pith. sign in

arxiv: 2605.18669 · v1 · pith:KLYROX6Znew · submitted 2026-05-18 · 🧮 math.OC

Robust Optimization Under Objective Functional Uncertainty

Yue Song , Yuxi Lu , Gang Li , Kairui Feng , Qi Liu This is my paper

Pith reviewed 2026-05-20 08:25 UTC · model grok-4.3

classification 🧮 math.OC
keywords robust optimizationobjective functional uncertaintymin-max problemoperator theorypiecewise linear approximationbattery charging scheduling
0
0 comments X

The pith

Robust optimization under objective functional uncertainty is solved by an alternating algorithm proven to converge to a semi-global saddle point using operator theory.

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

This paper establishes a new robust optimization problem called ObRO that treats uncertainty in the form of the objective function itself. The formulation sets up a min-max game where the maximizer picks the worst objective function from a continuous space and the minimizer picks the best decision. An iterative algorithm alternates between these two and is shown via operator theory to reach a semi-global saddle point. A piecewise linear approximation to the functions is introduced and proven numerically consistent, enabling practical computation. The method is demonstrated on scheduling battery charging in power networks while accounting for degradation.

Core claim

The ObRO formulation uses a min-max structure over objective functions in continuous space, solved by an alternating procedure that converges to the semi-global saddle point via operator theory, with the PWL version being numerically consistent.

What carries the argument

The alternating algorithm for finding the semi-global saddle point in the ObRO min-max problem, supported by operator theory for convergence proof.

Load-bearing premise

The objective functions belong to a continuous function space with properties that admit a semi-global saddle point and allow operator theory to establish convergence of the alternating procedure.

What would settle it

Applying the alternating algorithm to the battery charging scheduling example and observing whether it stabilizes at a point where neither the decision nor the objective function can be changed to further improve the cost.

Figures

Figures reproduced from arXiv: 2605.18669 by Gang Li, Kairui Feng, Qi Liu, Yue Song, Yuxi Lu.

Figure 1
Figure 1. Figure 1: An example of functional uncertainty. This paper investigates the RO where the objective function form is assumed completely unknown. The following threefold contributions are made. 1) We establish the minimax formulation of the robust optimization under objective functional uncertainty (ObRO), which finds the decision achieving the minimum cost under the worst-case objective function in a continuous funct… view at source ↗
Figure 2
Figure 2. Figure 2: for illustration). To sum up, using PWL approximation, the whole function fi(·) is captured by a set of points { ˆfi[p]} N p=1 and the values of xi and fi(xi) are estimated by the hatted notations xˆi , ˆfi(ˆxi) in (14). We now apply the PWL approximation to reformulating the subproblem (7). Assume we have xˆ k i at hand, which is the solution of the master problem in iteration k, such that xˆi[pk] ≤ xˆ k … view at source ↗
Figure 4
Figure 4. Figure 4: The solution information 0 0.01 0.02 0.03 0.04 |Pbi| 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Battery degradation reference function f i * for battery at node 2 f i * for battery at node 6 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Information of the 8-node test system. VI. CONCLUSION We have formulated the ObRO problem that finds the decision with the minimum cost under the worst-case objective function. We have designed an alternate iterative algorithm which converges to a semi-global saddle node of the ObRO problem. A PWL-based numerical solver has also been pro￾posed, which is consistent with the original ObRO problem. The obtain… view at source ↗
Figure 6
Figure 6. Figure 6: Worst-case functions under five schemes for BESS at node 2. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Worst-case functions under five schemes for BESS at node 6. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

This paper proposes a new robust optimization (RO) formulation namely the RO under objective functional uncertainty (ObRO). The ObRO adopts a min-max structure where the inner problem finds the worst-case objective function in a continuous function space to maximize the cost, and the outer problem finds the optimal decision in a Euclidean space to minimize the cost. A solution algorithm is designed to alternately generate the worst-case objective function at the current decision and the optimal decision for the current collection of objective functions. Using operator theory, we prove that this algorithm converges to the defined ``semi-global'' saddle point of the ObRO problem. In addition, we propose a numerical solver based on the piece-wise linearization (PWL) approximation of objective functions. The PWL approximate problem is proved to be numerically consistent with the original ObRO problem. The obtained results are applied to the degradation-aware battery charging scheduling in distribution networks.

