Modelling to Generate Continuous Alternatives: Enabling Real-Time Feasible Portfolio Generation in Convex Planning Models

Jesse D. Jenkins; Michael Lau; Neha Patankar; Xin Wang

REVIEW 5 major objections 8 minor 34 references

Modelling to Generate Continuous Alternatives: Enabling Real-Time Feasible Portfolio Generation in Convex Planning Models

T0 review · 5 major / 8 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that previously computed MGA results can be turned into a continuous, live-explorable space of feasible portfolios by interpolating over the convex hull of the iterates.

desk verdict A sound convex-combination post-processing method for MGA, but several claims about coverage and budget interpolation outrun the evidence. read the letter →

arxiv 2411.16887 v1 pith:WQ5FP4IO submitted 2024-11-25 math.OC cs.SYeess.SY

Michael Lau , Xin Wang , Neha Patankar , Jesse D. Jenkins This is my paper · ORCID

classification math.OCcs.SYeess.SY MSC 90C0590C2590C90

keywords modellingtogeneratealternativesconvexcombinationplanningmodelscapacityexpansionnear-optimalfeasiblespaceParetofrontierapproximationdimensionalityreductioninteractivedecisionsupport

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Capacity-expansion models that optimize for least cost return one plan, but real planning needs many viable alternatives. Modelling to Generate Alternatives (MGA) produces a scatter of near-optimal plans, and this paper adds a post-processing layer that turns that scatter into a continuous space users can explore live. The method, MGCA, projects MGA solutions onto capacity decisions and outcome metrics, forms their convex hull, and lets a user generate any interior portfolio by specifying weights, extra constraints, or a new objective. Because the resulting optimization problem is tiny, it solves in milliseconds to seconds on a laptop. The paper shows that capacity-related metrics are reproduced exactly by interpolation, operational metrics stay within feasible bounds and generally within 10 percent of an economic-dispatch calculation, and exported portfolios can be re-solved by the full model to recover exact operational details.

What carries the argument

The central object is the convex hull exploration problem (Eq. 6), a small linear or convex program over the MGA-iterate weights $\lambda$ that produces a point $z_k = \lambda^T Z$ inside the hull, subject to user-added inequalities $Az_k \le b$. Its companion identity is Eq. 7, the scalar-product property $c^T(\sum_i \lambda_i x_i) = \sum_i \lambda_i (c^T x_i)$, which lets affine capacity metrics be evaluated exactly by the same weights. Dimensionality reduction first projects each MGA solution onto selected capacity decisions and linear metrics, shrinking a model with $O(N \times T)$ operational variables to one with $O(N)$ entries, which is what makes interactive solve times possible.

What would settle it

Take a convex capacity-planning LP, run MGA once with a deliberately small or one-sided set of iterates that provably miss a known portion of the near-optimal region, then use MGCA to generate portfolios and Pareto frontiers there; if the omitted portfolios never appear or the frontiers stop short of known feasible extremes, the method is shown to be limited by the seed set. Alternatively, export the 50 interpolated portfolios to the full economic-dispatch model and tally errors: if substantially more than a small fraction of operational-metric interpolations exceed the reported 10 percent gap, the accuracy claim as stated fails.

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Extended reading notes

Core claim

MGCA's central claim is that the near-optimal feasible region of a convex capacity-planning model, once approximated by MGA iterates, can be treated as the convex hull of those iterates in a reduced capacity-and-metric subspace. Inside that hull, any convex combination of stored vertices is a feasible, within-budget portfolio, so a user can generate new plans anywhere in the space by solving a small optimization problem over the weights, with user-specified objectives and constraints, rather than re-running the large model. Affine functions of capacity decisions, including many outcome metrics, are then exactly recoverable from the weights, while generation-based operational metrics are interpolated to feasible values that a full economic dispatch would improve or match; the paper reports interpolated metric values generally within 10 percent of the optimal value. The method also supports synthesizing tighter-budget results, approximating Pareto frontiers between metrics, and exporting chosen capacity portfolios back to the full capacity-expansion model to obtain least-cost operational details.

