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Optimality Conditions for Nonlinear Second-Order Cone Programming and Symmetric Cone Programming

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Abstract

Nonlinear symmetric cone programming (NSCP) generalizes important optimization problems such as nonlinear programming, nonlinear semi-definite programming and nonlinear second-order cone programming (NSOCP). In this work, we present two new optimality conditions for NSCP without constraint qualifications, which implies the Karush–Kuhn–Tucker conditions under a condition weaker than Robinson’s constraint qualification. In addition, we show the relationship of both optimality conditions in the context of NSOCP, where we also present an augmented Lagrangian method with global convergence to a KKT point under a condition weaker than Robinson’s constraint qualification.

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References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)

    Article  MathSciNet  Google Scholar 

  2. Andersen, E.D., Roos, C., Terlaky, T.: Notes on duality in second order and p-order cone optimization. Optimization 51(4), 627–643 (2002)

    Article  MathSciNet  Google Scholar 

  3. Andersen, E.D., Roos, C., Terlaky, T.: On implementation a primal-dual interior-point method for conic quadratic programming. Math. Program. 95, 249–277 (2003)

    Article  MathSciNet  Google Scholar 

  4. Andreani, R., Fukuda, E.H., Haeser, G., Santos, D.O., Secchin, L.D.: On the use of Jordan algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming. Comput. Optim. Appl. 79, 633–648 (2021)

    Article  MathSciNet  Google Scholar 

  5. Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60(5), 627–641 (2011)

    Article  MathSciNet  Google Scholar 

  6. Andreani, R., Haeser, G., Mito, L.M., RamírezC H.: Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications. To appear in Set-Valued and Variational Analysis, (2022)

  7. Andreani, R., Haeser, G., Mito, L.M., RamírezC H.: Weak notions of nondegeneracy in nonlinear semidefinite programming. arXiv:2012.14810 (2022)

  8. Andreani, R., Haeser, G., Mito, L.M., Ramírez C, H., Santos, D.O., Silveira, T.P.: Naive constant rank-type constraint qualifications for multifold second-order cone programming and semidefinite programming. Optim. Lett. 16, 589–610 (2022)

  9. Andreani, R., Haeser, G., Mito, L.M., Ramírez C, H., Silveira, T.P.: Global convergence of algorithms under constant rank conditions for nonlinear second-order cone programming. J. Optim. Theory Appl. 195, 42–78 (2022)

  10. Andreani, R., Haeser, G., Ramos, A., Silva, P.J.S.: A second-order sequential optimality condition associated to the convergence of algorithms. IMA J. Numer. Anal. 37(4), 1902–1929 (2017)

    Article  MathSciNet  Google Scholar 

  11. Andreani, R., Haeser, G., Secchin, L.D., Silva, P.J.S.: New sequential optimality conditions for mathematical programs with complementarity constraints and algorithmic consequences. SIAM J. Optim. 29(4), 3201–3230 (2019)

    Article  MathSciNet  Google Scholar 

  12. Andreani, R., Haeser, G., Viana, D.S.: Optimality conditions and global convergence for nonlinear semidefinite programming. Math. Program. 180, 203–235 (2020)

    Article  MathSciNet  Google Scholar 

  13. Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26(1), 96–110 (2016)

    Article  MathSciNet  Google Scholar 

  14. Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: Strict constraint qualifications and sequential optimality conditions for constrained optimization. Math. Oper. Res. 43(3), 693–717 (2018)

    Article  MathSciNet  Google Scholar 

  15. Andreani, R., Martínez, J.M., Santos, L.T.: Newton’s method may fail to recognize proximity to optimal points in constrained optimization. Math. Program. 160, 547–555 (2016)

    Article  MathSciNet  Google Scholar 

  16. Andreani, R., Martínez, J.M., Svaiter, B.F.: A new sequencial optimality condition for constrained optimization and algorithmic consequences. SIAM J. Optim. 20(6), 3533–3554 (2010)

    Article  MathSciNet  Google Scholar 

  17. Baes, M.: Spectral Functions and Smoothing Techniques on Jordan Algebras. PhD thesis, Université Catholique de Louvain, France (2006)

  18. Baes, M.: Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras. Linear Algebra Appl. 422(2), 664–700 (2007)

    Article  MathSciNet  Google Scholar 

  19. Birgin, E., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM Publications, New Delhi (2014)

    Book  Google Scholar 

  20. Birgin, E.G., Haeser, G., Ramos, A.: Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points. Comput. Optim. Appl. 69(1), 51–75 (2018)

    Article  MathSciNet  Google Scholar 

  21. Birgin, E.G., Krejić, N., Martínez, J.M.: On the minimization of possibly discontinuous functions by means of pointwise approximations. Optim. Lett. 11(8), 1623–1637 (2017)

    Article  MathSciNet  Google Scholar 

  22. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  Google Scholar 

  23. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  24. Bueno, L.F., Haeser, G., Lara, F., Rojas, F.N.: An Augmented Lagrangian method for quasi-equilibrium problems. Comput. Optim. Appl. 76(3), 737–766 (2020)

    Article  MathSciNet  Google Scholar 

  25. Bueno, L.F., Haeser, G., Rojas, F.N.: Optimality conditions and constraint qualifications for generalized Nash equilibrium problems and their practical implications. SIAM J. Optim. 29(1), 31–54 (2019)

    Article  MathSciNet  Google Scholar 

  26. Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. J. Global Optim. 56(4), 1463–1499 (2013)

    Article  MathSciNet  Google Scholar 

  27. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs. Clarendon Press, Oxford (1994)