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

1 major / 2 minor

Summary. The manuscript proposes a new robust optimization formulation called ObRO under objective functional uncertainty. It adopts a min-max structure with the inner problem identifying the worst-case objective function from a continuous function space and the outer problem optimizing the decision variable in Euclidean space. An alternating algorithm is developed to solve this, with a proof via operator theory that it converges to a defined 'semi-global' saddle point. A piecewise-linear (PWL) approximation of the objective functions is introduced and shown to be numerically consistent with the original problem. The results are demonstrated on a degradation-aware battery charging scheduling application in distribution networks.

Significance. If the convergence and consistency claims hold, the work offers a meaningful extension of robust optimization to settings with uncertainty in the functional form of the objective rather than parameters alone. The operator-theoretic treatment of the alternating procedure and the numerical consistency result for the PWL solver are clear strengths that could support further development in infinite-dimensional robust optimization. The battery-scheduling example provides a concrete, relevant application that illustrates potential impact in energy systems.

major comments (1)
  1. [Convergence proof section (following algorithm presentation)] The convergence argument (invoked in the abstract and developed after the algorithm definition) casts the alternating procedure as an operator whose fixed point is the semi-global saddle point and appeals to operator theory for convergence. However, the required monotonicity, nonexpansiveness, or compactness properties of the worst-case map (the inner argmax over the continuous function space) are not verified via an inner-product inequality or level-set compactness argument in the chosen function space (e.g., C(X) with sup norm). This verification is load-bearing for the central convergence claim.
minor comments (2)
  1. [Problem formulation] The precise definition of the semi-global saddle point should be stated with explicit reference to the product space and the topology on the function space to remove any ambiguity in the convergence statement.
  2. [Numerical experiments] In the battery-charging numerical example, the concrete parametrization of the functional uncertainty set and the choice of basis for the PWL approximation should be described more explicitly so that the consistency result can be directly checked against the reported schedules.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the major comment below and will incorporate the suggested clarification into the revised version.

read point-by-point responses

  1. Referee: The convergence argument (invoked in the abstract and developed after the algorithm definition) casts the alternating procedure as an operator whose fixed point is the semi-global saddle point and appeals to operator theory for convergence. However, the required monotonicity, nonexpansiveness, or compactness properties of the worst-case map (the inner argmax over the continuous function space) are not verified via an inner-product inequality or level-set compactness argument in the chosen function space (e.g., C(X) with sup norm). This verification is load-bearing for the central convergence claim.

    Authors: We agree that an explicit verification of the monotonicity and nonexpansiveness properties of the worst-case map is necessary to fully support the operator-theoretic convergence claim. In the revised manuscript we will insert a new subsection immediately after the algorithm presentation that supplies an inner-product inequality argument establishing these properties for the map in the space C(X) equipped with the supremum norm, together with a brief compactness argument on the relevant level sets. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence proof invokes external operator theory on independently defined min-max structure

full rationale

The paper defines the ObRO problem as a min-max over a continuous function space for the inner worst-case objective and Euclidean decisions for the outer minimization. It then introduces an alternating algorithm that generates worst-case functions and optimal decisions iteratively. Convergence to the defined semi-global saddle point is established by applying operator theory (fixed-point or monotone-operator results) to this procedure. The PWL approximation is separately shown to be numerically consistent. No step reduces the claimed result to a fitted parameter, a self-referential definition of the saddle point, or a load-bearing self-citation whose content is itself unverified. The derivation remains self-contained against external mathematical benchmarks, with the operator-theoretic step providing independent grounding rather than tautological equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract alone, the central claim rests on domain assumptions about the function space and operator properties rather than fitted parameters or new entities.

axioms (2)
  • domain assumption Objective functions reside in a continuous function space permitting identification of a worst-case function that maximizes cost for any fixed decision.
    This enables the inner maximization step of the min-max structure.
  • domain assumption The operators arising from the alternating procedure satisfy conditions that guarantee convergence to a semi-global saddle point.
    Invoked for the convergence proof using operator theory.

pith-pipeline@v0.9.0 · 5682 in / 1298 out tokens · 42904 ms · 2026-05-20T08:25:32.242334+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Ben-Tal, L

    A. Ben-Tal, L. El Ghaoui, and A. Nemirovski,Robust Optimization. Princeton University Press, 2009

    work page 2009

  2. [2]

    Robust trajectory optimization over uncertain terrain with stochastic complementarity,

    L. Drnach and Y . Zhao, “Robust trajectory optimization over uncertain terrain with stochastic complementarity,”IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 1168–1175, 2021

    work page 2021

  3. [3]

