Skip to main content
SpringerLink
Log in

First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

The well known constant rank constraint qualification [Math. Program. Study 21:110–126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of our approach is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition that is stronger than the classical one obtained under Robinson’s constraint qualification, in the sense that it holds for every Lagrange multiplier, even though our condition is independent of Robinson’s condition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. The term “zero gap,” or “no gap,” is often used in NLP to refer to a second-order condition that does not require constraint qualifications to be necessary (using Fritz John/generalized Lagrange multipliers), and that becomes sufficient after replacing an inequality by a strict inequality. However, in this paper, we say that a condition has zero gap when it satisfies the latter, possibly subject to a constraint qualification, in the same way as [21].

References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Progr. Ser. B 95, 3–51 (2003). https://doi.org/10.1007/s10107-002-0339-5

    Article  MathSciNet  MATH  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). https://doi.org/10.1080/0233193021000030751

    Article  MathSciNet  MATH  Google Scholar 

  3. Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18(4), 1286–1309 (2008). https://doi.org/10.1137/060654797

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreani, R., Echagüe, C.E., Schuverdt, M.L.: Constant-rank condition and second-order constraint qualification. J. Optim. Theory Appl. 146(2), 255–266 (2010). https://doi.org/10.1007/s10957-010-9671-8

    Article  MathSciNet  MATH  Google Scholar 

  5. Andreani, R., Fukuda, E.H., Haeser, G., Ramírez, C., Santos, D.O., Silva, P.J.S., Silveira, T.P.: Erratum to: new constraint qualifications and optimality conditions for second order cone programs. Set-Valued Var. Anal. (2021). https://doi.org/10.1007/s11228-021-00573-5

    Article  MATH  Google Scholar 

  6. Andreani, R., Haeser, G., Mito, L.M., Ramırez, H., Santos, D.O., Silveira, T.P.: Naive constant rank-type constraint qualifications for multifold second-order cone programming and semidefinite programming. Optim. Lett. (2020). https://doi.org/10.1007/s11590-021-01737-w

    Article  MATH  Google Scholar 

  7. Andreani, R., Haeser, G., Mito, L. M., Ramírez C., H.: Weak notions of nondegeneracy in nonlinear semidefinite programming. Technical report (2020). Available at arXiv:2012.14810

  8. Andreani, R., Haeser, G., Mito, L. M., Ramírez C., H.: Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications. Technical report (2021). Available at arXiv:2106.00775v2

  9. Andreani, R., Haeser, G., Mito, L M., Ramírez C., H., Silveira, T. P.: Sequential constant rank for nonlinear second-order cone programming problems. Technical report, (2021)

  10. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135(1–2), 255–273 (2012). https://doi.org/10.1007/s10107-011-0456-0

    Article  MathSciNet  MATH  Google Scholar 

  11. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: Two new weak constraint qualifications and applications. SIAM J. Optim. 22(3), 1109–1135 (2012). https://doi.org/10.1137/110843939

    Article  MathSciNet  MATH  Google Scholar 

  12. Andreani, R., Martínez, J.M., Schuverdt, M.L.: On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125(2), 473–483 (2005). https://doi.org/10.1007/s10957-004-1861-9

    Article  MathSciNet  MATH  Google Scholar 

  13. Andreani, R., Martínez, J.M., Schuverdt, M.L.: On second-order optimality conditions for nonlinear programming. Optimization 56, 529–542 (2007). https://doi.org/10.1080/02331930701618617

    Article  MathSciNet  MATH  Google Scholar 

  14. Anitescu, M.: Degenerate nonlinear programming with a quadratic growth condition. SIAM J. Optim. 10(4), 1116–1135 (2000). https://doi.org/10.1137/S1052623499359178

    Article  MathSciNet  MATH  Google Scholar 

  15. Arutyunov, A.: Second-order conditions in extremal problems. The abnormal points. Trans. Am. Math. Soc. 350(11), 4341–4365 (1998). https://doi.org/10.1090/S0002-9947-98-01775-9

