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A new scenario-reduction cost function provably selects the best single scenario first, and about five scenarios suffice to approximate the full-distribution optimum within a few percent.

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2026-08-02 06:50 UTC pith:MYNWFNFG

load-bearing objection New cost function and an m=1 optimality theorem are real contributions; the paper is worth refereeing, but the 0.4% large-case claim is not resolved by experiments run at a 1% MIP gap. the 3 major comments →

arxiv 2607.11593 v2 pith:MYNWFNFG submitted 2026-07-13 math.OC

Scenario Reduction for Two-Stage Stochastic Mixed-Integer Programs

Yannick Werner , Juan Miguel Morales , Salvador Pineda , Line Roald , Sonja Wogrin This is my paper
classification math.OC MSC 90C1590C1149Q22
keywords scenario reductiontwo-stage stochastic mixed-integer programmingoptimal mass transportationcost functionForward Selectionrelative approximation errorunit commitmentstochastic programming
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes a new cost function, cPr, for the optimal mass transportation problem used in scenario reduction for two-stage stochastic mixed-integer programs. It proves that when this cost function is combined with Forward Selection, the first scenario drawn is the single scenario that minimizes the relative approximation error — the regret of making decisions on the reduced distribution instead of the full one. On stochastic unit commitment test systems, roughly five scenarios selected this way approximate the full-distribution optimum within about 2.1% (small) and 0.4% (large) error, whereas competing cost functions typically need 25 or more. Since solution time in two-stage stochastic MIPs grows sharply with scenario count, a handful of accurate scenarios can be the difference between tractable and intractable. The paper also gives a hybrid algorithm that preserves the quality while cutting computation.

Core claim

The central claim is that scenario reduction quality is governed by the choice of transportation cost function, and the paper's proposed asymmetric regret cost, cPr(ξi, ξj) = z(x*(ξj), ξi) − z(x*(ξi), ξi), makes the Forward Selection algorithm pick, on the first draw, the scenario whose single-scenario distribution minimizes the relative approximation error among all single-scenario distributions (Theorem 1). The proof exploits the observation that for m = 1 the distribution is a Dirac measure, so the two-stage problem reduces to the single-scenario problem, and the constant term z(x*(ξi), ξi) in cPr cancels in the minimization. Numerically, this first choice alone gives about 11.3% and 2.4%

What carries the argument

The central object is the transportation cost function cPr(ξi, ξj) = z(x*(ξj), ξi) − z(x*(ξi), ξi), the asymmetric regret of optimizing the first stage for scenario j and then observing scenario i, minus the cost when the decision is tailored to i. The load-bearing mechanism is Theorem 1: with Forward Selection's first draw, minimizing the transported mass under cPr is equivalent to minimizing the relative approximation error, because the subtractive term depends only on i and cancels. The paper also introduces a hybrid of an expected-value-problem-based cost function and cPr, where the former quickly prunes the scenario set and the latter refines it.

Load-bearing premise

The headline large-case accuracy (0.4% mean RAE at m = 5) is computed under a 1% MIP gap for every solve, including the reference solution, so the measured error is smaller than the solver's certified optimality tolerance.

What would settle it

Solve the large 300-bus case with the same scenario samples but a 0.01% MIP gap for the reference n = 1000 SAA and the m = 5 reduced problem; if the mean RAE is no longer around 0.4% or the ranking of cost functions changes, the headline number is an artifact of solver tolerance. Alternatively, on a different application, enumerate all single-scenario distributions and verify that Forward Selection's first pick under cPr attains the minimal RAE.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Replacing hundreds or thousands of scenarios with roughly five selected scenarios can bring two-stage stochastic MIPs within reach, since solution time typically scales badly with scenario count.
  • The first scenario drawn by Forward Selection with cPr is provably the best single-scenario choice, giving a warm start that other cost functions do not guarantee.
  • The method is consistent across random draws: cPr shows markedly lower variance in RAE across samples than competing cost functions.
  • The hybrid algorithm delivers the same approximation quality as cPr but at 18x less wall-clock time and 66x less solver work on the 300-bus case, making the approach practical.
  • Theorem 1's guarantee is limited to m = 1; the good behavior at m > 1 is empirical and not proven.

