Constrained Multi-Objective Reinforcement Learning with Max-Min Criterion

Amir Leshem; Giseung Park; Hyunyoung Nam; Woohyeon Byeon; Youngchul Sung

arxiv: 2605.31388 · v1 · pith:VSP2Y6H2new · submitted 2026-05-29 · 💻 cs.LG

Constrained Multi-Objective Reinforcement Learning with Max-Min Criterion

Giseung Park , Hyunyoung Nam , Woohyeon Byeon , Amir Leshem , Youngchul Sung This is my paper

Pith reviewed 2026-06-28 23:27 UTC · model grok-4.3

classification 💻 cs.LG

keywords multi-objective reinforcement learningmax-min criterionconstrained optimizationfairnessconvergence analysistabular reinforcement learning

0 comments

The pith

A new MORL framework integrates the max-min criterion with explicit constraints to produce fair and feasible policies.

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

The paper develops a multi-objective reinforcement learning method that applies the max-min criterion to promote fairness across conflicting objectives while enforcing explicit constraints on the learned policies. It supplies a theoretical foundation including convergence analysis for the resulting algorithm in tabular environments. Practical validation occurs through simulations in building thermal control, multi-objective locomotion, and greenhouse-gas-aware traffic management, where the approach balances fairness with constraint satisfaction.

Core claim

We propose a MORL framework that integrates the max-min criterion with explicit constraint satisfaction. We establish a theoretical foundation for the proposed framework and validate the resulting algorithm through convergence analysis and experiments in tabular settings.

What carries the argument

The max-min criterion combined with explicit constraint satisfaction, which enforces fairness by maximizing the worst-case objective while keeping policies inside the feasible set.

Load-bearing premise

The max-min criterion combined with explicit constraints yields policies that remain fair and feasible under the problem dynamics assumed in the convergence analysis.

What would settle it

A simple tabular multi-objective task where the trained policy either violates a stated constraint or produces unequal objective values despite the max-min objective.

read the original abstract

Multi-Objective Reinforcement Learning (MORL) extends standard RL by optimizing policies with respect to multiple, often conflicting, objectives. While max-min MORL has emerged as an effective approach for promoting fairness, its applicability remains limited, particularly when constraints must be incorporated. In this paper, we propose a MORL framework that integrates the max-min criterion with explicit constraint satisfaction. We establish a theoretical foundation for the proposed framework and validate the resulting algorithm through convergence analysis and experiments in tabular settings. We further demonstrate the practical relevance of our approach in simulated building thermal control, multi-objective locomotion control, and greenhouse-gas-emission-aware traffic management. Across these domains, our method effectively balances fairness and constraint satisfaction in multi-objective decision-making.

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 extends max-min MORL to handle explicit constraints with tabular convergence claims and a few application demos, but the evidence is mostly high-level.

read the letter

The paper's main move is to combine the max-min fairness criterion with hard constraints inside a MORL setup. They claim a theoretical foundation plus convergence analysis for the resulting algorithm in tabular cases, then show results on building thermal control, multi-objective locomotion, and emission-aware traffic management.

What stands out is the practical framing: max-min alone often ignores feasibility limits, so adding explicit constraints addresses a real gap in control and decision tasks. The tabular convergence check and the three simulated domains give some grounding that the approach can produce feasible, fair policies under the tested dynamics.

The soft spots are the lack of visible equations, proof outlines, or quantitative metrics. Without baselines, effect sizes, or details on how constraints are enforced during learning, it's difficult to judge whether the method improves on prior constrained MORL work or simply reproduces known behavior. The applications read as illustrative rather than rigorous stress tests.

The work is aimed at people already working on fairness-aware or constrained RL. A reader who needs a ready-to-use method for those settings could extract value once the full proofs and numbers are checked. The structure is internally consistent and does not rely on circular definitions or invented entities.

I would send this to peer review. The idea is straightforward enough that referees can evaluate the theory and experiments directly.

