some problems.

Disjunctive branch and bound

The idea behind the disjunctive branch and bound17,7 is to branch directly on the disjunc- tions, while using the continuous relaxation of the BM or HR of the remaining disjunc- tions. Let (R-BM) and (R-HR) be the continuous relaxation of (BM) and (HR), respec- tively. The disjunctive branch and bound is as follows:

For a nodeNp, letzpdenote the optimal value of (R-BM) or (R-HR) of the corresponding GDPp, and (xp, yp) its solution. LetLbe the set of nodes to be solved, andGDP0 be the

original GDP. Let zup be an upper bound for the optimal value of the objective function z∗.

0. Initialize.SetL=N0,zup =∞,(x∗, y∗) = ∅.

1. Terminate. IfL=∅, then(x∗, y∗)is optimal and algorithm terminates.

2. Node selection.Choose a nodeNp ∈L, and delete it fromL. Go either to 3a or to 3b.

3a. Bound.Solve the (R-HR) ofGDPp corresponding toNp. If it is infeasible, go to step

1. Else, letzp _{be its objective function and (x}p_{, y}p_{) its solution.}

3b. Bound.Solve the (R-BM) ofGDPpcorresponding toNp. If it is infeasible, go to step

1. Else, letzp _{be its objective function and (x}p_{, y}p_{) its solution.}

4. Prune.Ifzp ≥zup, go to step 1. If yp ∈ _Zq _{let z}up _{= z}p _{and (x}∗

, y∗) = (xp, yp). Delete all nodes Nr ∈ L in which zr_≥zup_{, and go to step 1. Else, go to step 5.}

5. Branch. Select a disjunctionk ∈K such thatyki ∈ {/ 0,1}for somei∈ Dk. For every i ∈Dk, construct the corresponding GDP (GDPpi) by setting the constraints correspond-

ing to the disjunctive termias global, and removing the Boolean variables and constraints corresponding to termi0 6= i;i0 ∈ Dk. Add|Dk|new nodes, corresponding toGDPpi, to L. Go to step 1.

It is easy to see that this algorithm terminates finitely, in the worst case evaluating every possible node. This algorithm can be trivially modified to consider a tolerance for termi- nation >0. The HR disjunctive branch and bound makes use of Step 3a at every node, while the BM disjunctive branch and bound makes use of Step 3b at every node. It is also possible to have a hybrid disjunctive branch and bound in which some nodes are solved us- ing the BM and others using the HR. It is important to note that, as the disjunctive branch and bound progresses, the GDP problems that correspond to each node become smaller. In particular, the constraints that correspond to the disjunctive terms that were not selected are removed from the problem formulation in the subsequent nodes.

Note that the worst case involvesQ

k∈Dk|Dk| leaf nodes, which is fewer than the worst

case number of leaf nodes in a binary branch and bound algorithm (bounded by2Pk∈K(|Dk|−1)₎

except when|Dk| = 2,∀k ∈ K (in which case the maximum number of leave nodes for

both algorithms is2|K|_{). The worst case for number of evaluated nodes in the disjunctive}

branch and bound depends on the sequence in which the disjunctions were branched, but it is smaller than the binary branch and bound except when|Dk|= 2,∀k(in which case it

CHAPTER 1. INTRODUCTION

is the same:2|K|+1−1).

It has been shown that the disjunctive branch and bound has advantages over the binary branch and bound17. Also, the disjunctive branch and bound can be used for linear, convex or nonconvex GDP problems. In the case of linear GDP problems, an LP is solved at every node. In nonlinear GDP problems, and NLP is solved to global optimality at every node.

Logic-based outer-approximation for nonlinear convex GDP

The logic-based outer-approximation18iteratively solves a master problem and a subprob- lem. The master problem is a linear GDP relaxation of the original GDP that seeks to find a lower bound and an alternative for the vector of Boolean variables (Y). The master problem in the first iteration is obtained by outer-approximating the nonlinear functions at certain solutions (xp_{). The subproblem is an NLP in which the Boolean variables are fixed}

(i.e. settingYki = T rue for the terms selected by the master problem). The subproblem

provides an upper bound when a feasible solution is found. If the subproblem is infeasible, then an alternative NLP subproblem is solved (the feasibility subproblem). The solution of the subproblem (xp_{) is used to perform further linearizations of the constraints, which}

are added to the master linear GDP. Note that for a given solutionxP_{, the linearizations are}

performed for the global constraints and for the constraints that correspond to the selected active terms(Yki =T rue).

In the outer-approximation method, the linearization is performed by generating a first- order Taylor series approximation of the constraints. For a vector of nonlinear constraint

(g(x) ≤ 0)and a given set of solutions (xp;p = 1, ..., P), the linearization is: g(xp) +

∇g(xp)T(x − xp) ≤ 0. The main limitation of this linearization is that it provides a valid linear relaxation only for convex functions. If the functiong(x)is nonconvex, this linearization can cut off regions that are feasible forg(x)≤ 0. In such a case, the master linear GDP is no longer a linear relaxation of the problem.

The convex logic-based outer-approximation guarantees convergence in finite iterations because the master problem and subproblem are equivalent for the discrete solutions in

which the subproblem has been evaluatedY ;p = 1, ..., P. This means that if the master problem selects an alternative Yp that was already evaluated in the subproblem (i.e. the linearization of this alternative is already included in the master problem), then the optimal objective values of the master problem and subproblem are the same. Clearly, if all the alternativesYp_{;p= 1, ..., Pthat are feasible for the master problem are evaluated, then the}

lower bound of the master problem and the upper bound of the subproblem are the same. The proof of convergence can be found in the original work by Turkay and Grossmann18_.