pith. sign in

arxiv: 2605.20212 · v1 · pith:NTMKJIHMnew · submitted 2026-04-23 · 🧮 math.GM

Robust Chance Constrained Complex Zero-Sum Games

Raneem Madani (L2S) , Abdel Lisser (L2S) , Zeno Toffano (L2S) This is my paper

Pith reviewed 2026-05-21 09:32 UTC · model grok-4.3

classification 🧮 math.GM
keywords complexzero-sumdistributionsframeworkgamemodelmomentsstrategies
0
0 comments X

The pith

Unified framework for complex zero-sum games with chance constraints that converts probabilistic constraints into convex second-order cone programs under various distribution assumptions.

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

The work starts by mapping complex numbers to pairs of real numbers so that standard convex analysis tools still apply. This mapping lets the authors prove that the classic minimax theorem still holds when payoffs and strategies are complex. They then define mixed strategies that can weight both the real and imaginary parts of the payoff matrix. To handle uncertainty, they introduce a chance-constrained version where the payoff must satisfy a probabilistic inequality. For complex elliptically symmetric distributions and for several moment-based ambiguity sets, they show that these chance constraints become deterministic second-order cone constraints. The resulting problems remain convex, so equilibria can be computed reliably. Numerical tests on a transmitter-jammer scenario illustrate how the complex mixed strategies behave under uncertainty.

Core claim

The probabilistic constraints in the complex chance-constrained zero-sum game model admit deterministic second-order cone representations, ensuring convex feasible strategy sets and enabling explicit characterization of the complex game value.

Load-bearing premise

The uncertainty in the payoff matrices is assumed to belong to the class of Complex Elliptically Symmetric random variables or to one of the three specified moment-based ambiguity sets; if the actual distribution falls outside these classes the second-order cone reformulations no longer hold.

Figures

Figures reproduced from arXiv: 2605.20212 by Abdel Lisser (L2S), Raneem Madani (L2S), Zeno Toffano (L2S).

Figure 1
Figure 1. Figure 1: Waveform-level interaction between a transmitter (Player 1) and a jammer (Player 2). [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical validation of chance constraints for Player 1 (top row) and Player 2 (bottom [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Euclidean norms of the real parts of the equilibrium strategies [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Euclidean norms of the real parts of the equilibrium strategies [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
read the original abstract

This paper develops a unified framework for zero-sum games in which both the pure strategies and the payoff matrices contain complex-valued entries. By leveraging a linear isomorphism between complex and real vector spaces, we extend key results from real-valued convex analysis to the complex domain, establishing the validity of the minimax theorem and the preservation of saddle-point structure. Building on this foundation, we formulate a complex zero-sum game model that enables mixed strategies to interact with the real and imaginary components of the payoff matrix, and we characterize its saddle-point equilibrium through associated primal and dual problems. To incorporate uncertainty, we introduce a complex chance-constrained zero-sum game model (3CP) that handles individual probabilistic constraints defined by complex linear functionals. We first study the 3CP formulation under known exact distributions, focusing on Complex Elliptically Symmetric random variables, which generalize the complex Gaussian family. The framework is then extended to moments-based ambiguity sets, including: (i) distributions with known first two moments, (ii) distributions with unknown second-order moments, and (iii) fully distributed with unknown moments. In all cases, the probabilistic constraints admit deterministic second-order cone representations, ensuring convex feasible strategy sets and enabling explicit characterization of the complex game value. Numerical experiments, including a transmitter--jammer waveform interaction model, show how the proposed framework captures the behavior of complex mixed strategies. Additionally, we evaluate out-of-sample rates and confirm that practical behavior closely aligns with the theoretical guarantees.

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 paper develops a unified framework for complex-valued zero-sum games with chance constraints. It leverages a linear isomorphism C^n ≅ R^{2n} to extend real-valued minimax theorems and saddle-point results to the complex domain, formulates a complex chance-constrained zero-sum game (3CP) with individual probabilistic constraints on complex linear functionals, and claims that these constraints admit exact deterministic second-order cone representations under Complex Elliptically Symmetric distributions and three moment-based ambiguity sets (known first two moments, unknown second-order moments, and fully unknown moments). This yields convex feasible strategy sets and explicit characterization of the complex game value. Numerical experiments on a transmitter-jammer waveform model are included to illustrate behavior and out-of-sample performance.

