DePICT: Decision-Preserving Interface for Constrained Downstream Tasks

Language models turn a worded situation into a numeric plan, and the dominant pipelines (NL4Opt, OptiMUS, ORLM, OR-LLM-Agent) commit to a single objective and point-valued coefficients, then solve once. For decisions that allocate real budget, effort, or clinical attention, that confidence is the failure mode: every objectified number is an assumption, and a plan optimal only if the guesses are exactly right is fragile -- mimicry of computation. YUKTI changes the target of autoformulation. Its representation is a typed-proposition graph whose relationships carry shape priors, coefficient uncertainty, and provenance. YUKTI routes each stage to an exact, nonlinear, or evolutionary solver; couples stages by a distributional Pareto hand-off; and introduces Assumption-Robust Pareto Frontiers (ARPF), resampling assumptions (including structural epsilon-contamination) to score how often each action survives (rho). We prove a bound making rho an exact factor of decision regret, add auditable traceability, and synthesize a benchmark-faithful data foundation when none exists (SRJANA). We validate three ways: under controlled misspecification the robust compromise cuts mean and tail regret by over 90% versus a naive point plan; on a regulated commercial decision we optimize inside a lawful action space and price the downside in euros; and on a real public dataset of 41,188 decisions an out-of-sample backtest beats the logged status quo by 34% and a naive point rule by 4% while reducing the optimizer's curse. The solvers are standard; we claim no benchmark-SOTA win. A head-to-head shows an LLM given the correct numbers, and single-objective optimization, both incur about 47x the held-out regret of YUKTI -- an LLM is a formulator, not a solver. Under long-range causal coupling, the forward hand-off becomes unsound, locating where it must become a backward-induction causal policy.

Decision-Focused Learning (DFL) trains predictors to improve downstream decision quality, but computing regret gradients typically requires differentiating through solvers or relying on surrogate losses, which can be computationally expensive or deviate from the true objective. We show that, under standard regularity with locally stable active constraints, the regret gradient admits a closed-form geometric characterization, equivalent to the prediction error projected onto the tangent space of active constraints, scaled by local curvature. This reveals that regret gradients can be obtained by filtering decision-irrelevant components from the MSE gradient, providing a simpler and more direct alternative to existing approaches. Based on this, we propose PEAR (Projected Error As Regret-gradient), which computes regret gradients via a reduced linear system over active constraints, avoiding differentiation through solver iterations or additional optimization solves. Experiments on LP benchmarks and a real-world QP task show that PEAR achieves the best decision quality among all baselines while being the most computationally efficient, with gains that persist under constraint shifts.

While optimal transport (OT) enforces a rigid constraint by requiring two measures to be matched exactly, partial optimal transport relaxes this requirement by allowing mass to remain unmatched through a global budget, scalar rebate, or uniform rejection rule. However, many applications call for more structured, pointwise rejection mechanisms, where the decision to leave mass unmatched depends on side-specific reliability, support geometry, or external information about which components should participate in the comparison. We introduce \emph{intent-controlled partial optimal transport} (IC-POT), a targeted generalization of partial transport that replaces the global rejection paradigm with pointwise rejection costs over both measures. We show that the resulting optimization problem admits a dual interpretation in terms of local acceptance thresholds and can be solved by recasting it as a balanced Kantorovich OT problem on an augmented support. Beyond theoretical analysis, we demonstrate the practical relevance of IC-POT in settings where rejection is driven by side information. In positive-unlabeled learning and open-partial domain adaptation, incorporating pointwise rejection rules that encode statistical structure improves fixed baseline pipelines. Finally, we motivate the use of IC-POT with a geophysical practical case: multi-modal satellite ocean measurements, for which physical and sensors priors naturally inform the rejection mechanism and define the retrieved comparable signal information.