    Article  MathSciNet  MATH  Google Scholar 

  16. Auslender, A., Ramírez, H.: Penalty and barrier methods for convex semidefinite programming. Math. Methods Oper. Res. 63, 195 (2006). https://doi.org/10.1007/s00186-005-0054-0

    Article  MathSciNet  MATH  Google Scholar 

  17. Baccari, A.: On the classical necessary second-order optimality conditions. J. Optim. Theory Appl. 123(1), 213–221 (2004). https://doi.org/10.1023/B:JOTA.0000043998.04008.e6

    Article  MathSciNet  MATH  Google Scholar 

  18. Behling, R., Haeser, G., Ramos, A., Viana, D.S.: On a conjecture in second-order optimality conditions. J. Optim. Theory Appl. 176(3), 625–633 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ben-Tal, A., Zowe, J.: A unified theory of first and second order conditions for extremum problems in topological vector spaces. In: Guignard, M. (ed) Optimality and stability in mathematical programming (Mathematical Programming Studies), pp. 39–76. Springer, (1982). https://doi.org/10.1007/BFb0120982

  20. Bonnans, J.F.: A semi-strong sufficiency condition for optimality in non convex programming and its connection to the perturbation problem. J. Optim. Theory Appl. 60, 7–18 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bonnans, J.F., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9(2), 466–492 (1999). https://doi.org/10.1137/S1052623496306760

    Article  MathSciNet  MATH  Google Scholar 

  22. Bonnans, J.F., Ramírez, H.: Perturbation analysis of second-order cone programming problems. Math. Program. 104(2), 205–227 (2005). https://doi.org/10.1007/s10107-005-0613-4

    Article  MathSciNet  MATH  Google Scholar 

  23. Bonnans, J. F., Ramírez, H.: Strong regularity of semidefinite programming problems. Technical report (2005). DIM-CMM N\(^{\circ }\) B-05/06-137

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

    Book  MATH  Google Scholar 

  25. Börgens, E., Kanzow, C., Mehlitz, P., Wachsmuth, G.: New constraint qualifications for optimization problems in banach spaces based on asymptotic KKT conditions. SIAM J. Optim. 30(4), 2956–2982 (2020). https://doi.org/10.1137/19M1306804

    Article  MathSciNet  MATH  Google Scholar 

  26. Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21(1), 265–287 (1990). https://doi.org/10.1007/BF01445166

    Article  MathSciNet  MATH  Google Scholar 

  27. Forsgren, A.: Optimality conditions for nonconvex semidefinite programming. Math. Program. 88(1), 105–128 (2000). https://doi.org/10.1007/PL00011370

    Article  MathSciNet  MATH  Google Scholar 

  28. Fukuda, E. H., Haeser, G., Mito, L. M.: Second-order analysis for semidefinite and second-order cone programming via sequential optimality conditions. Technical report (2020). Available at Optimization Online. http://www.optimization-online.org/DB_HTML/2020/08/7951.html

  29. Gfrerer, H., Mordukhovich, B.S.: Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions. SIAM J. Optim. 25(4), 2081–2119 (2015). https://doi.org/10.1137/15M1012608

    Article  MathSciNet  MATH  Google Scholar 

  30. Gould, F.J., Tolle, J.W.: Optimality conditions and constraint qualifications in banach space. J. Optim. Theory Appl. 15(6), 667–684 (1975). https://doi.org/10.1007/BF00935506

    Article  MathSciNet  MATH  Google Scholar 

  31. Guignard, M.: Generalized Kunh-Tucker conditions for mathematical programming in a banach space. SIAM J. Control 7, 232–241 (1969). https://doi.org/10.1137/0307016

    Article  MathSciNet  MATH  Google Scholar 

  32. Guo, L., Lin, H.-H., Ye: Second-order optimality conditions for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 158, 33–64 (2013). https://doi.org/10.1007/s10957-012-0228-x