    Robust optimization for emergency logistics planning: Risk mitigation in humanitarian relief supply chains,

    A. Ben-Tal, B. Do Chung, S. R. Mandala, and T. Yao, “Robust optimization for emergency logistics planning: Risk mitigation in humanitarian relief supply chains,”Transportation Research Part B: Methodological, vol. 45, no. 8, pp. 1177–1189, 2011

    work page 2011

  4. [4]

    X. A. Sun and A. J. Conejo,Robust Optimization in Electric Energy Systems. Springer, 2021, vol. 313

    work page 2021

  5. [5]

    Certifying some distributional ro- bustness with principled adversarial training,

    A. Sinha, H. Namkoong, and J. Duchi, “Certifying some distributional ro- bustness with principled adversarial training,” inInternational Conference on Learning Representations, 2018

    work page 2018

  6. [6]

    Solving two-stage robust optimization problems using a column-and-constraint generation method,

    B. Zeng and L. Zhao, “Solving two-stage robust optimization problems using a column-and-constraint generation method,”Operations Research Letters, vol. 41, no. 5, pp. 457–461, 2013

    work page 2013

  7. [7]

    Distributionally robust optimization under moment uncertainty with application to data-driven problems,

    E. Delage and Y . Ye, “Distributionally robust optimization under moment uncertainty with application to data-driven problems,”Operations Research, vol. 58, no. 3, pp. 595–612, 2010

    work page 2010

  8. [8]

    Data-driven risk-averse stochastic optimization with wasserstein metric,

    C. Zhao and Y . Guan, “Data-driven risk-averse stochastic optimization with wasserstein metric,”Operations Research Letters, vol. 46, no. 2, pp. 262–267, 2018

    work page 2018

  9. [9]

    Frameworks and results in distributionally robust optimization,

    H. Rahimian and S. Mehrotra, “Frameworks and results in distributionally robust optimization,”Open Journal of Mathematical Optimization, vol. 3, pp. 1–85, 2022

    work page 2022

  10. [10]

    Distributionally robust optimization,

    D. Kuhn, S. Shafiee, and W. Wiesemann, “Distributionally robust optimization,”Acta Numerica, vol. 34, pp. 579–804, 2025

    work page 2025

  11. [11]

    Maxmin under risk,

    F. Maccheroni, “Maxmin under risk,”Economic Theory, vol. 19, no. 4, pp. 823–831, 2002

    work page 2002

  12. [12]

    Decision making under uncertainty when preference information is incomplete,

    B. Armbruster and E. Delage, “Decision making under uncertainty when preference information is incomplete,”Management Science, vol. 61, no. 1, pp. 111–128, 2015

    work page 2015

  13. [13]

    Optimization with reference-based robust preference constraints,

    J. Hu and G. Stepanyan, “Optimization with reference-based robust preference constraints,”SIAM Journal on Optimization, vol. 27, no. 4, pp. 2230–2257, 2017

    work page 2017

  14. [14]

    Preference robust optimization for choice functions on the space of cdfs,

    W. B. Haskell, H. Xu, and W. Huang, “Preference robust optimization for choice functions on the space of cdfs,”SIAM Journal on Optimization, vol. 32, no. 2, pp. 1446–1470, 2022

    work page 2022

  15. [15]

    A two-layer energy management system for microgrids with hybrid energy storage considering degradation costs,

    C. Ju, P. Wang, L. Goel, and Y . Xu, “A two-layer energy management system for microgrids with hybrid energy storage considering degradation costs,”IEEE Trans. Smart Grid, vol. 9, no. 6, pp. 6047–6057, 2018

    work page 2018

  16. [16]

    A lithium-ion battery capacity degradation correction method for off-design cycling conditions,

    L. Kong, S. Fang, T. Niu, G. Chen, L. Yang, and R. Liao, “A lithium-ion battery capacity degradation correction method for off-design cycling conditions,”IEEE Trans. Ind. Appl., vol. 60, no. 4, pp. 5778–5789, 2024

    work page 2024

  17. [17]

    Some minimax theorems,

    F. Terkelsen, “Some minimax theorems,”Mathematica Scandinavica, vol. 31, no. 2, pp. 405–413, 1972

    work page 1972

  18. [18]

    Dempe, V

    S. Dempe, V . Kalashnikov, G. A. P ´erez-Vald´es, and N. Kalashnykova, Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks. Springer, 2015

    work page 2015

  19. [19]

