# (514a) Global Optimization Algorithm for Miqcps Featuring Spatial Branch-and-Bound and Multiparametric Disaggregation

#### AIChE Annual Meeting

#### 2016

#### 2016 AIChE Annual Meeting

#### Computing and Systems Technology Division

#### Advances in Optimization II

#### Wednesday, November 16, 2016 - 12:30pm to 12:50pm

A key element for global optimization is the computation of a tight lower bound. Three different strategies can essentially be used to achieve this goal: (i) find an equivalent formulation that differs for the better in the quality of the relaxation (perhaps the pooling problem is the most well-known [13]); (ii) rely on piecewise relaxation approaches [14-20], of which multiparametric disaggregation [19-20] has the advantage of adding a number of binary variables that scales logarithmically with the number of partitions (*N*) that is translated into a significantly better computational performance for *N*â?¥10; (iii) instead of reducing the variables domain simultaneously, do it one by one, an iterative procedure known as spatial branch-and-bound (SB&B) that can converge rather slowly.

The main novelty of this work is to integrate SB&B with the powerful mixed-integer linear programming relaxation derived from normalized multiparametric disaggregation [20]. Experiments are conducted after selecting one variable to discretize in every bilinear term and using two alternative settings for the number of partitions: *N*=10 and *N*=100. Optimality based bound tightening (OBBT) with the standard McCormick relaxation [21] is also performed at every node of the tree, after bisecting the domain of the variable contributing to the largest discrepancy, so as to compute tight lower and upper bounds for all bilinearly appearing variables.

Through the solution of a set of 43 NLP and MINLP benchmark problems from the literature, we show that the proposed global optimization algorithm achieves a better performance in terms of optimality gap compared to BARON 14.0.3 and GloMIQO 2.3 [22], being *N*=10 the best alternative. This is an indication that both piecewise relaxation approaches and OBBT should be used to a greater extent. On the other hand, GloMIQO led to the lowest computational time, benefiting from the faster solution time in the easiest problems, while BARON had some difficulties in the process network problems but excelled with the MINLP multiperiod blending problems. The latter result is an indication that we should explore branching on the binary variables in future work.

Acknowledgments: Financial support from Fundação para a Ciência e Tecnologia (FCT) through the Investigador FCT 2013 program and project UID/MAT/04561/2013.

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