Where Pith is reading between the lines

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

  • The proof structure suggests a recipe for other asymmetric regret functions: any cost of the form R(ξi, ξj) − const(ξi) with zero diagonal will inherit the m = 1 optimality property, so the result may transfer to other problem classes.
  • One could test the hybrid's Phase-1 choice r = 50 sensitivity: the paper fixes r for both cases, and a principled stopping rule for pruning could further reduce computation.
  • The reported 0.4% RAE on the large case is below the 1% MIP gap used for all solves; verifying with a tighter gap would separate the method's accuracy from solver tolerance.
  • For convex second stages, duality might let cPr be evaluated from dual prices of the n² − n second-stage LPs without solving them all, a route the paper notes as future work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper considers scenario reduction for two-stage stochastic mixed-integer programs, revisiting the optimal mass transportation framework and the Forward Selection Algorithm. It reviews input-data-driven and optimization-problem-driven transportation cost functions, proposes a new asymmetric cost function cPr(ξi, ξj) = z(x*(ξj), ξi) − z(x*(ξi), ξi), and proves (Theorem 1, Appendix B) that the first scenario selected by Forward Selection with cPr minimizes the relative approximation error (RAE) among all single-scenario reductions. To reduce the computational burden of evaluating cPr, the paper proposes a hybrid algorithm that first prunes scenarios using the cost function of Morales et al. (2009) and then applies cPr to the pre-selected subset. Numerical experiments on two-stage stochastic unit commitment for a 24-bus system (n=200) and a 300-bus system (n=1000) report that around five scenarios suffice for roughly 2.1% and 0.4% RAE, with the hybrid algorithm cutting wall-clock time by a factor of 18 and solver work by a factor of 66 on the large case.

Significance. If the results hold, the paper offers a practical and scalable scenario-reduction method for two-stage stochastic MIPs, with a first-scenario optimality guarantee and a hybrid algorithm that makes the expensive cPr cost function usable. The strengths of the paper include: the proof of Theorem 1 is correct (though, as discussed below, the result is essentially definitional); the comparison of cost functions is carried out under a common algorithmic framework; the small-case experiments use a 0.01% MIP gap and are credible; and the code and data are publicly available. The main limitation is that the headline large-case accuracy, 0.4% RAE, is below the 1% MIP gap used to obtain the reference solutions, so the quantitative claim is not numerically resolved. The theoretical contribution is also weaker than it appears: Theorem 1 follows directly from the construction of cPr. Overall, the paper has merit, but the advertised large-case performance needs additional numerical support.

major comments (3)
  1. [§5.2, §5.2.1, Eq. (12)] The reported large-case mean RAE of 0.4% is smaller than the 1% MIP gap used for all MIPs, including the n=1000 reference SAA and the reduced m=5 problems. Since z*(Pn) is only known to within roughly ±1% and x*(Qm) is itself an approximate solution, the computed RAE is within solver noise; the true RAE could be near zero or well above the reported value. This is load-bearing because the abstract and conclusions advertise the 0.4% figure. The authors should either solve the reference and reduced problems with a much tighter gap (as in the small case) for at least a subset of draws, or provide rigorous bounds/sensitivity analysis that show the RAE is insensitive to the 1% termination tolerance.
  2. [Appendix B, Eq. (B.3)] Theorem 1 is correct but is a direct algebraic consequence of the definition of cPr. In Eq. (B.3), the term z(x*(ξi), ξi) is subtracted and is independent of j, so the Forward Selection objective collapses to the RAE numerator for m=1. Thus the theorem essentially restates the cost-function construction rather than revealing a new structural property. The paper should explicitly acknowledge this and avoid presenting the result as a deep optimality guarantee; the contribution is better framed as the identification of a cost function that makes the greedy first selection optimal by design.
  3. [§5.2.1, Figures 1–2] The comparison of cPr with other cost functions for m=1 is partly a tautology: by Theorem 1, cPr is engineered to minimize RAE at m=1, so its superior first-draw performance is expected. The more meaningful claims are for m>1 and for the hybrid algorithm. These empirical claims, however, inherit the numerical tolerance problem described above. Please report results with a common, tighter MIP gap for the small and large cases, or at least demonstrate that the reported RAE differences are not attributable to solver stopping criteria.
minor comments (6)
  1. [Abstract / §6] The abstract says “roughly 2.1% and 0.4% error” for five scenarios, while Section 5.1 reports around 2% and Section 6 reports 11.3% and 2.4% for one scenario. Please make the numbers consistent and clarify which m corresponds to each RAE.
  2. [Theorem 1 statement] The notation Q1 = {Q_j^1 : j ∈ I} and later Q1 ∈ Q1 is confusing. Use a different symbol for the family of distributions, e.g., \mathcal{Q}_1, and state the assumption z*(Pn)>0 in the theorem, not only in a footnote in the proof.
  3. [Appendix B] There is a notational inconsistency: the proof uses both ζ_j and ξ_j for the candidate scenarios. Please unify to ξ_j throughout.
  4. [§3.2, Algorithm 1] In line 4, the set J is used before any scenario is selected (J = {}). While the expression is correct, a short remark explaining the convention for J = {} would improve readability.
  5. [§4.2.4 / Table 1] The claim that cPr requires n MIPs and n^2−n LPs assumes all entries of the cost matrix are computed. The paper later uses heuristics (pre-reduction with cID and warm-starting, Appendix C). Please state clearly that the reported computational costs in Table 2 include those heuristics, so that the comparison with the exact cost matrix is not misleading.
  6. [§5.2.2] The paper does not discuss the effect of multiple optimal first-stage solutions x*(Qm) on the RAE. If the reduced MIP has several optima, the RAE may depend on which optimum is returned. A brief comment or a check with a different solver seed would strengthen the numerical claims.