Significance. If the SOC reformulations are rigorously verified, the work would offer a convex-optimization approach to robust complex games with direct relevance to signal processing and communications applications. The extension to CES distributions (generalizing complex Gaussians) and multiple ambiguity sets strengthens the robustness analysis, while the numerical results provide concrete illustration of complex mixed strategies. The framework builds on external convex-analysis results, which is a positive feature for reproducibility of the core claims.

major comments (1)
  1. [Abstract and 3CP formulation] Abstract and the section introducing the 3CP model: the claim that probabilistic constraints defined by complex linear functionals (e.g., Re(w^H z) or |w^H z|) admit deterministic SOC representations for CES distributions and the three moment ambiguity sets is load-bearing for the convexity and explicit game-value results. The linear isomorphism to real space is invoked to import real-valued SOC results, but it is unclear whether the induced quadratic covariance terms properly inherit and enforce the Hermitian symmetry of the complex covariance matrix (including real/imaginary cross-block structure). Standard real SOC reformulations assume arbitrary positive-definite covariances; without explicit verification that the block structure preserves SOC-representability without extra constraints, the deterministic reformulation may not hold exactly as stated.
minor comments (2)
  1. The abstract refers to 'complex mixed strategies' interacting with real and imaginary components of the payoff matrix; a brief clarifying sentence on the precise definition of these mixed strategies (e.g., how support is defined over complex vectors) would improve readability.
  2. Numerical experiments section: while out-of-sample rates are mentioned, reporting the specific sample sizes used for the ambiguity-set constructions and the exact parameter values for the CES distributions would aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The single major comment raises a valid technical question about the preservation of Hermitian symmetry and SOC representability under the complex-to-real isomorphism. We address this point directly below and will incorporate clarifications in the revision.

read point-by-point responses

  1. Referee: [Abstract and 3CP formulation] Abstract and the section introducing the 3CP model: the claim that probabilistic constraints defined by complex linear functionals (e.g., Re(w^H z) or |w^H z|) admit deterministic SOC representations for CES distributions and the three moment ambiguity sets is load-bearing for the convexity and explicit game-value results. The linear isomorphism to real space is invoked to import real-valued SOC results, but it is unclear whether the induced quadratic covariance terms properly inherit and enforce the Hermitian symmetry of the complex covariance matrix (including real/imaginary cross-block structure). Standard real SOC reformulations assume arbitrary positive-definite covariances; without explicit verification that the block structure preserves SOC-representability without extra constraints, the deterministic reformulation may not hold exactly as stated.

    Authors: We appreciate the referee's observation on this foundational technical detail. The linear isomorphism φ: ℂⁿ → ℝ^{2n} maps z = x + iy to the stacked real vector (x; y). When the complex covariance Σ is Hermitian positive semidefinite, the induced real covariance takes the block form [Re(Σ), −Im(Σ); Im(Σ), Re(Σ)], which is symmetric and positive semidefinite. For the chance constraints involving Re(wᴴz) or |wᴴz|, the relevant quadratic term wᴴΣw translates exactly to a quadratic form on the real block matrix; the effective variance scalar remains identical. Consequently, the standard real-valued SOC reformulations for individual chance constraints under CES distributions and the three moment-based ambiguity sets carry over without requiring additional constraints beyond those already present in the real case. To make this inheritance fully explicit and address the concern, we will add a short remark (or appendix paragraph) in the revised manuscript that (i) states the precise block structure of the real covariance, (ii) verifies that the quadratic forms arising from the complex linear functionals lie in the SOC cone under the same conditions as the real case, and (iii) confirms that Hermitian symmetry is automatically enforced by construction of the isomorphism. This addition will strengthen the exposition without altering the main results. revision: yes

Circularity Check

0 steps flagged

Derivation relies on external convex-analysis results via isomorphism; no reduction to self-inputs or fitted predictions.

full rationale

The paper establishes the complex minimax theorem and SOC-representable chance constraints by mapping via linear isomorphism C^n ≅ R^{2n} to known real-valued results for elliptically symmetric distributions and moment ambiguity sets. This is an extension of independent external theory rather than a self-definitional loop, fitted-parameter prediction, or load-bearing self-citation chain. No equations in the abstract or described framework equate the game value or feasible sets to their own inputs by construction; the claims remain falsifiable against the stated distribution classes and Hermitian structure assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the linear isomorphism preserving saddle-point structure and on the chosen distribution families admitting SOC representations; no new free parameters or invented entities are introduced beyond standard convex and probabilistic assumptions.

