A Branch and Bound Algorithm for the Global Optimization and its Improvements

3 1 The term optimization 4 1.1 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Why global optimization? . . . . . . . . . . . . . . . . . . . . 10 1.3 The structure of Global Optimization Algorithms . . . . . . . 12 1.4 Local Optimization of NLPs . . . . . . . . . . . . . . . . . . . 12 1.4.1 Notions of convex analysis . . . . . . . . . . . . . . . . 12 1.4.2 Necessary and sufficient conditions for local optimality 13 1.5 A brief history of global optimization . . . . . . . . . . . . . . 17 2 Branch and Bound methods 19 2.1 A general B&B Framework . . . . . . . . . . . . . . . . . . . 20 2.2 Initialization step . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Node selection . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Lower bound update . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Upper bound update . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Fathoming rule . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Stopping rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.8 Branching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.9 Domain reduction . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Overlapping Branch and Bound (oBB) 28 3.1 Description of the Algorithm . . . . . . . . . . . . . . . . . . 28 3.1.1 Calculating Upper Bound: Discarding Balls and Feasible Points . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Splitting Rule . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Lipschitz lower bounds Improvements . . . . . . . . . . . . . 32 3.2.1 First order lower bounds . . . . . . . . . . . . . . . . . 33 3.2.2 Second order lower bounds . . . . . . . . . . . . . . . 35 3.3 Parallelization of oBB . . . . . . . . . . . . . . . . . . . . . . 37 3.3.1 Data Parallelism: Bounds in Parallel . . . . . . . . . . 37 3.3.2 Task Parallelism: Tree in Parallel . . . . . . . . . . . . 39