Load-bearing premise

The load-bearing premise is that the set of MGA iterates collected as vertices spans the near-optimal feasible region well enough that its convex hull stands in for the true region; if large parts of the region are missed, every generated portfolio, budget interpolant, and Pareto frontier is confined to an inner approximation.

Editorial extensions

If this is right

Stakeholders can explore near-optimal energy plans interactively, imposing constraints such as a cap on gas capacity or a farmland-preservation objective and seeing the feasible remainder in under a second.
Budget analysis at multiple cost slacks can be produced from one MGA run by interpolating each iterate toward the least-cost point, instead of re-running the planning model at each budget.
Pareto-frontier-style trade-off curves between any pair of outcome metrics can be approximated in post-processing with an epsilon-constraint method over the hull.
Any generated portfolio can be exported and re-solved with the full capacity-expansion model at fixed capacities, so operational metrics reflect least-cost dispatch rather than the interpolation estimate.
Because the exploration problem has only a few hundred variables, nonlinear convex objectives can be used in the live search.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

In our reading, the same convex-hull machinery transfers to any linear program with a large near-optimal set, such as water-resource or supply-chain planning, where stakeholders also need to see alternatives rather than a single optimum.
If the interpolation accuracy holds at scale, interactive decision-support dashboards could let opposing stakeholders co-design compromises directly, replacing the current workflow of pre-computing and hand-picking discrete MGA results.
A natural stress test would be to seed MGCA with deliberately adversarial or sparse MGA iterates and measure how much of the true near-optimal region is missed; the paper's guarantees are conditional on the seed set spanning the space.
The 10 percent accuracy figure is a demonstration on one test case, not a proven bound, so users would need case-specific validation before relying on interpolated operational metrics for regulatory decisions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core idea is simple and the paper executes it cleanly: take the convex hull of MGA iterates, project to capacity and metric space, and let users solve a tiny LP over convex combinations with custom objectives and constraints. The convexity argument is correct for LP planning models—convex combinations of feasible, within-budget iterates stay feasible and within budget, affine capacity metrics interpolate exactly, and operational metric estimates are feasible but not exact. The speed gains are real: 0.6 ms per interpolate and 0.01 s per constrained iterate on a laptop is genuinely useful for interactive decision support. It also improves on Pedersen's MAA by sidestepping Delaunay triangulation, which breaks down above ~10 dimensions, and by allowing user-directed rather than purely random sampling.

The soft spots are mostly about scope of claims. The set conv(Z) is an inner approximation of the near-optimal feasible region, and the paper acknowledges this in Section 3.5 and in its bracketing-run advice, but elsewhere it writes as if the hull were the region itself. No coverage diagnostic is offered—no check that the 200 iterates span the parts of the space a user might care about. The accuracy test is also narrow: the 50 interpolates are random convex combinations of just 4 exterior vertices, so they occupy a 3-simplex slice of a 12-dimensional hull. That supports 'interpolates are feasible and roughly within 10%' for that slice, but not for the hull generally. The budget-interpolation trick in Section 3.3 is pairwise homothetic scaling with the least-cost solution; it is a convenience for sketching lower-budget slices, not 'effectively performing an entirely new MGA problem,' as the text claims. Missing code and data make the runtime and accuracy numbers hard to verify independently, and there is no baseline comparison against MAA or a random-sampling alternative.

Those issues are fixable. The central method stands, and the paper is honest about the main limitation (inner approximation) even if some promotional sentences overshoot. For the MGA decision-support community this is a useful advance. A serious referee should see it, though the revision needs coverage diagnostics, a more representative accuracy evaluation, code and data, and toned-down claims about budget interpolation and full-region coverage. I would accept it for peer review and engage with the revised version.