axioms (2)
  • domain assumption Linear isomorphism between complex and real vector spaces extends key results from real-valued convex analysis to the complex domain
    Invoked to establish validity of the minimax theorem and saddle-point structure for complex games.
  • domain assumption Complex Elliptically Symmetric distributions and the listed moment-based ambiguity sets permit deterministic second-order cone representations of the chance constraints
    Required for the convex feasible sets and explicit game-value characterization.

pith-pipeline@v0.9.0 · 5805 in / 1357 out tokens · 37539 ms · 2026-05-21T09:32:05.712478+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

59 extracted references · 59 canonical work pages · 1 internal anchor

  1. [1]

    Nonlinear programming in complex space: Sufficient conditions and duality.Journal of Mathematical Analysis and Applications, 38:619–632, 1972

    Robert A Abrams. Nonlinear programming in complex space: Sufficient conditions and duality.Journal of Mathematical Analysis and Applications, 38:619–632, 1972. ISSN 0022- 247X. doi: https://doi.org/10.1016/0022-247X(72)90073-X

    work page doi:10.1016/0022-247x(72)90073-x 1972

  2. [2]

    A variable neighborhood search algorithm for massive mimo resource allocation

    Pablo Adasme and Abdel Lisser. A variable neighborhood search algorithm for massive mimo resource allocation. InMobile Web and Intelligent Information Systems, pages 3–15, Cham, 2019. Springer International Publishing. ISBN 978-3-030-27192-3

    work page 2019

  3. [3]

    A rewriting system for convex optimization problems.Journal of Control and Decision, 5:42–60, 2018

    Akshay Agrawal, Robin Verschueren, Steven Diamond, and Stephen Boyd. A rewriting system for convex optimization problems.Journal of Control and Decision, 5:42–60, 2018. doi: 10.1080/23307706.2017.1397554

    work page doi:10.1080/23307706.2017.1397554 2018

  4. [4]

    McGraw-Hill New York, 1979

    Lars Valerian Ahlfors and Lars V Ahlfors.Complex analysis, volume 3. McGraw-Hill New York, 1979

    work page 1979

  5. [5]

    The basic concepts and constructions of general topology

    AV Arkhangel’skiˇ ı et al. The basic concepts and constructions of general topology. In General Topology I: Basic Concepts and Constructions Dimension Theory, pages 1–90. Springer, 1990

    work page 1990

  6. [6]

    Random-payoff two-person zero-sum games.Operations Research, 22:1243– 1251, 1974

    Roger A Blau. Random-payoff two-person zero-sum games.Operations Research, 22:1243– 1251, 1974. doi: https://doi.org/10.1287/opre.22.6.1243

    work page doi:10.1287/opre.22.6.1243 1974

  7. [7]

    On distributionally robust chance-constrained linear programs.Journal of Optimization Theory and Applications, 130:1–22, 2006

    Giuseppe Carlo Calafiore and L El Ghaoui. On distributionally robust chance-constrained linear programs.Journal of Optimization Theory and Applications, 130:1–22, 2006. doi: https://doi.org/10.1007/s10957-006-9084-x

    work page doi:10.1007/s10957-006-9084-x 2006

  8. [8]

    Constrained games and linear programming.Proceedings of the National Academy of Sciences, 39:639–641, 1953

    Abraham Charnes. Constrained games and linear programming.Proceedings of the National Academy of Sciences, 39:639–641, 1953. doi: https://doi.org/10.1073/pnas.39.7.639

    work page doi:10.1073/pnas.39.7.639 1953

  9. [9]

    Chance-constrained programming.Management science, 6:73–79, 1959

    Abraham Charnes and William W Cooper. Chance-constrained programming.Management science, 6:73–79, 1959. doi: https://doi.org/10.1287/mnsc.6.1.73. 23

    work page doi:10.1287/mnsc.6.1.73 1959

  10. [10]

    A second-order cone programming approach for linear programs with joint probabilistic constraints.Operations Research Letters, 40:325–328,

    Jianqiang Cheng and Abdel Lisser. A second-order cone programming approach for linear programs with joint probabilistic constraints.Operations Research Letters, 40:325–328,

    work page

  11. [11]

    doi: https://doi.org/10.1016/j.orl.2012.06.008

    ISSN 0167-6377. doi: https://doi.org/10.1016/j.orl.2012.06.008

    work page doi:10.1016/j.orl.2012.06.008 2012

  12. [12]