    Book  Google Scholar 

  28. Feng, M., Li, S.: An approximate strong KKT condition for multiobjective optimization. Top 26(3), 489–509 (2018)

    Article  MathSciNet  Google Scholar 

  29. Fiacco, A.V., McCormick, G.P.: Nonlinear Programming Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)

    Google Scholar 

  30. Fukuda, E.H., Silva, P.J.S., Fukushima, M.: Differentiable exact penalty functions for nonlinear second-order cone programs. SIAM J. Optim. 22(4), 1607–1633 (2012)

    Article  MathSciNet  Google Scholar 

  31. Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12(2), 436–460 (2001)

    Article  MathSciNet  Google Scholar 

  32. Giorgi, G., Jiménez, B., Novo, V.: Approximate Karush–Kuhn–Tucker condition in multiobjective optimization. J. Optim. Theory Appl. 171(1), 70–89 (2016)

    Article  MathSciNet  Google Scholar 

  33. Haeser, G.: A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms. Comput. Optim. Appl. 70(2), 615–639 (2018)

    Article  MathSciNet  Google Scholar 

  34. Haeser, G., Melo, V.V.: Convergence detection for optimization algorithms: approximate-KKT stopping criterion when Lagrange multipliers are not available. Oper. Res. Lett. 43(5), 484–488 (2015)

    Article  MathSciNet  Google Scholar 

  35. Haeser, G., Schuverdt, M.L.: On approximate KKT condition and its extension to continuous variational inequalities. J. Optim. Theory Appl. 149(3), 528–539 (2011)

    Article  MathSciNet  Google Scholar 

  36. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  37. Kanzow, C., Ferenczi, I., Fukushima, M.: On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J. Optim. 20(41), 297–320 (2009)

    Article  MathSciNet  Google Scholar 

  38. Kanzow, C., Steck, D., Wachsmuth, D.: An augmented Lagrangian methods for optimization problems in Banach spaces. SIAM J. Control Optim. 56(1), 272–291 (2018)

    Article  MathSciNet  Google Scholar 

  39. Kanzow, D., Steck, D.: Approximated Lagrangian and exact penalty methods for quasi-variational inequalities. Comput. Optim. Appl. 60(3), 801–824 (2018)

    Article  Google Scholar 

  40. Kato, H., Fukushima, M.: An SQP-type algorithm for nonlinear second-order cone programs. Optim. Lett. 1(2), 129–144 (2007)

    Article  MathSciNet  Google Scholar 

  41. Liu, Y.J., Zhang, L.W.: Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cone. J. Optim. Theory Appl. 139(3), 557–575 (2008)

    Article  MathSciNet  Google Scholar 

  42. Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1–3), 193–228 (1998)

    Article  MathSciNet  Google Scholar 

  43. Lourenço, B.F., Fukuda, E.H., Fukushima, M.: Optimality conditions for problems over symmetric cones and a simple augmented Lagrangian method. Math. Oper. Res. 43, 1233–1251 (2018)

    Article  MathSciNet  Google Scholar 

  44. Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(4), 429–440 (2011)

    Article  MathSciNet  Google Scholar 

  45. Qi, L., Wei, Z.: On the constant positive linear dependence conditions and its application to SQP methods. SIAM J. Optim. 10(4), 963–981 (2000)

    Article  MathSciNet  Google Scholar 

  46. Ramos, A.: Mathematical programs with equilibrium constraints: a sequential optimality condition, new constraint qualifications and algorithmic consequences. Optim. Methods Softw. 36, 45–81 (2019)

    Article  MathSciNet  Google Scholar 

  47. Tuyen, N.V., Yao, J., Wen, C.: A note on approximate Karush–Kuhn–Tucker conditions in locally Lipschitz multiobjective optimization. Optim. Lett. 13, 163–174 (2019)

    Article  MathSciNet  Google Scholar 

  48. Yamashita, H., Yabe, H.: A primal-dual interior point method for nonlinear optimization over second-order cones. Optim. Methods Softw. 24(3), 407–426 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Grant-in-Aid for Scientific Research (C) (19K11840) from Japan Society for the Promotion of Science, FAPESP grants 2013/07375-0, 2017/18308-2 and 2018/24293-0, FAPES grant 116/2019, CNPq grants 309136/2021-0 and 306988/2021-6, CAPES, USP-Santander and PRONEX - CNPq/FAPERJ (grant E-26/010.001247/2016). We also would like to take the opportunity to thank an anonymous referee for insightful comments and suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Campinas, Campinas, SP, Brazil

    Roberto Andreani

  2. Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Ellen H. Fukuda

  3. Department of Applied Mathematics, University of São Paulo, São Paulo, SP, Brazil

    Gabriel Haeser

  4. Paulista School of Politics, Economics and Business, Federal University of São Paulo, Osasco, SP, Brazil

    Daiana O. Santos

  5. Department of Applied Mathematics, Federal University of Espírito Santo, São Mateus, ES, Brazil

    Leonardo D. Secchin

Authors
  1. Roberto Andreani
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  2. Ellen H. Fukuda
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  3. Gabriel Haeser
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  4. Daiana O. Santos
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  5. Leonardo D. Secchin
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Correspondence to Gabriel Haeser.

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Communicated by Florian Potra.

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Andreani, R., Fukuda, E.H., Haeser, G. et al. Optimality Conditions for Nonlinear Second-Order Cone Programming and Symmetric Cone Programming. J Optim Theory Appl 200, 1–33 (2024). https://doi.org/10.1007/s10957-023-02338-6

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