Decision-Diagram Convex Hull for Neural Gates

Integrate this as the discrete-constraint layer of a MILP for structured pruning or neural architecture search, not as a replacement for the forward pass of a normally trained dense network. Let x_v be binary keep/use variables for channels, experts, bit choices, or edges, and let q_e represent a higher-order interaction such as x_i x_j x_k that appears in a FLOP, memory, latency, or accuracy surrogate. First choose an ordering of the gate variables. Construct a layered DD: each layer assigns one x_v; an arc stores its 0/1 value and the subset E_1(a) of interaction monomials already forced to one. Merge nodes with identical future-relevant state to control width. Create one continuous flow variable y_a per arc and add the following constraints to the MILP: flow conservation, unit source/sink flow, x_j = sum_{a in Gamma_j^1} y_a, and q_e = sum_{a in Gamma_e^1} y_a. Pseudocode is: build_DD(order, interactions); for arc a create y[a]>=0; add_flow_conservation(); add source_flow=1; for each variable j add x[j]=sum(y[a] for a in Gamma1[j]); for each monomial e add q[e]=sum(y[a] for a in Gamma1[e]); optimize the task-loss surrogate plus resource penalties. The DD construction and flow constraints come from the mathematics; DD width, ordering quality, and the task-loss coefficients are empirical. The first experiment should use a small ResNet on CIFAR-10 with channel keep variables and pair/triple resource interactions. Compare independent McCormick linearizations, big-M constraints, and the DD formulation at equal Gurobi time and parameter budget. Success means a larger integrality gap closure, higher validation accuracy at the same FLOPs, or a better feasible compression solution before timeout.