Conic Optimization with Spectral Functions on Euclidean Jordan Algebras

Published Online:https://doi.org/10.1287/moor.2022.1324

References

  • [1] Andersen M, Dahl J, Liu Z, Vandenberghe L, Sra S, Nowozin S, Wright S (2011) Interior-point methods for large-scale cone programming. Sra S, Wright SJ, Nowozin S, eds. Optimization for Machine Learning, vol. 5583 (MIT Press, Cambridge, MA).CrossrefGoogle Scholar
  • [2] Baes M (2007) Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras. Linear Algebra Appl. 422(2–3):664–700.CrossrefGoogle Scholar
  • [3] Ben-Tal A, Nemirovski A (2001) Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MOS-SIAM Series on Optimization (SIAM, Philadelphia).CrossrefGoogle Scholar
  • [4] Borchers B (1999) CSDP, a C library for semidefinite programming. Optim. Methods Software 11(1–4):613–623.CrossrefGoogle Scholar
  • [5] Boyd S, Boyd SP, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, New York).CrossrefGoogle Scholar
  • [6] Carlen E (2010) Trace inequalities and quantum entropy: An introductory course. Sims R, Ueltschi D, eds. Entropy and the Quantum. Contemporary Mathematics, vol. 529 (American Mathematical Society, Providence, RI), 73–140.CrossrefGoogle Scholar
  • [7] Coey C (2022) Interior point and outer approximation methods for conic optimization. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
  • [8] Coey C, Kapelevich L, Vielma JP (2022a) Performance enhancements for a generic conic interior point algorithm. Math. Programming Comput.CrossrefGoogle Scholar
  • [9] Coey C, Kapelevich L, Vielma JP (2022b) Solving natural conic formulations with Hypatia.jl. INFORMS J. Comput. 34(5):2686–2699.LinkGoogle Scholar
  • [10] Davis C (1957) All convex invariant functions of Hermitian matrices. Archiv Math. 8(4):276–278.CrossrefGoogle Scholar
  • [11] Deng CY (2011) A generalization of the Sherman–Morrison–Woodbury formula. Appl. Math. Lett. 24(9):1561–1564.CrossrefGoogle Scholar
  • [12] Domahidi A, Chu E, Boyd S (2013) ECOS: An SOCP solver for embedded systems. 2013 Eur. Control Conf. (ECC) (IEEE), 3071–3076.Google Scholar
  • [13] Faraut J, Koranyi A (1994) Oxford mathematical monographs. Ball JM, Friedlander EM, Macdonald IG, Nirenberg L, Penrose R, Stuart JT, eds. Analysis on Symmetric Cones (Clarendon Press, New York).Google Scholar
  • [14] Fawzi H, Fawzi O (2018) Efficient optimization of the quantum relative entropy. J. Phys. A 51(15):154003.CrossrefGoogle Scholar
  • [15] Faybusovich L, Tsuchiya T (2017) Matrix monotonicity and self-concordance: How to handle quantum entropy in optimization problems. Optim. Lett. 11(8):1513–1526.CrossrefGoogle Scholar
  • [16] Faybusovich L, Zhou C (2021) Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications. Numer. Algebra Control Optim. 12(2):445–467.CrossrefGoogle Scholar
  • [17] Freund RW, Jarre F, Schaible S (1996) On self-concordant barrier functions for conic hulls and fractional programming. Math. Programming 74(3):237–246.CrossrefGoogle Scholar
  • [18] Furuta T (2008) Concrete examples of operator monotone functions obtained by an elementary method without appealing to Löwner integral representation. Linear Algebra Appl. 429(5–6):972–980.CrossrefGoogle Scholar
  • [19] Grant M, Boyd S (2014) CVX: MATLAB software for disciplined convex programming, version 2.1. http://cvxr.com/cvx/.Google Scholar
  • [20] Grant M, Boyd S, Ye Y (2006) Disciplined convex programming. Liberti L, Maculan N, eds. Global Optimization: From Theory to Implementation. Nonconvex Optimization and Its Applications, vol. 84 (Springer, Boston), 155–210.CrossrefGoogle Scholar
  • [21] Hildebrand R (2014) Analytic formulas for complete hyperbolic affine spheres. Beiträge Algebra Geom. 55(2):497–520.CrossrefGoogle Scholar
  • [22] Kwong MK (1989) Some results on matrix monotone functions. Linear Algebra Appl. 118:129–153.CrossrefGoogle Scholar