    Distributionally robust stochastic knap- sackproblem.SIAM Journal on Optimization, 24:1485–1506, 2014

    Jianqiang Cheng, Erick Delage, and Abdel Lisser. Distributionally robust stochastic knap- sackproblem.SIAM Journal on Optimization, 24:1485–1506, 2014. doi: 10.1137/130915315

    work page doi:10.1137/130915315 2014

  13. [13]

    Random-payoff two-person zero-sum game with joint chance constraints.European Journal of Operational Research, 252:213– 219, 2016

    Jianqiang Cheng, Janny Leung, and Abdel Lisser. Random-payoff two-person zero-sum game with joint chance constraints.European Journal of Operational Research, 252:213– 219, 2016. ISSN 0377-2217. doi: https://doi.org/10.1016/j.ejor.2015.12.024

    work page doi:10.1016/j.ejor.2015.12.024 2016

  14. [14]

    Modelling and simulation of non-rayleigh radar clutter

    Ernesto Conte, M Longo, and M Lops. Modelling and simulation of non-rayleigh radar clutter. InIEE Proceedings F (Radar and Signal Processing), volume 138, pages 121–130. IET, 1991. doi: https://doi.org/10.1049/ip-f-2.1991.0018

    work page doi:10.1049/ip-f-2.1991.0018 1991

  15. [15]

    B. D. Craven and B. Mond. On duality in complex linear programming.Journal of the Australian Mathematical Society, 16:172–175, 1973. doi: 10.1017/S144678870001418X

    work page doi:10.1017/s144678870001418x 1973

  16. [16]

    A proof of the equivalence of the programming problem and the game problem.Activity analysis of production and allocation, 13:330–335, 1951

    George B Dantzig. A proof of the equivalence of the programming problem and the game problem.Activity analysis of production and allocation, 13:330–335, 1951

    work page 1951

  17. [17]

    Linear programming under uncertainty.Management science, 1:197–206,

    George B Dantzig. Linear programming under uncertainty.Management science, 1:197–206,

    work page

  18. [18]

    doi: https://doi.org/10.1287/mnsc.1.3-4.197

    work page doi:10.1287/mnsc.1.3-4.197

  19. [19]

    Duality for a class of nondifferentiable mathematical programming problems in complex space.Journal of Mathematical Analysis and Appli- cations, 101:1–11, 1984

    Neelam Datta and Davinder Bhatia. Duality for a class of nondifferentiable mathematical programming problems in complex space.Journal of Mathematical Analysis and Appli- cations, 101:1–11, 1984. ISSN 0022-247X. doi: https://doi.org/10.1016/0022-247X(84) 90053-2

    work page doi:10.1016/0022-247x(84 1984

  20. [20]

    Cvxpy: A python-embedded modeling language for convex optimization.Journal of Machine Learning Research, 17:1–5, 2016

    Steven Diamond and Stephen Boyd. Cvxpy: A python-embedded modeling language for convex optimization.Journal of Machine Learning Research, 17:1–5, 2016. URLhttp: //jmlr.org/papers/v17/15-408.html

    work page 2016

  21. [21]

    On nonlinear programming in complex spaces.Journal of Mathematical Analysis and Applications, 164:399–416, 1992

    Oscar Ferrero. On nonlinear programming in complex spaces.Journal of Mathematical Analysis and Applications, 164:399–416, 1992. ISSN 0022-247X. doi: https://doi.org/10. 1016/0022-247X(92)90123-U

    work page 1992

  22. [22]

    Probability content of regions under spherical normal distributions, IV: The distribution of homogeneous and non-homogeneous quadratic functions of normal variables,

    NathanielRGoodman. Statisticalanalysisbasedonacertainmultivariatecomplexgaussian distribution(anintroduction).The Annals of mathematical statistics, 34:152–177, 1963. doi: 10.1214/aoms/1177704250

    work page doi:10.1214/aoms/1177704250 1963

  23. [23]

    The Complex Gradient Operator and the CR-Calculus

    Ken Kreutz-Delgado. The complex gradient operator and the cr-calculus.arXiv preprint arXiv:0906.4835, 2009

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  24. [24]

    Complex elliptically symmetric distributions.Communica- tions in Statistics-Theory and Methods, 15:3693–3718, 1986

