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arxiv: 2604.11533 · v2 · submitted 2026-04-13 · 🧮 math.OC

Decision-Aware Predictions for Right-Hand Side Parameters in Linear Programs

Jackson Forner , Miju Ahn , Harsha Gangammanavar This is my paper

Pith reviewed 2026-05-15 07:23 UTC · model grok-4.3

classification 🧮 math.OC MSC 90C05
keywords linear programmingright-hand side predictiondecision-aware learningfeasibility enforcementintegrated learning and optimizationpredictive optimization
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The pith

A prediction model for right-hand-side parameters in linear programs can be trained to ensure the predicted feasible region contains the true optimal solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops training formulations for a model that predicts the right-hand-side parameters of a linear program from contextual data. Rather than minimizing prediction error alone, the formulations minimize decision error while enforcing feasibility using a collection of historical primal and dual solutions. The resulting predicted constraint set is analyzed to determine when it contains the true optimal solution and when that solution remains optimal under the predicted parameters. Numerical tests on a synthetic linear program and a network optimization problem show the approach maintains feasibility more reliably than standard regression.

Core claim

The paper proposes formulations for training a prediction model by minimizing the decision error while accounting for feasibility, measured by a collection of historical primal and dual solutions. Analysis identifies conditions under which a resulting predicted feasible region contains the true solution, and whether the latter solution achieves optimality for the predicted problem. The methods are solved using existing LP and nonconvex programming techniques.

What carries the argument

Training formulations that minimize decision error subject to feasibility constraints derived from historical primal and dual solutions.

If this is right

  • The predicted feasible region contains the true solution under the identified conditions.
  • The true solution achieves optimality for the predicted problem in cases covered by the analysis.
  • The methods enforce the desired feasibility more effectively than standard regression models, as shown in experiments on synthetic LPs and network problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same training structure could be adapted to predict objective coefficients or other LP parameters while preserving decision validity.
  • In repeated decision settings, the approach might reduce the frequency of post-prediction repairs or re-optimizations.
  • Scaling the nonconvex training problems to larger instances would require specialized solvers or relaxations not explored here.

Load-bearing premise

A collection of historical primal and dual solutions is available and sufficient to measure and enforce feasibility and decision error during training.

What would settle it

Apply the trained model to new contextual data, solve the predicted linear program, and check whether the true optimal solution from the underlying right-hand-side parameters lies inside the predicted feasible region; if it does not, the feasibility claim fails.

Figures

Figures reproduced from arXiv: 2604.11533 by Harsha Gangammanavar, Jackson Forner, Miju Ahn.

Figure 1
Figure 1. Figure 1: Number of predicted constraints satisfied by the true solutions [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sparsity of the solution Wc from of Algorithm 1. components in the model Wc. We also observe the affect of the regularization parameter λ on the average in-sample optimality gap 1 N P i∈[N] (⟨c, x ⋆ i ⟩−⟨W ξ c i , yˆi ⟩) as well as the value r(Wc) = P j∈[m] P k∈[d] |Wcjk| using the solution (Wc,(yˆi )) obtained from Algorithm 1. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Component function values in primal-DAL problem (9) evaluated using the solution of Algo [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Affect of (λ, γ) on the component function values of primal-DAL problem (9). there is minimal impact on the regularization level r(Wc). We also investigate the affect of jointly varying (λ, γ) in the primal-DAL problem (9) on the average in-sample optimality gap 1 N P i∈[N] (⟨c, x ⋆ i ⟩−⟨W ξ c i , yˆi ⟩), the value r(Wc) = P j∈[m] P k∈[d] |Wcjk|, and the value ϕ(Wc) = P i∈[N] P j∈[m] max{0, bij − ⟨wj , ξi … view at source ↗
Figure 5
Figure 5. Figure 5: Affect of α on the solution (Wc,(xˆi)) of the dual-DAL problem (21) We see that the optimal value of (21) is the same regardless of α. However, the average in-sample optimality gap ˆˆf := 1 N P i∈[N] |⟨c, x ⋆ i ⟩ − ⟨W ξ c i , y ⋆ i ⟩| is large for small values of α, achieves a minimum at α = 2, and then slowly increases and tapers off as α increases. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
read the original abstract

