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A Note on Optimal Product Pricing

T0 review · 2 major / 2 minor · reviewed 2026-05-17 · grok-4.3

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Pith's one-line read The product pricing problem with self- and cross-elasticities is solved by maximizing a sum of convex and concave functions using local optimization methods.

desk verdict This note reformulates pricing with cross-elasticities as a DC program and shows three local solvers agree in examples, but the global-optimality claim rests only on that numerical agreement. read the letter →

arxiv 2511.06156 v4 submitted 2025-11-08 math.OC

Maximilian Schaller , Stephen Boyd This is my paper · ORCID
classification math.OC
keywords productpricingprofitmaximizationpriceelasticitycross-elasticityconvex-concaveprocedureminorization-maximizationnonlinearprogramming
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

This paper examines how to set prices for multiple products to maximize total profit when demand depends on both a product's own price and the prices of other products. It reformulates the resulting optimization problem as maximizing the sum of a convex function and a concave function. The authors then test three different computational methods for finding good solutions and observe that they all reach the same point regardless of where they start. This suggests the solutions may be the best possible ones in practice.

What carries the argument

The reformulation of the profit maximization objective as the sum of a convex function and a concave function, which enables the use of the convex-concave procedure, a minorization-maximization method, and general nonlinear programming to find local solutions.

What would settle it

An instance where starting any of the three methods from a different initial price vector produces a feasible price set with strictly higher profit than the previously found maximum.

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Extended reading notes

Core claim

We consider the problem of choosing prices of a set of products so as to maximize profit, taking into account self-elasticity and cross-elasticity, subject to constraints on the prices. We show that this problem can be formulated as maximizing the sum of a convex and concave function. We compare three methods for finding a locally optimal approximate solution. In numerical examples all three converge reliably to the same local maximum, independent of the starting prices, leading us to believe that the prices found are likely globally optimal.

Load-bearing premise

The assumption that consistent convergence to the same local maximum from varied starting points in numerical examples implies that the solution is globally optimal.

Editorial extensions

If this is right

  • The convex-concave procedure solves the problem through a short sequence of convex optimization problems.
  • The custom minorization-maximization method reduces each step to solving a quadratic program.
  • General purpose nonlinear programming solvers can be applied directly to the formulated problem.
  • All three methods produce the same prices independent of the initial guess in the tested numerical examples.

Reading between the lines

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

  • The observed reliability across methods may indicate that the profit landscape for such pricing problems has few or no poor local maxima in practice.
  • Similar reformulations could be explored for other allocation problems where objectives combine convex and concave terms in decision variables.
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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

2 major / 2 minor

Summary. The paper formulates the profit-maximization pricing problem with self- and cross-elasticities as maximizing a sum of a convex function and a concave function (a DC program). It compares three local solvers: the convex-concave procedure (CCCP), a custom minorization-maximization algorithm that reduces to quadratic programs, and a general-purpose nonlinear programming solver. Numerical examples show that all three methods converge to the identical local maximum from varied initial prices, which the authors interpret as evidence that the solution is likely globally optimal.

Significance. The DC reformulation is a direct application of standard convex-analysis techniques and the numerical experiments demonstrate reliable convergence behavior across the three methods. If the observed agreement holds more generally, the work supplies practical, easily implementable local solvers for this pricing model; however, the global-optimality conjecture rests entirely on empirical agreement rather than a proof or bound on the number of stationary points.

major comments (2)
  1. [Abstract] Abstract: the assertion that the prices found are 'likely globally optimal' is supported solely by the observation that CCCP, the custom MM method, and general NLP reach the same point from different starts. DC programs can possess multiple distinct local maxima, and the manuscript provides neither a proof of unimodality nor an upper bound on the number of stationary points; the numerical evidence therefore does not rigorously establish global optimality.
  2. [Numerical examples] Numerical examples section: while the three solvers agree on the reported instances, the paper does not report the number of distinct local maxima found across a broader set of random initializations or problem instances, nor does it supply a theoretical argument (e.g., via strict concavity of one term or a uniqueness result) that would guarantee a single global maximizer.
minor comments (2)
  1. [Introduction] The manuscript would benefit from a brief statement clarifying that the DC formulation is non-convex and that the reported solutions are local maxima whose global status is conjectural.
  2. [Problem formulation] Notation for the elasticity matrix and the profit function should be introduced with explicit definitions before the DC reformulation is stated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed review and valuable suggestions. We respond to the major comments point by point below. We agree that the global optimality claim requires qualification and will revise the manuscript accordingly to emphasize the empirical nature of our observations.

read point-by-point responses

  1. Referee: [Abstract] the assertion that the prices found are 'likely globally optimal' is supported solely by the observation that CCCP, the custom MM method, and general NLP reach the same point from different starts. DC programs can possess multiple distinct local maxima, and the manuscript provides neither a proof of unimodality nor an upper bound on the number of stationary points; the numerical evidence therefore does not rigorously establish global optimality.

