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arxiv: 2310.01848 · v1 · submitted 2023-10-03 · 🧮 math.OC

Uncertain random geometric programming problems

Tapas Mondal , Akshay Kumar Ojha , Sabyasachi Pani This is my paper

Pith reviewed 2026-05-24 06:09 UTC · model grok-4.3

classification 🧮 math.OC
keywords geometric programminguncertain random variableslinear-normal uncertain random variableoptimistic value criteriapessimistic value criteriaexpected value criteriastochastic geometric programmingdeterministic transformation
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The pith

Coefficients as linear-normal uncertain random variables let uncertain random geometric programs convert to deterministic forms via three value criteria.

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

The paper develops a deterministic formulation for geometric programming problems whose coefficients combine uncertainty and randomness. It defines linear-normal uncertain random variables and introduces three transformation techniques—optimistic value criteria, pessimistic value criteria, and expected value criteria—that map each such variable to an ordinary random variable. This mapping turns the original uncertain random geometric program into a stochastic geometric program, which then admits an equivalent deterministic representation. The approach preserves problem structure while making the optimization tractable, and a numerical example illustrates the steps.

Core claim

By modeling the coefficients of a geometric program as independent linear-normal uncertain random variables, the three proposed criteria convert the uncertain random geometric programming problem into a stochastic geometric programming problem that possesses an equivalent deterministic form.

What carries the argument

The linear-normal uncertain random variable together with the optimistic, pessimistic, and expected value criteria that map it to a standard random variable.

If this is right

  • The transformed stochastic geometric program can be solved with existing stochastic optimization methods.
  • Each of the three criteria yields its own equivalent deterministic geometric program.
  • The method supplies a systematic route from mixed uncertainty-randomness models to computable deterministic problems.
  • The numerical example confirms that the transformations produce concrete, solvable programs.

Where Pith is reading between the lines

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

  • The same criteria could be tested on other uncertain random variables beyond the linear-normal case.
  • The framework might carry over to linear or convex programs that share the same coefficient structure.
  • Different criteria will generally produce solutions with different degrees of conservatism.
  • Empirical comparison of the three resulting deterministic programs on benchmark instances would quantify their relative risk attitudes.

Load-bearing premise

The coefficients can be expressed as independent linear-normal uncertain random variables and the three criteria transform them into random variables without changing the underlying optimization structure.

What would settle it

An instance in which one of the three criteria applied to linear-normal uncertain random coefficients produces a stochastic program whose solution violates a constraint that holds under the original uncertain random model.

Figures

Figures reproduced from arXiv: 2310.01848 by Akshay Kumar Ojha, Sabyasachi Pani, Tapas Mondal.

Figure 1
Figure 1. Figure 1: Transformed probability distribution of a linear-normal uncertain random variable via [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Transformed probability distribution of a linear-normal uncertain random variable via [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Transformed probability distribution of a linear-normal uncertain random variable via expected [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Transformed probability distributions of a linear-normal uncertain random variable via opti [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Transformed probability distributions of a linear-normal uncertain random variable via pes [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Optimal objective values with respect to the parameter [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

In this paper, we introduce a deterministic formulation for the geometric programming problem, wherein the coefficients are represented as independent linear-normal uncertain random variables. To address the challenges posed by this combination of uncertainty and randomness, we introduce the concept of an uncertain random variable and present a novel framework known as the linear-normal uncertain random variable. Our main focus in this work is the development of three distinct transformation techniques: the optimistic value criteria, pessimistic value criteria, and expected value criteria. These approaches allow us to convert a linear-normal uncertain random variable into a more manageable random variable. This transition facilitates the transformation from an uncertain random geometric programming problem to a stochastic geometric programming problem. Furthermore, we provide insights into an equivalent deterministic representation of the transformed geometric programming problem, enhancing the clarity and practicality of the optimization process. To demonstrate the effectiveness of our proposed approach, we present a numerical example.

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

3 major / 2 minor

Summary. The paper introduces the concept of an uncertain random variable and a novel linear-normal uncertain random variable framework. It develops three transformation techniques (optimistic value criteria, pessimistic value criteria, and expected value criteria) to convert uncertain random geometric programming problems with independent linear-normal uncertain random coefficients into stochastic geometric programming problems, followed by equivalent deterministic representations. A numerical example illustrates the approach.