This paper studies an integrated learning and optimization problem in which a prediction model estimates the right-hand-side parameters of a linear program (LP) using a contextual vector. Considering that such a prediction alters the feasible region of the LP, we aim to estimate the constraint set to contain the optimal solution of the underlying LP, given by the true right-hand side parameters. We propose formulations for training a prediction model by minimizing the decision error while accounting for feasibility, measured by a collection of historical primal and dual solutions. Our analysis identifies conditions under which a resulting predicted feasible region contains the true solution, and whether the latter solution achieves optimality for the predicted problem. To solve the alternative training problems, we employ existing LP and nonconvex programming solution methods. We conduct numerical experiments on a synthetic LP and a network optimization problem. Our results indicate that the proposed methods effectively implement the desired feasibility, compared to standard regression models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops decision-aware methods to predict right-hand-side parameters of linear programs from contextual features. Training formulations minimize a decision-error objective subject to feasibility constraints constructed from a finite collection of historical primal and dual solutions; an accompanying analysis identifies conditions under which the resulting predicted feasible region contains the true optimal solution and the latter remains optimal for the predicted LP. The nonconvex training problems are solved with standard local solvers, and the approach is tested numerically on a synthetic LP and a network-flow problem.

Significance. If the derived containment and optimality conditions are reliably attained by the trained predictors, the work would offer a concrete mechanism for reducing downstream decision suboptimality while preserving feasibility guarantees, which is valuable in contextual optimization settings where pure regression can produce infeasible or highly suboptimal decisions.

major comments (2)
  1. [Training formulations and analysis] The analysis supplies conditions under which the predicted feasible region contains the true solution and preserves optimality, yet the training problems are explicitly nonconvex and solved with local solvers (IPOPT and similar). No argument is given that stationary points returned by these solvers satisfy the global containment conditions, so the practical training procedure may not inherit the theoretical guarantees.
  2. [Problem formulation] Feasibility and decision-error constraints in training are enforced using only a finite collection of historical primal/dual solutions. It is not shown that satisfaction on this collection implies the containment property for arbitrary new contextual vectors, leaving open whether the learned predictor generalizes the desired guarantees beyond the training data.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction would benefit from a concise statement of the precise decision-error metric employed in the numerical experiments.
  2. [Notation section] Notation for the true versus predicted right-hand-side vectors should be introduced once and used consistently to avoid ambiguity when describing the containment conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive comments. Below we provide point-by-point responses to the major comments and outline the revisions we intend to make to the manuscript.

read point-by-point responses

  1. Referee: [Training formulations and analysis] The analysis supplies conditions under which the predicted feasible region contains the true solution and preserves optimality, yet the training problems are explicitly nonconvex and solved with local solvers (IPOPT and similar). No argument is given that stationary points returned by these solvers satisfy the global containment conditions, so the practical training procedure may not inherit the theoretical guarantees.

    Authors: We agree that the nonconvexity of the training problems means that local solvers like IPOPT do not necessarily find points satisfying the global containment conditions derived in the analysis. In the revised version, we will add a remark acknowledging this gap between the theoretical conditions and the practical solutions obtained via local optimization. We will also report that in our numerical experiments, the solutions found by the local solvers did satisfy the containment and optimality properties for the test instances, providing empirical support. We plan to investigate convex relaxations or global solvers in future extensions. revision: partial

  2. Referee: [Problem formulation] Feasibility and decision-error constraints in training are enforced using only a finite collection of historical primal/dual solutions. It is not shown that satisfaction on this collection implies the containment property for arbitrary new contextual vectors, leaving open whether the learned predictor generalizes the desired guarantees beyond the training data.

    Authors: The referee is correct that the constraints are enforced only on the finite historical set, and our analysis provides conditions under which the containment holds when those constraints are satisfied, but does not prove that the learned predictor will satisfy the conditions for unseen contextual vectors. This is an inherent limitation when using a finite sample for training. We will revise the manuscript to explicitly state that the guarantees are conditional on the historical data being sufficiently representative, and that generalization relies on the capacity of the prediction model and the training procedure. No theoretical generalization bound is provided, as deriving one would require additional assumptions on the distribution of contexts and the function class. revision: partial

Circularity Check

0 steps flagged

No significant circularity; analysis relies on external historical data and standard solvers

full rationale

The paper trains predictors by minimizing decision error subject to feasibility constraints measured from a finite collection of historical primal and dual solutions obtained via standard LP solvers. The analysis derives conditions under which the predicted feasible region contains the true solution and preserves optimality, but these conditions are stated in terms of the external historical data and the true RHS parameters rather than reducing to fitted quantities by construction. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The approach remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the availability of representative historical solutions and on the existence of identifiable conditions for solution containment; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption A collection of historical primal and dual solutions is available to measure feasibility and decision error
    Invoked to define the training objectives and feasibility enforcement

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