Refinement strategies for piecewise linear functions utilized by reformulation-based techniques for global optimization

Abstract

The signomial global optimization algorithm is a method for solving nonconvex mixed-integer signomial problems to global optimality. A convex underestimation is produced by replacing nonconvex signomial terms with convex underestimators obtained through single-variable power and exponential transformations in combination with linearization techniques. The piecewise linear functions used in the linearizations are iteratively refined by adding breakpoints until the termination criteria are met. Depending on the strategy used for adding the breakpoints, the complexity of the reformulated problems as well as the solution time of these vary. One possibility is to initially add several breakpoints thus obtaining a tight convex underestimation in the first iteration at the cost of a more complex reformulated problem. This breakpoint strategy is compared to the normal strategies of iteratively adding more breakpoints through illustrative examples and test problems.