arxiv: 2510.23791 · v4 · pith:OE6ZBZSFnew · submitted 2025-10-27 · 🧮 math.OC

A Family of Convex Models to Achieve Fairness through Dispersion Control

Abhay Singh Bhadoriya , Deepjyoti Deka , Kaarthik Sundar This is my paper

Pith reviewed 2026-05-18 03:02 UTC · model grok-4.3

classification 🧮 math.OC

keywords convex optimizationdispersion controlcoefficient of variationfairnessload balancingassignment problemparameterized models

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The pith

A family of convex models uses one parameter to bound the coefficient of variation and enforce fairness from loose to strict equality.

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

The paper develops a family of convex optimization models to regulate dispersion among decision variables using the coefficient of variation as the dispersion measure. A single parameter in the interval from zero to one adjusts the constraint strength, imposing no real restriction at zero and forcing all variables to take the same value at one. The authors prove that the coefficient of variation stays below a monotonically decreasing function of this parameter. They further relate the feasible solution spaces for different parameter values and demonstrate the models on a modified assignment problem that controls dispersion in assignment costs.

Core claim

Building on the well-known equivalence of finite-dimensional norms, the article develops a family of parameterized convex models that regulate the dispersion of a vector of decision-variable values through its coefficient of variation. Each model has a single parameter taking values in the interval [0,1]. When the parameter is set to zero, the model imposes only a trivial constraint on the optimization problem; when set to one, it enforces equality of all the decision variables. As the parameter varies, the coefficient of variation is provably bounded above by a monotonic function of that parameter. The article also presents theoretical results relating the space of feasible solutions across

What carries the argument

Parameterized convex constraints derived from finite-dimensional norm equivalences that upper-bound the coefficient of variation of a vector of decision variables.

If this is right

As the parameter increases from zero to one the upper bound on the coefficient of variation decreases monotonically.
At parameter value one the model forces all relevant decision variables to be identical.
The feasible solution sets for different parameter values are related through inclusion or comparability relations.
The models preserve convexity while providing a continuous range of dispersion control on problems such as the assignment variant.

Where Pith is reading between the lines

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

The single-parameter structure allows practitioners to sweep through a spectrum of fairness levels by solving a sequence of convex programs rather than redesigning constraints each time.
The norm-equivalence approach may extend to controlling other relative dispersion measures in allocation or balancing problems outside the assignment setting.
Intermediate parameter values could be used in iterative solvers to locate efficient trade-offs between objective value and dispersion without solving separate problems for each target level.

Load-bearing premise

The equivalence of finite-dimensional norms can be used to construct convex constraints that regulate dispersion specifically through the coefficient of variation in a monotonic way controlled by one parameter.

What would settle it

Solving one of the models for an intermediate parameter value and finding that the coefficient of variation of the resulting solution exceeds the claimed monotonic upper bound function.

Figures

Figures reproduced from arXiv: 2510.23791 by Abhay Singh Bhadoriya, Deepjyoti Deka, Kaarthik Sundar.

**Figure 1.** Figure 1: Graphical illustration for εp1 (x) > εp2 (x). Theorem 3.2 shows that the size of the feasible sets of requiring x to be at least (ε, p)-fair decreases with p when ε ∈ (0, 1) and Theorem 3.1 shows that the feasible sets are equal for any value of integer p ⩾ 2 when ε ∈ {0, 1}. Next, we consider an example application to illustrate the use of the models presented thus far in this note. 4 Multi-Agent Assignme… view at source ↗

**Figure 2.** Figure 2: Coefficient of variation CV(t ∗ (ε, p)) versus ε for different p for one instance. 5.2 Price of Fairness The term “Price of Fairness” (PoF) is defined as the percentage increase in the objective value of the fairness-constrained MAAP of Section 4.1 relative to that of the unconstrained problem (17). This definition is standard in the literature [3, 27]. Formally, PoF(ε, p) ≜ t ∗ (ε, p) ⊺e t ∗(0)⊺e − 1 … view at source ↗

**Figure 3.** Figure 3: illustrates this trend for one representative instance; the other 0 2 4 6 0.2 0.4 0.6 0.8 0.99 p = 2 p = 3 p = 5 p = 10 p = ∞ ε [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Box-plot of computation times in seconds for all the 50 instances for [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

read the original abstract

Controlling the dispersion of a subset of decision variables in an optimization problem is crucial for enforcing fairness or load-balancing across a wide range of applications. Building on the well-known equivalence of finite-dimensional norms, the article develops a family of parameterized convex models that regulate the dispersion of a vector of decision-variable values through its coefficient of variation. Each model has a single parameter taking values in the interval $[0,1]$. When the parameter is set to zero, the model imposes only a trivial constraint on the optimization problem; when set to one, it enforces equality of all the decision variables. As the parameter varies, the coefficient of variation is provably bounded above by a monotonic function of that parameter. The article also presents theoretical results relating the space of feasible solutions across all models. Finally, it compares the models' solution quality on a variant of the assignment problem that regulates the dispersion in the assignment costs.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean, parameterized convex way to bound the coefficient of variation of decision variables for fairness or load balancing, built from standard norm equivalences.