    PR Krishnaiah and Jugan Lin. Complex elliptically symmetric distributions.Communica- tions in Statistics-Theory and Methods, 15:3693–3718, 1986. doi: https://doi.org/10.1080/ 03610928608829341

    work page 1986

  25. [25]

    Linear programming in complex space.Journal of Mathematical Anal- ysis and Applications, 14:44–62, 1966

    Norman Levinson. Linear programming in complex space.Journal of Mathematical Anal- ysis and Applications, 14:44–62, 1966. ISSN 0022-247X. doi: https://doi.org/10.1016/ 0022-247X(66)90061-8

    work page 1966

  26. [26]

    Cogni- tive radar waveform design against signal-dependent modulated jamming.Progress in Electromagnetics Research B, 80:59–77, 2018

    Shuping Lu, Guolong Cui, Xianxiang Yu, Lingjiang Kong, and Xiaobo Yang. Cogni- tive radar waveform design against signal-dependent modulated jamming.Progress in Electromagnetics Research B, 80:59–77, 2018. URLhttps://api.semanticscholar.org/ CorpusID:115960839

    work page 2018

  27. [27]

    Chanceconstrainedoptimizationwithcomplexvariables

    RaneemMadaniandAbdelLisser. Chanceconstrainedoptimizationwithcomplexvariables. 24 InOptimization and Learning, pages 279–290. Springer Nature Switzerland, 2026. ISBN 978-3-032-13589-6. doi: https://doi.org/10.1007/978-3-032-13589-6_21

    work page doi:10.1007/978-3-032-13589-6_21 2026

  28. [28]

    Middleton

    D. Middleton. Man-made noise in urban environments and transportation systems: Models and measurements.IEEE Transactions on Communications, 21:1232–1241, 1973. doi: 10.1109/TCOM.1973.1091566

    work page doi:10.1109/tcom.1973.1091566 1973

  29. [29]

    Game theory in complex space.Opsearch, 19:1–11, 1982

    B Mond and GJ Murray. Game theory in complex space.Opsearch, 19:1–11, 1982

    work page 1982

  30. [30]

    A minimax theorem for matrix games in complex space.Opsearch, 20:25–34, 1983

    B Mond and GJ Murray. A minimax theorem for matrix games in complex space.Opsearch, 20:25–34, 1983

    work page 1983

  31. [31]

    John J. Murray. A solution technique for complex matrix games. Technical report, De- fense Technical Information Center, 1983. URLhttps://apps.dtic.mil/sti/html/tr/ ADA137787/

    work page 1983

  32. [32]

    Equilibrium points in n-person games.Proceedings of the national academy of sciences, 36:48–49, 1950

    John F Nash Jr. Equilibrium points in n-person games.Proceedings of the national academy of sciences, 36:48–49, 1950. doi: https://doi.org/10.1073/pnas.36.1.48

    work page doi:10.1073/pnas.36.1.48 1950

  33. [33]

    von Neumann

    J. von Neumann. Zur theorie der gesellschaftsspiele.Mathematische Annalen, 100:295–320,

    work page

  34. [34]

    URLhttp://eudml.org/doc/159291

    work page

  35. [35]

    Random games under ellipti- cally distributed dependent joint chance constraints.Journal of Optimization Theory and Applications, 195:249–264, 2022

    Hoang Nam Nguyen, Abdel Lisser, and Vikas Vikram Singh. Random games under ellipti- cally distributed dependent joint chance constraints.Journal of Optimization Theory and Applications, 195:249–264, 2022. doi: https://doi.org/10.1007/s10957-022-02077-0

    work page doi:10.1007/s10957-022-02077-0 2022

  36. [36]

    Distributionally robust chance- constrained markov decision processes with random payoff.Applied Mathematics & Opti- mization, 90:25, 2024

    Hoang Nam Nguyen, Abdel Lisser, and Vikas Vikram Singh. Distributionally robust chance- constrained markov decision processes with random payoff.Applied Mathematics & Opti- mization, 90:25, 2024. doi: https://doi.org/10.1007/s00245-024-10167-w

    work page doi:10.1007/s00245-024-10167-w 2024

  37. [37]

    A complex generalized gaussian distribu- tion—characterization, generation, and estimation.IEEE Transactions on Signal Process- ing, 58:1427–1433, 2009