    Constraint generation for two- stage robust network flow problems,

    D. Simchi-Levi, H. Wang, and Y . Wei, “Constraint generation for two- stage robust network flow problems,”INFORMS Journal on Optimization, vol. 1, no. 1, pp. 49–70, 2019

    work page 2019

  20. [20]

    Adaptive robust optimal control of constrained continuous-time linear systems: A functional constraint generation approach,

    Y . Song, T. Liu, and G. Li, “Adaptive robust optimal control of constrained continuous-time linear systems: A functional constraint generation approach,”IEEE Trans. Autom. Control, vol. 70, no. 2, pp. 1312–1319, 2025

    work page 2025

  21. [21]

    Multistage adaptive robust optimization for the unit commitment problem,

    ´A. Lorca, X. A. Sun, E. Litvinov, and T. Zheng, “Multistage adaptive robust optimization for the unit commitment problem,”Operations Research, vol. 64, no. 1, pp. 32–51, 2016

    work page 2016

  22. [22]

    Sufficient conditions for the convergence of monotonic mathematical programming algorithms,

    R. R. Meyer, “Sufficient conditions for the convergence of monotonic mathematical programming algorithms,”Journal of Computer and System Sciences, vol. 12, no. 1, pp. 108–121, 1976

    work page 1976

  23. [23]

    Exact convex relaxation of optimal power flow in radial networks,

    L. Gan, N. Li, U. Topcu, and S. H. Low, “Exact convex relaxation of optimal power flow in radial networks,”IEEE Trans. Autom. Control, vol. 60, no. 1, pp. 72–87, 2015

    work page 2015

  24. [24]

    Strong SOCP relaxations for the optimal power flow problem,

    B. Kocuk, S. S. Dey, and X. A. Sun, “Strong SOCP relaxations for the optimal power flow problem,”Operations Research, vol. 64, no. 6, pp. 1177–1196, 2016

    work page 2016

  25. [25]

    Energy-optimal pump scheduling and water flow,

    D. Fooladivanda and J. A. Taylor, “Energy-optimal pump scheduling and water flow,”IEEE Trans. Control Netw. Syst., vol. 5, no. 3, pp. 1016–1026, 2018

    work page 2018

  26. [26]

    Online distributed MPC- based optimal scheduling for EV charging stations in distribution systems,

    Y . Zheng, Y . Song, D. J. Hill, and K. Meng, “Online distributed MPC- based optimal scheduling for EV charging stations in distribution systems,” IEEE Trans. Ind. Informat., vol. 15, no. 2, pp. 638–649, 2019

    work page 2019

  27. [27]

    Natural gas flow equations: Uniqueness and an MI-SOCP solver,

    M. K. Singh and V . Kekatos, “Natural gas flow equations: Uniqueness and an MI-SOCP solver,” inProc. Amer. Control Conf., 2019, pp. 2114–2120

    work page 2019

  28. [28]

    Learning combinatorial optimization algorithms over graphs,

    H. Dai, E. B. Khalil, Y . Zhang, B. Dilkina, and L. Song, “Learning combinatorial optimization algorithms over graphs,” inProc. NeurIPS, 2017, pp. 6351–6361

    work page 2017

  29. [29]

    Machine learning for combinatorial optimization: A methodological tour d’horizon,

    Y . Bengio, A. Lodi, and A. Prouvost, “Machine learning for combinatorial optimization: A methodological tour d’horizon,”European Journal of Operational Research, vol. 290, no. 2, pp. 405–421, 2021

    work page 2021

  30. [30]

    Equilibrium and dynamics of local voltage control in distribution systems,

    M. Farivar, L. Chen, and S. Low, “Equilibrium and dynamics of local voltage control in distribution systems,” inProc. IEEE Conf. Dec. Control, 2013, pp. 4329–4334

    work page 2013

  31. [31]

    case8adn.m

    “case8adn.m.” [Online]. Available: https://drive.google.com/file/d/ 1cyqHKITynbqEjdwFYNWN1APS6a0VQWMe/view?usp=sharing

    work page

  32. [32]

    Rudin,Functional Analysis

    W. Rudin,Functional Analysis. McGraw-Hill, 1991

    work page 1991

  33. [33]

    Canonical piecewise-linear approximations,

    J.-N. Lin and R. Unbehauen, “Canonical piecewise-linear approximations,” IEEE Trans. Circuits Syst. I, vol. 39, no. 8, pp. 697–699, 1992

    work page 1992