Referee Report

0 major / 2 minor

Summary. The paper proposes a MORL framework integrating the max-min criterion with explicit constraint satisfaction. It claims to establish a theoretical foundation for the framework, provide convergence analysis, and validate the algorithm via experiments in tabular settings as well as three application domains (building thermal control, multi-objective locomotion control, and GHG-emission-aware traffic management), showing effective balancing of fairness and constraint satisfaction.

Significance. If the claimed convergence analysis and experimental results hold, the work addresses a relevant gap in fair constrained MORL by combining max-min fairness with explicit constraints, with potential practical relevance in the demonstrated domains. The tabular convergence analysis and multi-domain experiments would be strengths if they include rigorous metrics and baselines.

minor comments (2)

[Abstract] Abstract: the claim of a 'theoretical foundation' and 'convergence analysis' is stated without any equations, proof sketches, or specific metrics; this makes the strength of the central claims difficult to evaluate from the provided text.
[Abstract] Abstract: no baselines, quantitative results, or constraint classes are mentioned despite the emphasis on explicit constraint satisfaction and experimental validation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the work's relevance to fair constrained MORL, and recommendation for minor revision. We appreciate the acknowledgment of the theoretical foundation, convergence analysis, and multi-domain experiments.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes a MORL framework integrating max-min criterion with constraints, supported by convergence analysis in tabular settings and experiments. No equations, fitted parameters, self-citations, or ansatzes are visible in the provided abstract or claims that reduce any prediction or result to its own inputs by construction. The derivation chain relies on standard RL theory and empirical validation without self-referential definitions or load-bearing self-citations. This is the expected outcome for a framework paper whose central claims remain independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; ledger left empty.

pith-pipeline@v0.9.1-grok · 5658 in / 975 out tokens · 20034 ms · 2026-06-28T23:27:03.607707+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 12 canonical work pages · 2 internal anchors

[1]

URLhttp://proceedings.mlr

PMLR, 2019. URLhttp://proceedings.mlr. press/v97/abels19a.html. Achiam, J., Held, D., Tamar, A., and Abbeel, P. Constrained policy optimization. InInternational conference on ma- chine learning, pp. 22–31. PMLR, 2017. Alegre, L. N., Bazzan, A. L., Roijers, D. M., Now´e, A., and da Silva, B. C. Sample-efficient multi-objective learning via generalized poli...

work page doi:10.1613/jair.575 2019
[2]

Che, F., Xiao, C., Mei, J., Dai, B., Gummadi, R., Ramirez, O

URL https://openreview.net/forum? id=8tzjEMF0Vq. Che, F., Xiao, C., Mei, J., Dai, B., Gummadi, R., Ramirez, O. A., Harris, C. K., Mahmood, A. R., and Schuurmans, D. Target networks and over-parameterization stabi- lize off-policy bootstrapping with function approxima- tion. InForty-first International Conference on Machine Learning, ICML 2024, Vienna, Aus...

work page doi:10.5555/3545946 2024
[3]

URL http://proceedings

PMLR, 2017. URL http://proceedings. mlr.press/v70/haarnoja17a.html. Haarnoja, T., Zhou, A., Abbeel, P., and Levine, S. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In Dy, J. G. and Krause, A. (eds.),Proceedings of the 35th International Conference on Machine Learn- ing, ICML 2018, Stockholmsm ¨assan, S...

