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A Flexible Inexact-Restoration Method for Constrained Optimization

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Abstract

We introduce a new flexible inexact-restoration algorithm for constrained optimization problems. In inexact-restoration methods, each iteration has two phases. The first phase aims at improving feasibility and the second phase aims to minimize a suitable objective function. In the second phase, we also impose bounded deterioration of the feasibility, obtained in the first phase. Here, we combine the basic ideas of the Fischer-Friedlander approach for inexact-restoration with the use of approximations of the Lagrange multipliers. We present a new option to obtain a range of search directions in the optimization phase, and we employ the sharp Lagrangian as merit function. Furthermore, we introduce a flexible way to handle sufficient decrease requirements and an efficient way to deal with the penalty parameter. Global convergence of the new inexact-restoration method to KKT points is proved under weak constraint qualifications.

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References

  1. Abadie, J., Carpentier, J.: Generalization of the Wolfe reduced-gradient method to the case of nonlinear constraints. In: Fletcher, R. (ed.) Optimization, pp. 37–47. Academic Press, New York (1968)

    Google Scholar 

  2. Lasdon, L.S.: Reduced gradient methods. In: Powell, M.J.D. (ed.) Nonlinear Optimization, pp. 235–242. Academic Press, New York (1982)

    Google Scholar 

  3. Miele, A., Huang, H.Y., Heideman, J.C.: Sequential gradient-restoration algorithm for the minimization of constrained functions, ordinary and conjugate gradient version. J. Optim. Theory Appl. 4, 213–246 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  4. Miele, A., Levy, A.V., Cragg, E.E.: Modifications and extensions of the conjugate-gradient restoration algorithm for mathematical programming problems. J. Optim. Theory Appl. 7, 450–472 (1971)

    Article  MathSciNet  Google Scholar 

  5. Miele, A., Sims, E.M., Basapur, V.K.: Sequential gradient-restoration algorithm for mathematical programming problem with inequality constraints, Part 1, Theory. Aero-Astronautics Report No. 168, Rice University (1983).

  6. Rom, M., Avriel, M.: Properties of the sequential gradient-restoration algorithm (SGRA), Part 1: introduction and comparison with related methods. J. Optim. Theory Appl. 62, 77–98 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Rom, M., Avriel, M.: Properties of the sequential gradient-restoration algorithm (SGRA), Part 2: convergence analysis. J. Optim. Theory Appl. 62, 99–126 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rosen, J.B.: The gradient projection method for nonlinear programming, Part 1: linear constraints. SIAM J. Appl. Math. 8, 181–217 (1960)

    Article  MATH  Google Scholar 

  9. Rosen, J.B.: The gradient projection method for nonlinear programming, Part 2: nonlinear constraints. SIAM J. Appl. Math. 9, 514–532 (1961)

    Article  MATH  Google Scholar 

  10. Martínez, J.M., Pilotta, E.A.: Inexact restoration algorithms for constrained optimization. J. Optim. Theory Appl. 104, 135–163 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Martínez, J.M.: Inexact restoration method with Lagrangian tangent decrease and new merit function for nonlinear programming. J. Optim. Theory Appl. 111, 39–58 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gonzaga, C.C., Karas, E.W., Vanti, M.: A globally convergent filter method for nonlinear programming. SIAM J. Optim. 14, 646–669 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Birgin, E.G., Martínez, J.M.: Local convergence of an Inexact-Restoration method and numerical experiments. J. Optim. Theory Appl. 127, 229–247 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Martínez, J.M., Pilotta, E.A.: Inexact restoration methods for nonlinear programming: advances and perspectives. In: Qi, L.Q., Teo, K., Yang, X.Q. (eds.) Optimization and Control with Applications, pp. 271–292. Springer, New York (2005)

