Lexiographic Extension (Phase 2)

Our convex program has two tiers so we can minimize cost to buy qualified volume

At the optimum of the phase 1 primal problem, there may be many different allocation vectors x_i that deliver the same maximum volume T*. For example, several miners may be equally eligible and interchangeable within the budget and diversity constraints, and different mixes of x_i can all achieve the same T, though at different total payout costs.

To address this ambiguity, we extend the primal problem into a second phase. This is known as lexiographic goal programming. The term lexiographic simply refers to a priority ordering, where one goal is considered more important than those below it. This approach is particularly useful when there are multiple, potentially conflicting objectives: by solving them in sequence, we ensure that the most important goal (maximizing qualified volume) is always satisfied first, while subsequent goals (minimizing payout cost) refine the solution without undermining the primary objective. Once we’ve locked in the maximum feasible T*, we then re-optimize over the same feasible set:

Where epsilon is a small tolerance (e.g. 0.0001) that ensures Phase 2 keeps the total qualified volume essentially the same as the maximum found in Phase 1. In essence this is a tolerance which relaxes exactness and accepts solutions that are 99.99% accurate allowing for convergence. The goal of Phase 2 ensures that, among all the possible max volume allocations, we pick the one that is cheapest in payout cost. The consequence is that the system doesn’t waste emissions budget on unnecessarily costly allocations and payouts are minimized subject to still delivering the maximum possible flow of qualified volume. Intuitively, when multiple miners are interchangeable in terms of volume, Phase 2 breaks the tie by preferring the mix that keeps payout costs down. This prevents “over-paying” miners who don’t add unique incremental flow. Furthermore, Phase 2 ensures that volume maximization is always the top priority (Phase 1), but payout efficiency is guaranteed as a second priority. This maintains clear incentive structure where miners must provide both volume and efficient inference quality to secure long term rewards.

A cost minimization routine is just like this. The convex program strives to pay just enough for the nuggets of information provided by the miner and no more.

The output of Phase 2 that we can expect on top of Phase 1 outputs are:

• The minimal payout cost C*. This is the value of the second-phase objective function. It represents the cheapest possible spend needed to sustain the maximum feasible volume T* (within the small tolerance ϵ).

• A cost-efficient allocation vector These are the miner allocations chosen in Phase 2 which are refined allocation vectors from initial X* computed in Phase 1.

By construction, this vector still achieves essentially the same maximum volume but it does so with the lowest payout cost. Compared to Phase 1, this vector is unique and refined, since the tie is broken by cost minimization.