Rigorous bounds on the analytic S-matrix

<!--HTML-->In this talk I will introduce a complementary approach to the non-perturbative S-matrix Bootstrap based on duality in optimization theory. In particular, I will focus on the problem of bounding the quartic coupling for gapped theories in 4-dimensions, both from above and below. The bounds obtained rely solely on proven analyticity properties and the numerical problem can be efficiently solved using SDPB.