Optimization Letters

, Volume 10, Issue 4, pp 655–665 | Cite as

Maximizing concave piecewise affine functions on the unitary group

  • Stéphane Gaubert
  • Zheng Qu
  • Srinivas Sridharan
Original Paper


We show that a convex relaxation, introduced by Sridharan, McEneaney, Gu and James to approximate the value function of an optimal control problem arising from quantum gate synthesis, is exact. This relaxation applies to the maximization of a class of concave piecewise affine functions over the unitary group.


Convex relaxation Unitary group Optimal control Quantum control Approximate dynamic programming 



Z. Qu carried out parts of this work when she was with INRIA and CMAP, École Polytechnique, CNRS and subsequently with the School of Mathematics of University of Edinburgh. S. Sridharan carried out part of this work when he was with UMA, ENSTA, Palaiseau, France. S. Gaubert and Z. Qu were partially supported by the PGMO Program of FMJH and EDF, and by the program “Ingénierie Numérique & Sécurité” of the French National Agency of Research, Project “MALTHY”, Number ANR-13-INSE-0003.


  1. 1.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. In: MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM)/Mathematical Programming Society (MPS), Philadelphia (2001)Google Scholar
  3. 3.
    Gaubert, S., McEneaney, W.M., Qu, Z.: Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 11), pp. 1054–1061. IEEE (2011)Google Scholar
  4. 4.
    Gaubert, S., Qu, Z., Sridharan, S.: Bundle-based pruning in the max-plus curse of dimensionality free method. In: 21st International Symposium on Mathematical Theory of Networks and Systems, Groningen, The Netherlands (2014)Google Scholar
  5. 5.
    McEneaney, W.M.: A curse-of-dimensionality-free numerical method for solution of certain HJB PDEs. SIAM J. Control Optim. 46(4), 1239–1276 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    McEneaney, W.M.: Complexity reduction, cornices and pruning. In: Tropical and Idempotent Mathematics. Contemporary Mathematics, vol. 495, pp. 293–303. American Mathematical Society, Providence (2009)Google Scholar
  7. 7.
    McEneaney, W.M., Deshpande, A., Gaubert, S.: Curse-of-complexity attenuation in the curse-of-dimensionality-free method for HJB PDEs. In: Proceedings of the 2008 American Control Conference, pp. 4684–4690, Seattle, Washington, USA (2008)Google Scholar
  8. 8.
    McEneaney, W.M., Kluberg, L.J.: Convergence rate for a curse-of-dimensionality-free method for a class of HJB PDEs. SIAM J. Control Optim. 48(5), 3052–3079 (2009/2010)Google Scholar
  9. 9.
    Pereira, M.V.F., Pinto, L.M.V.G.: Multi-stage stochastic optimization applied to energy planning. Math. Program. 52, 359375 (1991)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Qu, Z.: Contraction of Riccati flows applied to the convergence analysis of a max-plus curse-of-dimensionality-free method. SIAM J. Control Optim. 52(5), 2677–2706 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Sridharan, S., Gu, M., James, M.R., McEneaney, W.M.: Reduced-complexity numerical method for optimal gate synthesis. Phys. Rev. A 82, 042319 (2010)CrossRefGoogle Scholar
  13. 13.
    Shapiro, A.: Analysis of stochastic dual dynamic programming method. Eur. J. Oper. Res. 209(1), 63–72 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sridharan, S., McEneaney, W.M.: Deterministic filtering for optimal attitude estimation on SO(3) using max-plus methods. In: Proceedings of the European Control Conference ECC 2013, Zurich, pp. 2220–2225 (2013)Google Scholar
  15. 15.
    Sridharan, S., McEneaney, W., Gu, M., James, M.R.: A reduced complexity min-plus solution method to the optimal control of closed quantum systems. Appl. Math. Optim. 70(3), 469–510 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.INRIA and CMAP, École Polytechnique, CNRSPalaiseau CedexFrance
  2. 2.Department of MathematicsThe University of Hong KongHong KongHong Kong
  3. 3.Informatics Department at University of SussexBrightonUK

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