The bleak and all-too-common spectacle of roadkill was upsetting to Vedant Srinivas -- particularly when his uncle and cousin's beloved German Shepherd-Rottweiler mix was fatally hit by a car. More importantly, the losses made the high school student wonder if he could do something about it. What if Srinivas could stop the pet owners' broken hearts, save wildlife and deflect the economic impacts caused by the collisions? This month his efforts were rewarded. The sophomore from Eastlake High School in Sammamish, Wash., brought home a $5,000, first place grand award for the category of Environmental Engineering from the Regeneron International Science and Engineering Fair (ISEF).

There has been much recent interest in hierarchies of progressively stronger convexifications of polynomial optimisation problems (POP). These often converge to the global optimum of the POP, asymptotically, but prove challenging to solve beyond the first level in the hierarchy for modest instances. We present a finer-grained variant of the Lasserre hierarchy, together with first-order methods for solving the convexifications, which allow for efficient warm-starting with solutions from lower levels in the hierarchy.