    Mike Novey, Tülay Adali, and Anindya Roy. A complex generalized gaussian distribu- tion—characterization, generation, and estimation.IEEE Transactions on Signal Process- ing, 58:1427–1433, 2009. doi: https://doi.org/10.1109/TSP.2009.2036049

    work page doi:10.1109/tsp.2009.2036049 2009

  38. [38]

    Ollila and V

    E. Ollila and V. Koivunen. Generalized complex elliptical distributions. InProcessing Workshop Proceedings, 2004 Sensor Array and Multichannel Signal, pages 460–464, 2004. doi: 10.1109/SAM.2004.1502990

    work page doi:10.1109/sam.2004.1502990 2004

  39. [39]

    Tyler, Visa Koivunen, and H

    Esa Ollila, David E. Tyler, Visa Koivunen, and H. Vincent Poor. Complex elliptically symmetricdistributions: Survey, newresultsandapplications.IEEE Transactions on Signal Processing, 60:5597–5625, 2012. doi: 10.1109/TSP.2012.2212433

    work page doi:10.1109/tsp.2012.2212433 2012

  40. [40]

    General sum games with joint chance constraints.Operations Research Letters, 46:482–486, 2018

    Shen Peng, Vikas Vikram Singh, and Abdel Lisser. General sum games with joint chance constraints.Operations Research Letters, 46:482–486, 2018. ISSN 0167-6377. doi: https: //doi.org/10.1016/j.orl.2018.07.003

    work page doi:10.1016/j.orl.2018.07.003 2018

  41. [41]

    Chance-constrained games with mixture distributions.Mathematical Methods of Operations Research, 94:71– 97, 2021

    Shen Peng, Navnit Yadav, Abdel Lisser, and Vikas Vikram Singh. Chance-constrained games with mixture distributions.Mathematical Methods of Operations Research, 94:71– 97, 2021. doi: https://doi.org/10.1007/s00186-021-00747-9

    work page doi:10.1007/s00186-021-00747-9 2021

  42. [42]

    Dobre, and Jiangzhou Wang

    Zhangjie Peng, Zhibo Zhang, Cunhua Pan, Marco Di Renzo, Octavia A. Dobre, and Jiangzhou Wang. Beamforming optimization for active ris-aided multiuser communica- tions with hardware impairments.IEEE Transactions on Wireless Communications, 23: 9884–9898, 2024. doi: 10.1109/TWC.2024.3367131

    work page doi:10.1109/twc.2024.3367131 2024

  43. [43]

    Frameworks and results in distributionally robust optimization.Open Journal of Mathematical Optimization, 3:1–85, 2022

    Hamed Rahimian and Sanjay Mehrotra. Frameworks and results in distributionally robust optimization.Open Journal of Mathematical Optimization, 3:1–85, 2022. doi: https://doi. org/10.5802/ojmo.15. 25

    work page doi:10.5802/ojmo.15 2022

  44. [44]

    Complementarity formulation of games with random payoffs.Computational Management Science, 20:35,

    Rossana Riccardi, Giorgia Oggioni, Elisabetta Allevi, and Abdel Lisser. Complementarity formulation of games with random payoffs.Computational Management Science, 20:35,

    work page

  45. [45]

    doi: https://doi.org/10.1007/s10287-023-00467-x

    work page doi:10.1007/s10287-023-00467-x

  46. [46]

    Convex analysis.Princeton Mathematical Series, 28, 1970

    R Rockafellar. Convex analysis.Princeton Mathematical Series, 28, 1970

    work page 1970

  47. [47]

    Schreier and L.L

    P.J. Schreier and L.L. Scharf. Second-order analysis of improper complex random vectors and processes.IEEE Transactions on Signal Processing, 51:714–725, 2003. doi: 10.1109/ TSP.2002.808085

    work page arXiv 2003

  48. [48]

    Jieqiu Shao, Mantas Naris, John Hauser, and Marco M. Nicotra. Solving quantum optimal control problems using projection-operator-based newton steps.Phys. Rev. A, 109:012609, Jan 2024. doi: 10.1103/PhysRevA.109.012609

    work page doi:10.1103/physreva.109.012609 2024

  49. [49]

    A characterization of nash equilibrium for the games withrandompayoffs.Journal of Optimization Theory and Applications, 178:998–1013, 2018

    Vikas Vikram Singh and Abdel Lisser. A characterization of nash equilibrium for the games withrandompayoffs.Journal of Optimization Theory and Applications, 178:998–1013, 2018. doi: https://doi.org/10.1007/s10957-018-1343-0

    work page doi:10.1007/s10957-018-1343-0 2018

  50. [50]