2017
[4]

Hayes, C

URL http://proceedings.mlr.press/ v80/haarnoja18b.html. Hayes, C. F., Radulescu, R., Bargiacchi, E., K¨allstr¨om, J., Macfarlane, M., Reymond, M., Verstraeten, T., Zintgraf, L. M., Dazeley, R., Heintz, F., Howley, E., Irissappane, A. A., Mannion, P., Now ´e, A., de Oliveira Ramos, G., Restelli, M., Vamplew, P., and Roijers, D. M. A practi- cal guide to mu...
[5]

A practical guide to multi-objective reinforcement learning and planning,

doi: 10.1007/S10458-022-09552-Y. URL https: //doi.org/10.1007/s10458-022-09552-y. Horn, R. A. and Johnson, C. R.Matrix analysis. Cambridge university press, 2012. 10 Constrained Multi-Objective Reinforcement Learning with Max-Min Criterion Huang, S. H., Abdolmaleki, A., Vezzani, G., Brakel, P., Mankowitz, D. J., Neunert, M., Bohez, S., Tassa, Y ., Heess, ...

work page doi:10.1007/s10458-022-09552-y 2012
[6]

Hung, W., Huang, B., Hsieh, P., and Liu, X

URL https://proceedings.mlr.press/ v164/huang22a.html. Hung, W., Huang, B., Hsieh, P., and Liu, X. Q-pensieve: Boosting sample efficiency of multi-objective RL through memory sharing of q-snapshots. InThe Eleventh In- ternational Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenRe- view.net, 2023. URL https://openrevie...

work page doi:10.1016/j.jnca.2016.07.008 2023
[7]

Liu, Y ., Ding, J., and Liu, X

URL https://openreview.net/forum? id=fDGPIuCdGi. Liu, Y ., Ding, J., and Liu, X. Ipo: Interior-point policy optimization under constraints, 2019. URL https:// arxiv.org/abs/1910.09615. Lu, H., Herman, D., and Yu, Y . Multi-objective reinforce- ment learning: Convexity, stationarity and pareto opti- mality. InThe Eleventh International Conference on Learni...

work page arXiv 2019
[8]

Miryoosefi, S., Brantley, K., Daume III, H., Dudik, M., and Schapire, R

URL https://proceedings.mlr.press/ v139/malik21a.html. Miryoosefi, S., Brantley, K., Daume III, H., Dudik, M., and Schapire, R. E. Reinforcement learning with convex constraints. In Wallach, H., Larochelle, H., Beygelzimer, A., d'Alch ´e-Buc, F., Fox, E., and Garnett, R. (eds.),Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc.,
[9]

cc/paper_files/paper/2019/file/ 873be0705c80679f2c71fbf4d872df59-Paper

URL https://proceedings.neurips. cc/paper_files/paper/2019/file/ 873be0705c80679f2c71fbf4d872df59-Paper. pdf. M¨uller, A., Alatur, P., Cevher, V ., Ramponi, G., and He, N. Truly no-regret learning in constrained MDPs. InForty- first International Conference on Machine Learning,

2019
[10]

Nesterov, Y

URL https://openreview.net/forum? id=hrWte3nlzr. Nesterov, Y . E. and Spokoiny, V . G. Random gradient- free minimization of convex functions.Found. Com- put. Math., 17(2):527–566, 2017. doi: 10.1007/ S10208-015-9296-2. URL https://doi.org/10. 1007/s10208-015-9296-2. Park, G. and Sung, Y . Reward dimension reduction for scal- able multi-objective reinforc...

work page doi:10.1002/9780470316887 2017
[11]

doi: 10.1609/AAAI.V24I1

AAAI Press, 2010. doi: 10.1609/AAAI.V24I1

work page doi:10.1609/aaai.v24i1 2010
[12]

Speech recognition meets large language model: benchmarking, models, and exploration,

URL https://doi.org/10.1609/aaai. v24i1.7740. Roijers, D. M., Vamplew, P., Whiteson, S., and Dazeley, R. A survey of multi-objective sequential decision-making. J. Artif. Intell. Res., 48:67–113, 2013. doi: 10.1613/JAIR

work page doi:10.1609/aaai 2013
[13]