    Article  MathSciNet  MATH  Google Scholar 

  33. Guo, L., Lin, H.-H., Ye, J.J., Zhang, J.: Sensitivity analysis of the value function for parametric mathematical programs with equilibrium constraints. SIAM J. Optim. 24(3), 1206–1237 (2014). https://doi.org/10.1137/130929783

    Article  MathSciNet  MATH  Google Scholar 

  34. Henrion, R., Kruger, A.Y., Outrata, J.V.: Some remarks on stability of generalized equations. J. Optim. Theory Appl. 159, 681–697 (2013). https://doi.org/10.1007/s10957-012-0147-x

    Article  MathSciNet  MATH  Google Scholar 

  35. Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137, 257–288 (2013). https://doi.org/10.1007/s10107-011-0488-5

    Article  MathSciNet  MATH  Google Scholar 

  36. Janin, R.: Directional derivative of the marginal function in nonlinear programming. In: Fiacco, A.V. (ed) Sensitivity, Stability and Parametric Analysis (Mathematical Programming Studies), pp. 110–126. Springer Berlin Heidelberg, (1984). https://doi.org/10.1007/BFb0121214

  37. Jarre, F.: Elementary optimality conditions for nonlinear SDPs. In: Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, (2012). https://doi.org/10.1007/978-1-4614-0769-0_16

  38. Jiang, H., Ralph, D.: Smooth SQP methods for mathematical programs with nonlinear complementarity constraints. SIAM J. Optim. 10(3), 779–808 (2000). https://doi.org/10.1137/S1052623497332329

    Article  MathSciNet  MATH  Google Scholar 

  39. Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1995)

    Book  MATH  Google Scholar 

  40. Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems. Math. Program. 41(1), 73–96 (1988). https://doi.org/10.1007/BF01580754

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu, T.W.: A reduced Hessian SQP method for inequality constrained optimization. Comput. Optim. Appl. 49, 31–59 (2011). https://doi.org/10.1007/s10589-009-9285-y

    Article  MathSciNet  MATH  Google Scholar 

  42. Lourenço, B.F., Fukuda, E.H., Fukushima, M.: Optimality conditions for nonlinear semidefinite programming via squared slack variables. Math. Program. 166, 1–24 (2016). https://doi.org/10.1007/s10107-016-1040-4

    Article  MATH  Google Scholar 

  43. Maciel, M.C., Santos, S.A., Sottosanto, G.N.: On second-order optimality conditions for vector optimization. J. Optim. Theory Appl. 149, 332–351 (2011). https://doi.org/10.1007/s10957-010-9793-z

    Article  MathSciNet  MATH  Google Scholar 

  44. Mehlitz, P., Minchenko, L.I.: R-regularity of set-valued mappings under the relaxed constant positive linear dependence constraint qualification with applications to parametric and bilevel optimization. Set-Valued Var. Anal. (2021). https://doi.org/10.1007/s11228-021-00578-0

    Article  MATH  Google Scholar 

  45. Minchenko, L., Leschov, A.: On strong and weak second-order necessary optimality conditions for nonlinear programming. Optimization 65(9), 1693–1702 (2016). https://doi.org/10.1080/02331934.2016.1179300

    Article  MathSciNet  MATH  Google Scholar 

  46. Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(4), 429–440 (2011). https://doi.org/10.1080/02331930902971377

    Article  MathSciNet  MATH  Google Scholar 

  47. Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. (2011). https://doi.org/10.1137/090761318

    Article  MathSciNet  MATH  Google Scholar 

  48. Monte, M.R.C., de Oliveira, V.A.: A constant rank constraint qualification in continuous-time nonlinear programming. Set-Valued Var. Anal. 29, 61–81 (2021). https://doi.org/10.1007/s11228-020-00537-1