    Authors: We acknowledge that our statement in the abstract relies on numerical evidence rather than a theoretical guarantee. DC programs indeed can have multiple local optima in general. Our intent was to report the observed behavior in the examples considered. We will revise the abstract to state that the methods converge to the same point from different initial prices in the numerical examples, which leads us to conjecture that the solution is globally optimal for the instances tested. We will also add a brief discussion noting the absence of a general proof. revision: yes

  2. Referee: [Numerical examples] while the three solvers agree on the reported instances, the paper does not report the number of distinct local maxima found across a broader set of random initializations or problem instances, nor does it supply a theoretical argument (e.g., via strict concavity of one term or a uniqueness result) that would guarantee a single global maximizer.

    Authors: We agree that reporting results from a larger number of random initializations and instances would provide stronger support. In the current manuscript, we tested several starting points and observed consistent convergence, but we did not exhaustively enumerate all possible local maxima. As this is a concise note focused on the DC formulation and solver comparisons, a comprehensive theoretical analysis of uniqueness is beyond its scope. We will expand the numerical examples section to include additional tests with random initializations and explicitly state that the agreement suggests but does not prove global optimality in general. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; DC formulation and solver comparisons are independently derived

full rationale

The paper starts from a standard profit-maximization objective with linear elasticity demand and algebraically rewrites it as the sum of a convex function and a concave function, which is a direct mathematical decomposition rather than a self-referential definition. The three solution methods (convex-concave procedure, custom minorization-maximization, and general NLP) are then applied as off-the-shelf algorithms to this DC program; none of the methods or the convergence observation in the numerical examples is obtained by fitting parameters to the target result or by renaming an input. No load-bearing self-citation chain is invoked to justify uniqueness or global optimality; the authors simply report that the three local solvers agree across random starts and therefore conjecture global optimality on heuristic grounds. Because the core derivation remains self-contained and externally verifiable against the original profit expression, the analysis contains no circular reduction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work relies on standard convexity properties of the profit function under linear elasticity models; elasticities are treated as given inputs rather than fitted parameters.

assumptions (1)
  • domain assumption Profit can be expressed as the sum of a convex function and a concave function based on the elasticity demand model.
    Invoked in the initial problem formulation to enable the convex-concave procedure.

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Cite this review

Pith. "Pith review of A Note on Optimal Product Pricing." pith.science (2026). https://pith.science/paper/2511.06156

@misc{pith2026251106156,
  author       = {Pith},
  title        = {Pith review of: A Note on Optimal Product Pricing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2511.06156}},
  note         = {Machine review of arXiv:2511.06156}
}
read the original abstract

We consider the problem of choosing prices of a set of products so as to maximize profit, taking into account self-elasticity and cross-elasticity, subject to constraints on the prices. We show that this problem can be formulated as maximizing the sum of a convex and concave function. We compare three methods for finding a locally optimal approximate solution. The first is based on the convex-concave procedure, and involves solving a short sequence of convex problems. Another one uses a custom minorization-maximization method, and involves solving a sequence of quadratic programs. The final method is to use a general purpose nonlinear programming method. In numerical examples all three converge reliably to the same local maximum, independent of the starting prices, leading us to believe that the prices found are likely globally optimal.

Figures

Figures reproduced from arXiv: 2511.06156 by the authors.

Figure 1
Figure 1. Modeling and solving the PPP with CVXPY. The dimensions n, m and the data pi min, pi max, rnom, knom, E, C are given. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Profit and price changes for n = 1280 and m = 256. groups of related products that might be substitutes or complements. We sample the self￾elasticities Eii between −3.0 and −1.0, and the cross-elasticities Eij (within each block) between −0.05 and 0.05. We set the nominal revenue per product r nom i between 1.0 and 5.0, and the nominal cost to κ nom i = 0.9r nom i , i.e., a nominal profit margin of 10%. Constraints.… view at source ↗
Figure 3
Figure 3. Solve times for different problem sizes, with CCP, QMM, and NLP. effect of dealing with a quadratic program at each iteration of QMM (and being able to use a specialized solver) appears to outweigh the effect of larger approximation errors. In fact, QMM took around 3–5 iterations for all problem sizes, just slightly more than the 3–4 iterations required by CCP. These scaling results were insensitive to the seed used… view at source ↗

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Forward citations

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