URL https://doi.org/10.1613/jair. 3987. Saifullah, A., Ferry, D., Li, J., Agrawal, K., Lu, C., and Gill, C. D. Parallel real-time scheduling of dags.IEEE Trans. Parallel Distributed Syst., 25(12):3242–3252, 2014. doi: 10.1109/TPDS.2013.2297919. Schmidt, M., Roux, N. L., and Bach, F. Convergence rates of inexact proximal-gradient methods for convex opti- m...

work page doi:10.1613/jair 2014
[14]

Reward Constrained Policy Optimization

PMLR, 2020. URL http://proceedings. mlr.press/v119/siddique20a.html. Silver, D. Lectures on reinforcement learning. URL: https://www.davidsilver.uk/ teaching/, 2015. Subramanian, S. G., Liu, G., Elmahgiubi, M., Rezaee, K., and Poupart, P. Confidence aware inverse constrained reinforcement learning. InForty-first International Con- ference on Machine Learn...

work page internal anchor Pith review Pith/arXiv arXiv 2020
[15]

Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application

URL https://openreview.net/forum? id=ZJ7Lrtd12x_. Wang, K., Jiang, X., Guan, N., Liu, D., Liu, W., and Deng, Q. Real-time scheduling of DAG tasks with arbitrary deadlines.ACM Trans. Design Autom. Electr. Syst., 24 (6):66:1–66:22, 2019. doi: 10.1145/3358603. Wang, W. and Carreira-Perpi˜n´an, M. ´A. Projection onto the probability simplex: An efficient algo...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1145/3358603 2019
[16]

URL https: //doi.org/10.1109/TSP.2013.2262278

doi: 10.1109/TSP.2013.2262278. URL https: //doi.org/10.1109/TSP.2013.2262278. 13 Constrained Multi-Objective Reinforcement Learning with Max-Min Criterion A. Proof on Optimality Gap Proof. With a slight abuse of notation, let J(π) := [J 1(π),· · ·, J K(π)]⊤ ∈R K and let H(π) denote the expected cumulative entropy ofπ. We express the optimization of (2) an...

work page doi:10.1109/tsp.2013.2262278 2013
[17]

Stationarity condition gives ∀(s, a),−βlog ρ(s, a)P a′ ρ(s, a′) +ξ(s, a) +η u,v,w(s, a) = 0(21) and 1− KX k=1 wk = 0.(22)
[18]

Complementary slackness condition gives ∀(s, a), ξ(s, a)ρ(s, a) = 0.(23) From (21), we derive ∀(s, a), ρ(s, a)P a′ ρ(s, a′) = exp ξ(s, a) +η u,v,w(s, a) β (24) soρ(s, a)>0andξ(s, a) = 0from (23). Therefore, ∀(s, a), ρ(s, a)P a′ ρ(s, a′) = exp ηu,v,w(s, a) β .(25) Inserting (22) and (25), we obtain: min u∈RL + min v,w X s µ0(s)v(s)− LX l=1 ulC(l) (26) ∀s, ...

2004

[1] [1]

URLhttp://proceedings.mlr

PMLR, 2019. URLhttp://proceedings.mlr. press/v97/abels19a.html. Achiam, J., Held, D., Tamar, A., and Abbeel, P. Constrained policy optimization. InInternational conference on ma- chine learning, pp. 22–31. PMLR, 2017. Alegre, L. N., Bazzan, A. L., Roijers, D. M., Now´e, A., and da Silva, B. C. Sample-efficient multi-objective learning via generalized poli...

work page doi:10.1613/jair.575 2019

[2] [2]

Che, F., Xiao, C., Mei, J., Dai, B., Gummadi, R., Ramirez, O

URL https://openreview.net/forum? id=8tzjEMF0Vq. Che, F., Xiao, C., Mei, J., Dai, B., Gummadi, R., Ramirez, O. A., Harris, C. K., Mahmood, A. R., and Schuurmans, D. Target networks and over-parameterization stabi- lize off-policy bootstrapping with function approxima- tion. InForty-first International Conference on Machine Learning, ICML 2024, Vienna, Aus...

work page doi:10.5555/3545946 2024

[3] [3]

URL http://proceedings

PMLR, 2017. URL http://proceedings. mlr.press/v70/haarnoja17a.html. Haarnoja, T., Zhou, A., Abbeel, P., and Levine, S. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In Dy, J. G. and Krause, A. (eds.),Proceedings of the 35th International Conference on Machine Learn- ing, ICML 2018, Stockholmsm ¨assan, S...