    Chapter  Google Scholar 

  15. Kaya, C.Y., Martínez, J.M.: Euler discretization and inexact restoration for optimal control. J. Optim. Theory Appl. 134, 191–206 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Andreani, R., Castro, S.L., Chela, J.L., Friedlander, A., Santos, S.A.: An inexact-restoration method for nonlinear bilevel programming problems. Comput. Optim. Appl. 43, 307–328 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gomes-Ruggiero, M.A., Martínez, J.M., Santos, S.A.: Spectral projected gradient method with inexact restoration for minimization with nonconvex constraints. SIAM J. Sci. Comput. 31, 1628–1652 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Karas, E.W., Pilotta, E.A., Ribeiro, A.A.: Numerical comparison of merit function with filter criterion in inexact restoration algorithms using Hard-Spheres Problems. Comput. Optim. Appl. 44, 427–441 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fischer, A., Friedlander, A.: A new line search inexact restoration approach for nonlinear programming. Comput. Optim. Appl. 46, 333–346 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Karas, E.W., Gonzaga, C.C., Ribeiro, A.A.: Local convergence of filter methods for equality constrained nonlinear programming. Optimization 59, 1153–1171 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kaya, C.Y.: Inexact Restoration for Runge–Kutta discretization of optimal control problems. SIAM J. Numer. Anal. 48, 1492–1517 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Francisco, J.B., Martínez, J.M., Martínez, L., Pisnitchenko, F.I.: Inexact Restoration method for minimization problems arising in electronic structure calculations. Comput. Optim. Appl. 50, 555–590 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Bueno, L.F., Friedlander, A., Martínez, J.M., Sobral, F.: Inexact restoration method for derivative-free optimization with smooth constraints. SIAM J. Optim. 23, 1189–1213 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl. 118, 117–133 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Barzilai, J., Borwein, J.M.: Two point step size gradient method. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. Dai, Y.H., Fletcher, R.: On the asymptotic behaviour of some new gradient methods. Math. Prog. 103, 541–559 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Friedlander, A., Martínez, J.M., Molina, B., Raydan, M.: Gradient method with retards and generalizations. SIAM J. Numer. Anal. 36, 275–289 (1998)

    Article  MATH  Google Scholar 

  29. Raydan, M.: On the Barzilai and Borwein choice of steplength for the gradient method. IMA J. Numer. Anal. 13, 321–326 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  30. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Birgin, E.G., Martínez, J.M., Raydan, M.: Algorithm 813: SPG Software for convex-constrained optimization. ACM Trans. Math. Softw. 27, 340–349 (2001)

    Article  MATH  Google Scholar 

  33. Birgin, E.G., Martínez, J.M., Raydan, M.: Inexact spectral projected gradient methods on convex sets. IMA J. Numer. Anal. 23, 539–559 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Andreani, R., Birgin, E.G., Martínez, J.M., Yuan, J.-Y.: Spectral projected gradient and variable metric methods for optimization with linear inequalities. IMA J. Numer. Anal. 25, 221–252 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  36. Burachik, R.S., Gasimov, R.N., Ismayilova, A.N., Kaya, C.Y.: On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J. Global Optim. 34, 55–78 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Burachik, R.S., Iusem, A.N., Melo, J.G.: A primal dual modified subgradient algorithm with sharp Lagrangian. J. Global Optim. 46, 347–361 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Burachik, R.S., Kaya, C.Y., Mammadov, M.: An inexact modified subgradient algorithm for nonconvex optimization. Comput. Optim. Appl. 45, 1–24 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  39. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  40. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: Two new constraint qualifications and applications. SIAM J. Optim. 22, 1109–1135 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  41. Andreani, R., Martínez, J.M., Schuverdt, M.L.: On the relation between the Constant Positive Linear Dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125, 473–485 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Prog. 135, 255–273 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  45. Bueno, L.F.: Otimização com restries LOVO, Restauração Inexata e o Equilíbrio Inverso de Nash. Ph.D. dissertation, Departamento de Matemática Aplicada, Universidade Estadual de Campinas (2011)

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Acknowledgments

This work was supported by PRONEX-CNPq/FAPERJ (Grant E-26/171.164/ 2003 - APQ1), FAPESP (Grants 2010/19720-5 and 2013/05475-7), CEPID-Cemeai-Fapesp Industrial Mathematics 201307375-0, and CNPq. The authors would like to thank the three anonymous referees for their insightful comments and suggestions. We also thank Alene Alder-Rangel and William Stanton for reviewing the English grammar and style of this manuscript.

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Bueno, L.F., Haeser, G. & Martínez, J.M. A Flexible Inexact-Restoration Method for Constrained Optimization. J Optim Theory Appl 165, 188–208 (2015). https://doi.org/10.1007/s10957-014-0572-0

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  • DOI: https://doi.org/10.1007/s10957-014-0572-0

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