    Article  MathSciNet  MATH  Google Scholar 

  49. Mordukhovich, B.S., Nghia, T.T.A.: Second-order characterizations of tilt stability with applications to nonlinear programming. Math. Program. 149, 83–104 (2015). https://doi.org/10.1007/s10107-013-0739-8

    Article  MathSciNet  MATH  Google Scholar 

  50. Pataki, G.: The geometry of semidefinite programming. In: Saigal, R., Vandenberghe, L., Wolkowicz, H. (eds.) Handbook of Semidefinite Programming, pp. 29–65. Kluwer Academic Publishers, Waterloo (2000). https://doi.org/10.1007/978-1-4615-4381-7_3

    Chapter  MATH  Google Scholar 

  51. 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). https://doi.org/10.1137/S1052623497326629

    Article  MathSciNet  MATH  Google Scholar 

  52. Qiu, S.: A globally convergent regularized interior point method for constrained optimization. Optim. Methods Softw. (2021). https://doi.org/10.1080/10556788.2021.1908283

    Article  MATH  Google Scholar 

  53. Robinson, S.M.: First-order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30(4), 597–610 (1976). https://doi.org/10.1137/0130053

    Article  MathSciNet  MATH  Google Scholar 

  54. Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. SIAM J. Optim. 77(1), 301–320 (1997). https://doi.org/10.1007/BF02614439

    Article  MathSciNet  MATH  Google Scholar 

  55. Shapiro, A.: On uniqueness of Lagrange multipliers in optimization problems subject to cone constraints. SIAM J. Optim. 7, 508–518 (1997). https://doi.org/10.1137/S1052623495279785

    Article  MathSciNet  MATH  Google Scholar 

  56. Shapiro, A., Fan, M.K.H.: On eigenvalue optimization. SIAM J. Optim. 5(3), 552–569 (1995). https://doi.org/10.1137/0805028

    Article  MathSciNet  MATH  Google Scholar 

  57. Steffensen, S., Ulbrich, M.: A new relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 20(5), 2504–2539 (2010). https://doi.org/10.1137/090748883

    Article  MathSciNet  MATH  Google Scholar 

  58. Wright, S.J.: Modifying SQP for degenerate problems. SIAM J. Optim. 13(2), 470–497 (2002). https://doi.org/10.1137/S1052623498333731

    Article  MathSciNet  MATH  Google Scholar 

  59. Xu, M., Ye, J.J.: Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs. J. Glob. Optim. 78, 181–205 (2020). https://doi.org/10.1007/s10898-020-00907-x

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhang, Y., Zhang, L.: New constraint qualifications and optimality conditions for second order cone programs. Set-Valued Var. Anal. 27, 693–712 (2019). https://doi.org/10.1007/s11228-018-0487-2

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work has received financial support from CEPID-CeMEAI (FAPESP 2013/07375-0), FAPESP (grants 2018/24293-0, 2017/18308-2, 2017/17840-2, 2017/12187-9, and 2020/00130-5), CNPq (grants 301888/2017-5, 303427/2018-3, and 404656/2018-8), PRONEX - CNPq/FAPERJ (grant E-26/010.001247/2016), and FONDECYT grant 1201982 and Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence, both from ANID (Chile).

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, State University of Campinas, São Paulo, Brazil

    Roberto Andreani

  2. Department of Applied Mathematics, University of São Paulo, São Paulo, Brazil

    Gabriel Haeser, Leonardo M. Mito & Thiago P. Silveira

  3. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS IRL 2807), Universidad de Chile, Santiago, Chile

    Héctor Ramírez

Authors
  1. Roberto Andreani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriel Haeser
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leonardo M. Mito
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Héctor Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Thiago P. Silveira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel Haeser.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Andreani, R., Haeser, G., Mito, L.M. et al. First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition. Math. Program. 202, 473–513 (2023). https://doi.org/10.1007/s10107-023-01942-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-023-01942-8

Keywords

Mathematics Subject Classification

Search

Navigation