    Existence of nash equilibrium for chance-constrained games.Operations Research Letters, 44:640–644, 2016

    Vikas Vikram Singh, Oualid Jouini, and Abdel Lisser. Existence of nash equilibrium for chance-constrained games.Operations Research Letters, 44:640–644, 2016. ISSN 0167-6377. doi: https://doi.org/10.1016/j.orl.2016.07.013

    work page doi:10.1016/j.orl.2016.07.013 2016

  51. [51]

    Distributionally robust chance- constrained games: existence and characterization of nash equilibrium.Optimization Let- ters, 11:1385–1405, 2017

    Vikas Vikram Singh, Oualid Jouini, and Abdel Lisser. Distributionally robust chance- constrained games: existence and characterization of nash equilibrium.Optimization Let- ters, 11:1385–1405, 2017. doi: https://doi.org/10.1007/s11590-016-1077-6

    work page doi:10.1007/s11590-016-1077-6 2017

  52. [52]

    An equivalent mathematical program for games with random constraints.Statistics & Probability Letters, 174:109092, 2021

    Vikas Vikram Singh, Abdel Lisser, and Monika Arora. An equivalent mathematical program for games with random constraints.Statistics & Probability Letters, 174:109092, 2021. ISSN 0167-7152. doi: https://doi.org/10.1016/j.spl.2021.109092

    work page doi:10.1016/j.spl.2021.109092 2021

  53. [53]

    On general minimax theorems.Pacific J

    Maurice Sion. On general minimax theorems.Pacific J. Math., 8:171–176, 1958. URL http://dml.mathdoc.fr/item/1103040253

    work page arXiv 1958

  54. [54]

    Unconstrained optimization of real functions in complex variables.SIAM Journal on Optimization, 22:879–898, 2012

    Laurent Sorber, Marc Van Barel, and Lieven De Lathauwer. Unconstrained optimization of real functions in complex variables.SIAM Journal on Optimization, 22:879–898, 2012. doi: 10.1137/110832124

    work page doi:10.1137/110832124 2012

  55. [55]

    Predicting nash equilibria in bimatrix games using a robust bichannel convolutional neural network.IEEE Transactions on Artificial Intelligence, 5: 2358–2370, 2024

    Dawen Wu and Abdel Lisser. Predicting nash equilibria in bimatrix games using a robust bichannel convolutional neural network.IEEE Transactions on Artificial Intelligence, 5: 2358–2370, 2024. doi: 10.1109/TAI.2023.3321584

    work page doi:10.1109/tai.2023.3321584 2024

  56. [56]

    Distributionally robust chance constrained games under wasserstein ball.Operations Research Letters, 51:315–321, 2023

    Tian Xia, Jia Liu, and Abdel Lisser. Distributionally robust chance constrained games under wasserstein ball.Operations Research Letters, 51:315–321, 2023. ISSN 0167-6377. doi: https://doi.org/10.1016/j.orl.2023.03.015

    work page doi:10.1016/j.orl.2023.03.015 2023

  57. [57]

    Variational inequality for n-player strategic chance-constrained games.SN Computer Science, 4:82, 2022

    Shangyuan Zhang, Makhlouf Hadji, Abdel Lisser, and Yacine Mezali. Variational inequality for n-player strategic chance-constrained games.SN Computer Science, 4:82, 2022. doi: https://doi.org/10.1007/s42979-022-01488-0

    work page doi:10.1007/s42979-022-01488-0 2022

  58. [58]

    A complex-valued neural dynamical optimization approach and its stability analysis.Neural Networks, 61:59–67, 2015

    Songchuan Zhang, Youshen Xia, and Weixing Zheng. A complex-valued neural dynamical optimization approach and its stability analysis.Neural Networks, 61:59–67, 2015. ISSN 0893-6080. doi: https://doi.org/10.1016/j.neunet.2014.10.003

    work page doi:10.1016/j.neunet.2014.10.003 2015

  59. [59]

    Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method.npj Quantum Infor- mation, 10:127, 2024

    Muqing Zheng et al. Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method.npj Quantum Infor- mation, 10:127, 2024. doi: https://doi.org/10.1038/s41534-024-00916-8. 26

    work page doi:10.1038/s41534-024-00916-8 2024