2017

[4] [4]

Hayes, C

URL http://proceedings.mlr.press/ v80/haarnoja18b.html. Hayes, C. F., Radulescu, R., Bargiacchi, E., K¨allstr¨om, J., Macfarlane, M., Reymond, M., Verstraeten, T., Zintgraf, L. M., Dazeley, R., Heintz, F., Howley, E., Irissappane, A. A., Mannion, P., Now ´e, A., de Oliveira Ramos, G., Restelli, M., Vamplew, P., and Roijers, D. M. A practi- cal guide to mu...

[5] [5]

A practical guide to multi-objective reinforcement learning and planning,

doi: 10.1007/S10458-022-09552-Y. URL https: //doi.org/10.1007/s10458-022-09552-y. Horn, R. A. and Johnson, C. R.Matrix analysis. Cambridge university press, 2012. 10 Constrained Multi-Objective Reinforcement Learning with Max-Min Criterion Huang, S. H., Abdolmaleki, A., Vezzani, G., Brakel, P., Mankowitz, D. J., Neunert, M., Bohez, S., Tassa, Y ., Heess, ...

work page doi:10.1007/s10458-022-09552-y 2012

[6] [6]

Hung, W., Huang, B., Hsieh, P., and Liu, X

URL https://proceedings.mlr.press/ v164/huang22a.html. Hung, W., Huang, B., Hsieh, P., and Liu, X. Q-pensieve: Boosting sample efficiency of multi-objective RL through memory sharing of q-snapshots. InThe Eleventh In- ternational Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenRe- view.net, 2023. URL https://openrevie...

work page doi:10.1016/j.jnca.2016.07.008 2023

[7] [7]

Liu, Y ., Ding, J., and Liu, X

URL https://openreview.net/forum? id=fDGPIuCdGi. Liu, Y ., Ding, J., and Liu, X. Ipo: Interior-point policy optimization under constraints, 2019. URL https:// arxiv.org/abs/1910.09615. Lu, H., Herman, D., and Yu, Y . Multi-objective reinforce- ment learning: Convexity, stationarity and pareto opti- mality. InThe Eleventh International Conference on Learni...

work page arXiv 2019

[8] [8]

Miryoosefi, S., Brantley, K., Daume III, H., Dudik, M., and Schapire, R

URL https://proceedings.mlr.press/ v139/malik21a.html. Miryoosefi, S., Brantley, K., Daume III, H., Dudik, M., and Schapire, R. E. Reinforcement learning with convex constraints. In Wallach, H., Larochelle, H., Beygelzimer, A., d'Alch ´e-Buc, F., Fox, E., and Garnett, R. (eds.),Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc.,

[9] [9]

cc/paper_files/paper/2019/file/ 873be0705c80679f2c71fbf4d872df59-Paper

URL https://proceedings.neurips. cc/paper_files/paper/2019/file/ 873be0705c80679f2c71fbf4d872df59-Paper. pdf. M¨uller, A., Alatur, P., Cevher, V ., Ramponi, G., and He, N. Truly no-regret learning in constrained MDPs. InForty- first International Conference on Machine Learning,

2019

[10] [10]

Nesterov, Y

URL https://openreview.net/forum? id=hrWte3nlzr. Nesterov, Y . E. and Spokoiny, V . G. Random gradient- free minimization of convex functions.Found. Com- put. Math., 17(2):527–566, 2017. doi: 10.1007/ S10208-015-9296-2. URL https://doi.org/10. 1007/s10208-015-9296-2. Park, G. and Sung, Y . Reward dimension reduction for scal- able multi-objective reinforc...