  • [23] Lasserre JB (1998) Homogeneous functions and conjugacy. J. Convex Anal. 5(2):397–404.Google Scholar
  • [24] Löwner K (1934) Über monotone matrixfunktionen. Math. Zeitschrift 38(1):177–216.CrossrefGoogle Scholar
  • [25] MOSEK ApS (2020) Modeling Cookbook Release 3.3.0. https://docs.mosek.com/modeling-cookbook/index.html.Google Scholar
  • [26] Nesterov Y (2006) Constructing self-concordant barriers for convex cones. CORE discussion paper No. 2006/30. Preprint, submitted August 4, https://dx.doi.org/10.2139/ssrn.921790.Google Scholar
  • [27] Nesterov Y (2012) Toward non-symmetric conic optimization. Optim. Methods Software 27(4–5):893–917.CrossrefGoogle Scholar
  • [28] Nesterov Y (2018) Lectures on Convex Optimization. Springer Optimization and Its Applications, vol. 137 (Springer, Cham).CrossrefGoogle Scholar
  • [29] Nesterov Y, Nemirovskii A (1994) Interior-Point Polynomial Algorithms in Convex Programming. Studies in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia).CrossrefGoogle Scholar
  • [30] Nesterov YE, Todd MJ (1997) Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22(1):1–42.LinkGoogle Scholar
  • [31] Nesterov Y, Todd MJ, Ye Y (1999) Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems. Math. Programming 84(2):227–267.CrossrefGoogle Scholar
  • [32] Papp D, Alizadeh F (2013) Semidefinite characterization of sum-of-squares cones in algebras. SIAM J. Optim. 23(3):1398–1423.CrossrefGoogle Scholar
  • [33] Papp D Yildiz S (2017) On “A homogeneous interior-point algorithm for non-symmetric convex conic optimization.” Preprint, submitted December 1, https://doi.org/10.48550/arXiv.1712.00492.Google Scholar
  • [34] Parrilo PA (2012) Polynomial optimization, sums of squares, and applications. Blekherman G, Parrilo PA, Thomas RR, eds. Semidefinite Optimization and Convex Algebraic Geometry. MOS-SIAM Series on Optimization, vol. 13 (SIAM, Philadelphia), 47–157.CrossrefGoogle Scholar
  • [35] Permenter F, Friberg HA, Andersen ED (2017) Solving conic optimization problems via self-dual embedding and facial reduction: A unified approach. SIAM J. Optim. 27(3):1257–1282.CrossrefGoogle Scholar
  • [36] Rockafellar RT (1970) Convex Analysis. Princeton Mathematical Series, vol. 28 (Princeton University Press, Princeton, NJ).CrossrefGoogle Scholar
  • [37] Sendov HS (2007) The higher-order derivatives of spectral functions. Linear Algebra Appl. 424(1):240–281.CrossrefGoogle Scholar
  • [38] Serrano SA (2015) Algorithms for unsymmetric cone optimization and an implementation for problems with the exponential cone. PhD thesis, Stanford University, Stanford, CA.Google Scholar
  • [39] Skajaa A, Ye Y (2015) A homogeneous interior-point algorithm for nonsymmetric convex conic optimization. Math. Programming 150(2):391–422.CrossrefGoogle Scholar
  • [40] Sun D, Sun J (2008) Löwner’s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33(2):421–445.LinkGoogle Scholar
  • [41] Sutter D, Sutter T, Esfahani PM, Renner R (2015) Efficient approximation of quantum channel capacities. IEEE Trans. Inform. Theory 62(1):578–598.CrossrefGoogle Scholar
  • [42] Vieira MV (2007) Jordan Algebraic approach to symmetric optimization. PhD thesis, Delft University of Technology, Delft, Netherlands.Google Scholar
  • [43] Vieira MV (2016) Derivatives of eigenvalues and Jordan frames. Numer. Algebra Control Optim. 6(2):115–126.CrossrefGoogle Scholar
  • [44] Yamashita M, Fujisawa K, Kojima M (2003) Implementation and evaluation of SDPA 6.0 (semidefinite programming algorithm 6.0). Optim. Methods Software 18(4):491–505.CrossrefGoogle Scholar
  • [45] Zhang S (2004) A new self-dual embedding method for convex programming. J. Global Optim. 29(4):479–496.CrossrefGoogle Scholar
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