Circularity Check

1 steps flagged

Theorem 1 is an algebraic consequence of the proposed cost function's definition; the m>1 empirical comparison is independent, but the headline 0.4% large-case claim is not numerically resolved.

specific steps
  1. self definitional [Section 4.2.4, Eq. (18); Theorem 1; Appendix B, Eqs. (B.2)-(B.3)]
    "cPr(ξi, ξj) = z(x∗(ξj), ξi) − z(x∗(ξi), ξi) ... j∗ = arg min_{j∈I} Σ_i p_i cPr(ξ_i, ζ_j) = arg min_{j∈I} Σ_i p_i(z(x∗(ζ_j), ξ_i) − z(x∗(ξ_i), ξ_i)) = arg min_{j∈I} Σ_i p_i z(x∗(ζ_j), ξ_i), which is equivalent to the minimization of the RAE in Equation (B.1) and completes the proof."

    The cost function is defined as the per-scenario regret term appearing in the RAE numerator, minus a term independent of the candidate scenario j. Appendix B shows the first Forward Selection draw minimizes Σ_i p_i z(x*(ξ_j), ξ_i), which is exactly the only j-dependent part of RAE(Pn, δ_{ξ_j}) because the denominator and z*(Pn) are constants. Theorem 1 therefore asserts that the scenario minimizing an objective defined to be the RAE numerator also minimizes the RAE. This is a valid but definitional equivalence: the optimality is built into Eq. (18) by construction, not discovered independently.

full rationale

The main theoretical result is self-definitional: cPr is designed as the pointwise RAE regret, and the proof in Appendix B (B.2)-(B.3) reduces the Forward Selection objective to the RAE numerator. The numerical experiments with m>1 and the hybrid algorithm are genuine external comparisons and mitigate the circularity; the small-case 0.01% MIP gap supports those results. The large-case 0.4% RAE is computed with a 1% MIP gap (Section 5.2), so the headline accuracy is numerically unresolved and not certified against solver tolerance, but this is a correctness/robustness issue rather than circularity. No load-bearing self-citation chain or imported uniqueness theorem was found; the citations to Morales et al. (2009) are prior-work references, not circular support.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The method rests on standard stochastic programming assumptions (fixed recourse, relatively complete recourse, RHS-only uncertainty), which are stated. The main additional load is the hand-chosen pre-selection size r and the numerical tolerance used to compute the reference optima, both of which affect the headline numbers.

free parameters (2)
  • r (pre-selection size) = 50
    Hybrid Algorithm Phase 1 pre-selects r=50 scenarios using cMo; chosen by hand and reported as performing well, without sensitivity analysis. Section 5.2.1 and Algorithm 3.
  • MIP gap for large case = 1%
    All MIPs in the 300-bus study are solved to 1% gap. The claimed RAE of 0.4% at m=5 is smaller than this tolerance, so the reported accuracy is not numerically resolved. Section 5.2.
axioms (6)
  • domain assumption The two-stage problem has relatively complete recourse and fixed recourse matrix W and technology matrix T independent of uncertainty.
    Stated in Section 2; restricts applicability but is common in stochastic programming.
  • domain assumption Uncertainty appears only in the right-hand sides of second-stage constraints.
    Section 2; the cost functions are developed for this setting.
  • domain assumption The reduced distribution Q must be supported on atoms of Pn (subset selection).
    Section 3.1 restricts to discrete scenario reduction where selected scenarios come from the original sample.
  • domain assumption z*(Pn) > 0 for the RAE to be well-defined.
    Footnote 1 in Appendix B; can be ensured by adding a large constant to the objective.
  • standard math The Monge-Kantorovich distance and the Forward Selection algorithm of Dupačová et al. (2003) provide a valid greedy selection framework.
    Recapped in Section 3; the m=1 optimality of Forward Selection is a cited result.
  • domain assumption Single-scenario MIPs and second-stage LPs used to evaluate cost functions are solved exactly (or within the stated MIP gap).
    The proof of Theorem 1 assumes exact x*(ξj); numerical implementation uses MIP gaps of 0.01% or 1%.

