Guided dive for the spatial branch-and-bound

Abstract

We study the spatial Brand-and-Bound algorithm for the global optimization of nonlinear problems. In particular we are interested in a method to find quickly good feasible solutions. Most spatial Branch-and-Bound-based solvers use a non-global solver at a few nodes to try to find better incumbents. We show that it is possible to improve the branching rules and the node priority by exploiting the solutions from the non-global solver. We also propose several smart adaptive strategies to choose when to run the non-global solver. We show that despite the time spent in solving more NLP problems in the nodes, the new strategies enable the algorithm to find the first good incumbents faster and to prove the global optimality faster. Numerous easy, medium size as well as hard NLP instances from the Coconut library are benchmarked. All experiments are run using the open source solver Couenne.

Calculating Global Minimum Points to Binary Polynomial Optimization Problem: Optimizing the Optimal PMU Localization Problem as a Case-Study

“…We study a combinatorial optimization problem related to some binary (Boolean) optimization model with application in the field of synchronized measurements for monitoring the smart grids. The innovation of this study is proved in practice because a polynomial problem with continuous variables is able to attain locally optimal solutions [8]- [12], [27]- [30].…”

“…(11) (12) Where ℎ element determines a polynomial observability constraint for the ℎ bus [27]- [31], [44]: (13) An auxiliary binary-valued variable is an extra term incorporated into the polynomial model to reflect yes or no decision investment [8]- [9]. Hence, such variables are binary adding to our proposed model with the aim to declare it as a pure Boolean optimization model [15].…”

“…Ιn spatial branch-and-bound implementation tree, the NLP problem is solved by the FMINCON or IPOPT optimizer routines using the solution of the convexification at the upper bound of the B&B process tree [8]. Therefore, a local search is implemented to come across the first best integral optimal solution [8], [12]- [13], [24]- [25].…”

A Geometric Branch and Bound Method for a Class of Robust Maximization Problems of Convex Functions

“…The geometric branch-and-bound (GB2) method developed in this paper for solving (RM) has some similarity with the spatial branch-and-bound (SB2) method used in nonconvex optimization [24,34,36]. Both have the common idea of sequentially partitioning the feasible region into sub-regions, and finding a lower and upper bound from the restricted problem (a convex optimization problem) in a sub-region [12,16,20]. However, the G2B2 method developed in this paper differs from the spatial branch-and-bound approach in how it partitions the feasible set, and its evaluation of the lower and upper bounds.…”

Security-Constrained Unit Commitment for Electricity Market: Modeling, Solution Methods, and Future Challenges

“…Primal heuristics are usually used to provide high-quality feasible solutions for MILPs at a relatively low computational expense [58]- [60]. Most primal heuristics can be divided into two folds of LP-based heuristics [60], [62], [63] and MILPbased heuristics [61], [64]- [67]. Metaheuristic methods perform well in finding high-quality solutions for warm starts with primal heuristics [68]- [74].…”