Referee Report

5 major / 8 minor

Summary. The paper proposes 'Modelling to Generate Continuous Alternatives' (MGCA), a post-processing method for linear-programming capacity expansion models. After an MGA exploration has produced a finite set of near-optimal capacity solutions, MGCA projects those solutions onto a subspace of capacity decisions and outcome metrics, then treats the resulting points as vertices of a convex hull. The method (i) generates new portfolios as convex combinations of these vertices, (ii) solves a small optimization problem (Eq. 6) over this hull so that users can impose objectives and additional constraints, (iii) computes affine capacity metrics exactly and operational metrics by convex interpolation, and (iv) exports chosen portfolios back to the full CEM for least-cost dispatch. The demonstrations on a three-zone GenX ISONE case show sub-second solve times for the reduced problems, including budget interpolation, constrained exploration, Pareto-front approximation, and re-dispatch of 50 interpolated portfolios. The paper also reports that interpolated operational metrics are generally within about 5-10% of the full-CEM re-solve values for the tested points.

Significance. If the claims are properly qualified, MGCA is a useful and practical methods contribution. The central convexity argument is correct for LP planning models: convex combinations of feasible, within-budget iterates are feasible and within budget, and affine capacity metrics are exactly interpolated. The dimensionality reduction to capacity decisions and metrics is simple but effective, and the reported speedups (sub-second post-processing versus hundreds of seconds per MGA iterate) are credible and potentially valuable for interactive stakeholder tools. The paper uses an open-source model (GenX) and openly describes the demonstration case, which supports reproducibility. The main weaknesses are overstatements about the convex hull of computed iterates representing the near-optimal feasible region, the claim in Section 3.3 that pairwise budget interpolation represents the lower-budget convex hull, and the limited scope of the accuracy test that underlies the abstract's 'within 10%' claim. These issues are fixable in revision and do not invalidate the core method.

major comments (5)

[Section 2.2 and 2.3] The paper does not consistently state that all MGCA outputs live in conv(Z), the convex hull of the computed MGA iterates, and that conv(Z) is generally an inner approximation of the full near-optimal feasible region. Section 2.2 says the projected points 'can then be used as vertices for the convex hull of the near-optimal feasible region of the planning problem,' but this is only true if the MGA iterates actually span that region. Equation (6) optimizes over conv(Z), so if the user's true optimum lies outside this hull, the returned point is the best hull point, not the best near-optimal portfolio. The paper does acknowledge this later in Section 3.5 ('likely to not include the true optimal solution') and recommends bracketing runs in Section 3.2, but the abstract and introduction would mislead readers who do not reach those sections. Please add an explicit statement in the abstract and in Section 2 that the method operates on the convex hull of computed iterates and that coverage of the near-optimal region is an input assumption that must be verified.
[Section 3.3, Figures 5-6] The budget-interpolation demonstration is overstated. The procedure of combining each MGA iterate with the least-cost solution at weight 0.6 produces only the union of line segments between the least-cost point and each iterate, a homothetic subset of the 10% hull. This does not, in general, 'represent the convex hull of the feasible region within this tighter constraint,' because other constraints besides the cost budget shape the lower-budget slice of conv(Z). A point with cost at 6% slack can lie outside the union of these segments. To explore the full 6% budget slice, one should solve Eq. (6) with the additional cost constraint; the pairwise interpolation should be described as a rapid approximation or a partial sample, not as the convex hull of the lower-budget feasible region.
[Section 3.6 and Abstract] The headline accuracy claim is not supported by the experiment as described. The n=50 interpolates in Section 3.2 are random convex combinations of only 4 exterior points, i.e., points in a 3-simplex inside the 14-dimensional reduced space. Section 3.6 compares the metric values of these interpolates against the full-CEM re-solve of the same portfolios, which measures interpolation error for those specific points but does not compare against the true optimum of any user objective, nor does it test the hull's coverage of the near-optimal region. The abstract's phrase 'within 10% of the optimal value' should be replaced with a more precise statement, for example 'within 10% of the full-CEM re-solve value for the tested interpolated portfolios,' and the text should note that the test is confined to a small simplex and cannot bound errors elsewhere in the hull.
[Section 2.3, Eq. (6)] Equation (6) contains undefined notation: the term 'lambda^T Z X' uses 'X' without definition, and the preceding text says 'z = lambda^T Z' while earlier Z is defined as a subspace. The model size stated ('2m variables and 3n+d+1 constraints') also does not match the displayed problem, which has n+m variables and n+1+d+m+n constraints (depending on how variable bounds are counted). Please correct the equation, define all symbols, and fix the complexity statement so readers can rely on the claimed dimensionality reduction.
[Section 2.4] The statement that an interpolated operational metric is 'a feasible value within the convex hull of those found through the full model with economic dispatch' is imprecise. A convex combination of scalar metric values is not itself a feasible decision in the original model; it is simply an interpolation of the vertex values. The paper already notes that operational metrics 'will not match economic dispatch' and that 'no guarantee of lower overall system metric values is possible,' which is the correct caveat. The wording about feasibility of the interpolated metric should be revised to avoid implying that the interpolated scalar is a feasible outcome of the optimization model.

