Comparison between SOS approach and Slack Variable approach for non-negativity check of polynomial functions: Multiple variable case

This note addresses the comparison between Sums-Of-Squares (SOS) approach and Slack Variable (SV) approach for the non-negativity check of multiple variable polynomial functions. In the process to compare these methods, we derive a new method of SV approach with use of multiple slack variables, which encompasses the previously proposed method of SV approach with use of a single slack variable. The obtained result is as follows: For the unbounded case, SOS approach and our new SV approach are equivalent; for the bounded case, our new SV approach is less conservative with increased number of decision variables compared to SOS approach. Several numerical examples are shown to illustrate our conclusions.