Weight Certificates for Convex Multi-Objective MPC: Geometric Characterization, $\ell^1$ Construction, and $\ell^2$ Foreclosure

What the math gives to ML

The paper gives a constructive criterion for replacing a lexicographic cascade of convex priority objectives by one weighted solve: the augmented vector of weights, with the performance coefficient fixed to one, must lie in the outward normal cone of the upper image at the lexicographic solution. For hinge penalties this condition can be converted into a projected KKT linear program, yielding weights with a certified interior margin rather than relying on exponentially separated heuristic weights. A second practically important result is that squared-hinge penalties generally cannot reproduce lexicographic priorities at any finite weight, although their violation decreases only as O(1/w). The transferable opportunity is a priority-aware loss or constrained neural module that computes local certificate weights for L1 hinge penalties and falls back to sequential optimization when the certificate margin or active-set persistence is poor.