minor comments (8)

[Section 2.1] Typo: 'constraint coefficient matricies' should be 'matrices'.
[Section 3.2] Typo: 'analagous' should be 'analogous'.
[Section 3.1] Please state explicitly the dimension of the reduced space used in the demonstration (12 technology capacity aggregates plus 2 system metrics = 14) and clarify that the convex hull and all interpolations are performed in this full 14-dimensional space, while Figures 2-7 show only pairwise 2D projections.
[Section 3.3] The text says the budget interpolation 'of all solutions computed in 0.35 seconds'; for reproducibility, please clarify how many solutions were interpolated and whether this includes the time to generate Figure 5 and Figure 6.
[Section 3.6, Figures 10-11] The vertical axes are labeled 'Percent Difference' and 'Difference in Zonal Percent Share'; please state clearly whether the latter is in percentage points, and report the n=50 sample size in the figure captions.
[Section 2.3] The claim that the exploration problem 'will, by definition, always find an exterior solution' is acceptable for a linear objective over a polytope if 'exterior' means a vertex, but the wording is confusing because vertices of conv(Z) are the original MGA iterates only if the iterates are not redundant; please clarify the intended meaning.
[Section 3.5] The sentence 'This Pareto frontier is may not be identical' contains a grammatical error; please rephrase as 'This Pareto frontier may not be identical to a similar Pareto frontier calculated on the original space.'
[General] The paper relies on the companion paper Lau, Patankar, and Jenkins (2024) for the quality of the MGA exploration; please cite a published version if one exists, or add a note that the companion is under review, since the coverage assumptions in the main paper depend on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MGCA's outputs are convex combinations of externally supplied MGA iterates, and the claimed metric properties follow from convexity and are validated against full CEM re-solves.

full rationale

The derivation chain is: (i) a set of MGA iterates Z is generated externally and projected; (ii) the exploration problem (Eq. 6) restricts z to conv(Z), so every output is by construction a convex combination; (iii) Eq. 7 is the linearity identity for affine capacity metrics, not a fitted prediction; (iv) operational-metric estimates are explicitly acknowledged as interpolations, and Section 3.6 compares them against full GenX re-solves, an external benchmark. The budget-interpolation example constructs 6%-cost points by choosing weights so the affine cost equals the target; the individual cost values are therefore exact by construction, but the paper's broader claim to represent the lower-budget hull is a coverage/approximation limitation, not a circular reduction. Self-citations to Lau, Patankar and Jenkins [11] describe how to obtain diverse MGA inputs and are not used to prove the convexity results; the central mathematical content is independent of that citation. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported, and the paper explicitly discloses that the Pareto frontier and hull are constrained by the computed iterates and may not contain the true optimum. Thus no circular step is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No fundamentally new entities are postulated. The method's burden is carried by the convex hull of precomputed MGA solutions, whose completeness with respect to the true near-optimal feasible region is assumed rather than verified. The LP convexity of the planning model is also a structural requirement; the paper notes models are linear or mixed-integer but does not flag that discrete variables break the convex combination argument.

free parameters (3)