work page doi:10.1002/9780470316887 2017

[11] [11]

doi: 10.1609/AAAI.V24I1

AAAI Press, 2010. doi: 10.1609/AAAI.V24I1

work page doi:10.1609/aaai.v24i1 2010

[12] [12]

Speech recognition meets large language model: benchmarking, models, and exploration,

URL https://doi.org/10.1609/aaai. v24i1.7740. Roijers, D. M., Vamplew, P., Whiteson, S., and Dazeley, R. A survey of multi-objective sequential decision-making. J. Artif. Intell. Res., 48:67–113, 2013. doi: 10.1613/JAIR

work page doi:10.1609/aaai 2013

[13] [13]

URL https://doi.org/10.1613/jair. 3987. Saifullah, A., Ferry, D., Li, J., Agrawal, K., Lu, C., and Gill, C. D. Parallel real-time scheduling of dags.IEEE Trans. Parallel Distributed Syst., 25(12):3242–3252, 2014. doi: 10.1109/TPDS.2013.2297919. Schmidt, M., Roux, N. L., and Bach, F. Convergence rates of inexact proximal-gradient methods for convex opti- m...

work page doi:10.1613/jair 2014

[14] [14]

Reward Constrained Policy Optimization

PMLR, 2020. URL http://proceedings. mlr.press/v119/siddique20a.html. Silver, D. Lectures on reinforcement learning. URL: https://www.davidsilver.uk/ teaching/, 2015. Subramanian, S. G., Liu, G., Elmahgiubi, M., Rezaee, K., and Poupart, P. Confidence aware inverse constrained reinforcement learning. InForty-first International Con- ference on Machine Learn...

work page internal anchor Pith review Pith/arXiv arXiv 2020

[15] [15]

Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application

URL https://openreview.net/forum? id=ZJ7Lrtd12x_. Wang, K., Jiang, X., Guan, N., Liu, D., Liu, W., and Deng, Q. Real-time scheduling of DAG tasks with arbitrary deadlines.ACM Trans. Design Autom. Electr. Syst., 24 (6):66:1–66:22, 2019. doi: 10.1145/3358603. Wang, W. and Carreira-Perpi˜n´an, M. ´A. Projection onto the probability simplex: An efficient algo...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1145/3358603 2019

[16] [16]

URL https: //doi.org/10.1109/TSP.2013.2262278

doi: 10.1109/TSP.2013.2262278. URL https: //doi.org/10.1109/TSP.2013.2262278. 13 Constrained Multi-Objective Reinforcement Learning with Max-Min Criterion A. Proof on Optimality Gap Proof. With a slight abuse of notation, let J(π) := [J 1(π),· · ·, J K(π)]⊤ ∈R K and let H(π) denote the expected cumulative entropy ofπ. We express the optimization of (2) an...

work page doi:10.1109/tsp.2013.2262278 2013

[17] [17]

Stationarity condition gives ∀(s, a),−βlog ρ(s, a)P a′ ρ(s, a′) +ξ(s, a) +η u,v,w(s, a) = 0(21) and 1− KX k=1 wk = 0.(22)

[18] [18]

Complementary slackness condition gives ∀(s, a), ξ(s, a)ρ(s, a) = 0.(23) From (21), we derive ∀(s, a), ρ(s, a)P a′ ρ(s, a′) = exp ξ(s, a) +η u,v,w(s, a) β (24) soρ(s, a)>0andξ(s, a) = 0from (23). Therefore, ∀(s, a), ρ(s, a)P a′ ρ(s, a′) = exp ηu,v,w(s, a) β .(25) Inserting (22) and (25), we obtain: min u∈RL + min v,w X s µ0(s)v(s)− LX l=1 ulC(l) (26) ∀s, ...

2004