Significance. If the transformations are shown to correctly map the original uncertain-random optimum to the transformed stochastic GP optimum (accounting for monomial structure), the work could extend hybrid uncertainty modeling in geometric programming, offering practical deterministic equivalents for applications in engineering optimization under combined randomness and uncertainty.

major comments (3)
  1. [§4] §4 (transformations): The optimistic/pessimistic/expected value criteria are applied coefficient-wise to the linear-normal uncertain random variables, but the manuscript provides no explicit argument or proof that this preserves the essential ordering or expectation properties of the posynomial objective and constraints under the product/power structure of geometric programs; the joint distribution after transformation is not shown to be equivalent.
  2. [§3] §3 (linear-normal uncertain random variable definition): The independence assumption for the coefficients is stated but not connected to the uncertainty measure or random measure in a way that justifies applying the criteria independently; this is load-bearing for the reduction to a stochastic GP.
  3. [Numerical example] Numerical example section: The example reports results for the transformed problem but contains no comparison to the original uncertain-random formulation, no sensitivity analysis on the criteria, and no verification that the surrogate optimum bounds or coincides with the original problem's solution.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'insights into an equivalent deterministic representation' is vague; the specific form of the deterministic GP (e.g., after taking logs or applying standard GP transformations) should be stated explicitly.
  2. [§3] Notation: The definition of the linear-normal uncertain random variable introduces new symbols without a clear table or comparison to standard uncertain random variables (e.g., from Liu's theory), which hinders readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each of the major comments in detail below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: [§4] §4 (transformations): The optimistic/pessimistic/expected value criteria are applied coefficient-wise to the linear-normal uncertain random variables, but the manuscript provides no explicit argument or proof that this preserves the essential ordering or expectation properties of the posynomial objective and constraints under the product/power structure of geometric programs; the joint distribution after transformation is not shown to be equivalent.

    Authors: We agree that an explicit argument is needed to justify the coefficient-wise application. Given the independence of the uncertain random variables, the criteria can be applied separately, resulting in independent random variables for the coefficients. The posynomial structure is preserved because products and powers of the transformed variables maintain the form. We will add a detailed remark or subsection in §4 explaining why the ordering and expectation properties are preserved under this transformation, and clarify the equivalence of the joint distribution for the purpose of the optimization problem. revision: yes

  2. Referee: [§3] §3 (linear-normal uncertain random variable definition): The independence assumption for the coefficients is stated but not connected to the uncertainty measure or random measure in a way that justifies applying the criteria independently; this is load-bearing for the reduction to a stochastic GP.

    Authors: The independence assumption is crucial and we will strengthen the connection in §3 by explaining how the product measure for independent uncertain random variables allows the criteria to be applied independently without affecting the overall measure. This justifies the reduction to a stochastic geometric program with independent random coefficients. revision: yes

  3. Referee: [Numerical example] Numerical example section: The example reports results for the transformed problem but contains no comparison to the original uncertain-random formulation, no sensitivity analysis on the criteria, and no verification that the surrogate optimum bounds or coincides with the original problem's solution.

    Authors: We acknowledge the lack of direct comparison and verification in the numerical example. Direct solving of the original uncertain-random problem is not straightforward with existing methods, which is why the transformation is proposed. However, we will add a sensitivity analysis on the parameters of the criteria (such as the confidence levels in optimistic/pessimistic values) to the example. We will also include a discussion on the relationship between the surrogate and original optima based on the properties of the criteria. revision: partial

Circularity Check

0 steps flagged

No circularity: new definitions and criteria are introduced as primitives, not reduced to inputs by construction

full rationale

The paper defines an uncertain random variable and the linear-normal subclass as novel concepts, then directly specifies the optimistic-value, pessimistic-value, and expected-value criteria as transformation rules that map these objects to ordinary random variables. These steps are definitional proposals rather than derivations that presuppose the target stochastic GP or its optimum. No self-citation chain, fitted-parameter renaming, or ansatz smuggling is present in the provided text; the subsequent conversion to a deterministic geometric program follows from applying the stated criteria coefficient-wise. The framework is therefore self-contained as an original modeling approach.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition and properties of the newly introduced linear-normal uncertain random variable and the validity of the transformation criteria.

axioms (1)
  • domain assumption Coefficients are independent linear-normal uncertain random variables
    Stated as the representation for the coefficients in the geometric programming problem.
invented entities (1)
  • linear-normal uncertain random variable no independent evidence
    purpose: To represent coefficients combining uncertainty and randomness
    Newly introduced concept in the paper to model the specific type of uncertainty.

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