MGA budget slack epsilon = 10% (0.10)
User-chosen budget above least-cost cost; defines the near-optimal region that the MGA vertices sample. The paper uses 10% for the demonstration.
Number of MGA iterates = 200
Determines the size and shape of the convex hull. With only 200 points in a 12-dimensional subspace, the hull may be a coarse inner approximation of the true feasible set.
Metric aggregation scheme = 12 technology capacities (summed across zones) plus system cost and emissions
Dimensionality reduction choice; affects which decisions and outcomes can be explored and the accuracy of operational metric interpolation.

assumptions (4)

domain assumption The capacity expansion and dispatch model (1) is a linear program, so its feasible set is convex.
Invoked in Section 2.3 when stating that the feasible space is 'by necessity convex and affine as all constraints are linear', and required for convex combinations of iterates to be feasible.
domain assumption The set of MGA iterates adequately explores the near-optimal feasible region, so their convex hull approximates it well.
Section 2.2 instructs users to run MGA algorithms 'which explore as much of the near-optimal feasible space as possible' and the rest of the method assumes this hull is a faithful proxy for planning decisions.
domain assumption Metrics of interest are linear (affine) functions of capacity and operational decisions.
Section 2.2 defines metrics as f^T y and g^T x; exact interpolation in Eq. (7) relies on linearity.
standard math The economic dispatch value function is convex, so interpolated operational cost overestimates the true least-cost value.
Used in Section 2.4 to claim that MGCA system cost estimates are an upper bound; true for convex (LP) dispatch problems.

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Cite this review

Pith. "Pith review of Modelling to Generate Continuous Alternatives: Enabling Real-Time Feasible Portfolio Generation in Convex Planning Models." pith.science (2026). https://pith.science/paper/WQ5FP4IO

@misc{pith2026241116887,
  author       = {Pith},
  title        = {Pith review of: Modelling to Generate Continuous Alternatives: Enabling Real-Time Feasible Portfolio Generation in Convex Planning Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQ5FP4IO}},
  note         = {Machine review of arXiv:2411.16887}
}

read the original abstract

Decarbonization provides new opportunities to plan energy systems for improved health, resilience, equity, and environmental outcomes, but challenges in siting and social acceptance of transition goals and targets threaten progress. Modelling to Generate Alternatives (MGA) provides an optimization method for capturing many near-cost-optimal system configurations, and can provide insights into the tradeoffs between objectives and flexibility available in the system. However, MGA is currently limited in interactive applicability to these problems due to a lack of methods for allowing users to explore near-optimal feasible spaces. In this work we describe Modelling to Generate Continuous Alternatives (MGCA), a novel post-processing algorithm for convex planning problems which enables users to rapidly generate new interior solutions, incorporate new constraints, and solve within the space with convex objectives. MGCA begins with a dimensionality reduction to capacity decisions and metric values. We then take advantage of convex combinations to generate interior points by allowing user weight specification and encoding convex combinations in an optimization problem with user-defined additional constraints and objective. Dimensionality reduction enables this problem to solve in tenths of a second, suitable for analysis in interactive settings. We discuss the interpolation of capacity and operational metric values, finding capacity metrics can be perfectly interpolated while operational metrics remain within the feasible range of the points used to create them. We demonstrate interpolated solutions can be exported and re-solved with an economic dispatch model to provide operational metric values consistent with least-cost decision-making and show interpolated metric values are generally within 10% of the optimal value.

Figures

Figures reproduced from arXiv: 2411.16887 by the authors.

**Figure 2.** Aggregate capacities for all MGA solutions found for 3-Zone GenX ISONE [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Page 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 3.** Figure 3: Capacities of 50 randomly generated interior MGCA interpolates (orange) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png]

**Figure 4.** Figure 4: Capacities of solution found through exploration problem with objective [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: MGA capacity space with 6% budget (gray) superimposed over 10% bud [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: MGA metric space with 6% budget interpolates (gray) superimposed over [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: MGA capacity space with constraint keeping natural gas capacity [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: MGA capacity space with cost-emission Pareto frontier (set of non [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: MGA metric space with Pareto frontier interpolates (orange) superim [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: Boxplot of percent difference between metrics computed by MGCA and [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]

**Figure 11.** Figure 11: Boxplot of the difference in each zone’s percentage share as calculated [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

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