read the letter

The main point is that the authors turn finite-dimensional norm equivalences into a single-parameter family of convex models that directly regulate dispersion through the coefficient of variation. The parameter sits in [0,1]: zero adds essentially no constraint, one forces all variables equal, and values in between produce a monotonic upper bound on the CV. They also map out how the feasible sets nest as the parameter moves and test the models on a dispersion-constrained assignment problem.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a family of parameterized convex optimization models, indexed by a scalar α ∈ [0,1], that regulate dispersion of a vector of decision variables via an upper bound on the coefficient of variation. The construction rests on finite-dimensional norm equivalences. At α = 0 the constraint is trivial; at α = 1 all variables are forced equal. The paper proves that the coefficient of variation is bounded above by a monotonic function of α, derives inclusion relations among the feasible sets of the different models, and illustrates the approach on a dispersion-constrained variant of the assignment problem.

Significance. If the convexity and monotonicity claims hold, the parameterization supplies a clean, single-parameter mechanism for trading off dispersion against objective value in convex programs. This is potentially useful for fairness and load-balancing applications. The explicit boundary behavior and the feasible-set relations are attractive features; the assignment-problem experiment provides a concrete test case.

major comments (2)

[§3.1] §3.1, the model formulation: the passage from the norm-equivalence identity to the explicit convex constraint that bounds the coefficient of variation is the central technical step. The text should state explicitly whether the resulting program remains convex for every α ∈ [0,1] without additional assumptions on the sign or magnitude of the decision variables.
[§4] §4, Theorem 2 on feasible-set inclusions: the claimed nested structure of feasible regions is load-bearing for the family interpretation. The proof sketch relies on the monotonicity of the bound; please supply the complete argument, including any dependence on problem dimension.

minor comments (3)

[Abstract] Abstract: the statement that the coefficient of variation is 'provably bounded above by a monotonic function' should be accompanied by the explicit form of that function (or a reference to the theorem that supplies it).
Notation: the coefficient of variation is used throughout; confirm that the definition (standard deviation divided by mean) is stated once, together with the convention adopted when the mean is zero or negative.
[Experimental section] Experimental section: the comparison of solution quality on the assignment problem would benefit from an additional baseline that uses a different dispersion measure (e.g., range or Gini coefficient) to illustrate the specific effect of the coefficient-of-variation control.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: §3.1, the model formulation: the passage from the norm-equivalence identity to the explicit convex constraint that bounds the coefficient of variation is the central technical step. The text should state explicitly whether the resulting program remains convex for every α ∈ [0,1] without additional assumptions on the sign or magnitude of the decision variables.

Authors: We appreciate the referee's suggestion to make this explicit. The resulting optimization program remains convex for every α ∈ [0,1] because the constraint is formulated using convex functions derived from the equivalence of norms, specifically bounding the ratio of the Euclidean norm to the L1 norm in a way that preserves convexity. This holds without additional assumptions on the magnitude of the decision variables, provided they are non-negative (as required for the coefficient of variation to be meaningfully defined in this context). We will add a clarifying sentence in §3.1 stating this explicitly. revision: yes
Referee: §4, Theorem 2 on feasible-set inclusions: the claimed nested structure of feasible regions is load-bearing for the family interpretation. The proof sketch relies on the monotonicity of the bound; please supply the complete argument, including any dependence on problem dimension.

Authors: We thank the referee for this request. The nested structure follows from the monotonicity of the upper bound on the coefficient of variation as a function of α. Specifically, let f(α) be the monotonic increasing function such that CV(x) ≤ f(α). For α' > α, f(α') ≥ f(α), so any x feasible for α' (i.e., CV(x) ≤ f(α')) is also feasible for α. This argument is independent of the problem dimension n, as it relies only on the scalar monotonicity and not on the specific constants from norm equivalences (which may depend on n but do not affect the inclusion direction). We will provide the complete proof in the revised §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops a parameterized family of convex models for dispersion control by invoking the standard, externally known equivalence of finite-dimensional norms. The central results consist of proving that the coefficient of variation is bounded above by a monotonic function of the single parameter in [0,1], together with relations among feasible sets and boundary behaviors (trivial constraint at 0, equality at 1). These statements follow directly from the norm-based construction and standard convex-analysis arguments; no step reduces a claimed prediction or theorem to a fitted quantity, a self-referential definition, or a load-bearing self-citation whose validity is assumed without external support. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard equivalence of finite-dimensional norms to derive the convex models and the monotonic bound; no free parameters are fitted to data and no new entities are postulated.

axioms (1)

standard math Equivalence of finite-dimensional norms
Invoked in the abstract as the foundation for developing the family of parameterized convex models.

pith-pipeline@v0.9.0 · 5690 in / 1233 out tokens · 48078 ms · 2026-05-18T03:02:46.479459+00:00 · methodology

discussion (0)

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