pith-pipeline@v1.3.0-alltime-deepseek · 17896 in / 16971 out tokens · 151107 ms · 2026-08-02T06:50:20.865293+00:00 · methodology

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read the original abstract

Two-stage stochastic mixed-integer programs are important tools for decision-making under uncertainty. Representing the uncertainty with many scenarios, however, can make them challenging to solve. Scenario reduction addresses this by finding a distribution supported on fewer scenarios that still yields similar optimal first-stage decisions. In this paper, we revisit the classical scenario reduction theory based on distances between probability distributions and the optimal mass transportation problem. The transportation problem's cost function captures scenario similarity and is central to the effectiveness of scenario reduction. We then review and compare various transportation cost functions from the literature and propose a new one. Using the Forward Selection Algorithm, we prove that our proposed cost function selects the best possible scenario from a given sample on the first draw with respect to the relative approximation error. To reduce the computational cost of evaluating this cost function, we further propose a hybrid algorithm with a scenario pre-selection phase. We assess solution quality and computational complexity on the two-stage stochastic unit commitment problem for small 24-bus and large 300-bus case studies. With only around five scenarios, the proposed cost function approximates the full-distribution optimum to within roughly 2.1% and 0.4% error for the small and large cases, respectively. In contrast, prevalent cost functions often need 25 scenarios or more to achieve that solution quality. The hybrid algorithm achieves similar solution quality while reducing wall-clock time by a factor of 18 and work (per Gurobi solver) by a factor of 66 on the large case study.

Figures

Figures reproduced from arXiv: 2607.11593 by Juan Miguel Morales, Line Roald, Salvador Pineda, Sonja Wogrin, Yannick Werner.

Figure 1
Figure 1. Figure 1: RAE for the cost functions c ∈ C using the Scenario Reduction Algorithm 2 and the Hybrid Algorithm 3 with r = 50 depending on the size m of the reduced scenario set J for the modified IEEE RTS 24-bus system described in Section 5.1. Box plots represent variation across 50 randomly sampled distributions Pn of size n = 200, drawn from the probabilistic forecast in Pinson (2013). The dashed line indicates the… view at source ↗
Figure 1
Figure 1. Figure 1: RAE for the cost functions c ∈ C using the Scenario Reduction Algorithm 2 and the Hybrid Algorithm 3 with r = 50 depending on the size m of the reduced scenario set J for the modified IEEE RTS 24-bus system described in Section 5.1. Box plots represent variation across 50 randomly sampled distributions Pn of size n = 200, drawn from the probabilistic forecast in Pinson (2013). The dashed line indicates the… view at source ↗
Figure 2
Figure 2. Figure 2: RAE for the cost functions c ∈ C using the Scenario Reduction Algorithm 2 and the Hybrid Algorithm 3 with r = 50 depending on the size m of the reduced scenario set J for the modified IEEE 300-bus system described in Section 5.1. Box plots represent variation across 10 randomly sampled distributions Pn of size n = 1000, drawn from the probabilistic forecast in Pinson (2013). The dashed line indicates the S… view at source ↗
Figure 2
Figure 2. Figure 2: RAE for the cost functions c ∈ C using the Scenario Reduction Algorithm 2 and the Hybrid Algorithm 3 with r = 50 depending on the size m of the reduced scenario set J for the modified IEEE 300-bus system described in Section 5.1. Box plots represent variation across 10 randomly sampled distributions Pn of size n = 1000, drawn from the probabilistic forecast in Pinson (2013). The dashed line indicates the S… view at source ↗
Figure 3
Figure 3. Figure 3: Wall-clock times in s (mean over draws) required for solving the optimization problems in Step 1 of Algorithm 2 and Phases 1 and 2 of Hybrid Algorithm 3 and solving the reduced two-stage stochastic MIP (11a) for the modified IEEE 300-bus system described in Section 5.2 with n = 1000 scenarios. Whiskers indicate variation over draws of wall-clock time for computing the requirements. 16 [PITH_FULL_IMAGE:fig… view at source ↗
Figure 3
Figure 3. Figure 3: Wall-clock times in s (mean over draws) required for solving the optimization problems in Step 1 of Algorithm 2 and Phases 1 and 2 of Hybrid Algorithm 3 and solving the reduced two-stage stochastic MIP (11a) for the modified IEEE 300-bus system described in Section 5.2 with n = 1000 scenarios. Whiskers indicate variation over draws of wall-clock time for computing the requirements. 17 [PITH_FULL_IMAGE:fig… view at source ↗

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