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First- and Second-Order Necessary Conditions Via Exact Penalty Functions

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Abstract

In this paper, we study first- and second-order necessary conditions for nonlinear programming problems from the viewpoint of exact penalty functions. By applying the variational description of regular subgradients, we first establish necessary and sufficient conditions for a penalty term to be of KKT-type by using the regular subdifferential of the penalty term. In terms of the kernel of the subderivative of the penalty term, we also present sufficient conditions for a penalty term to be of KKT-type. We then derive a second-order necessary condition by assuming a second-order constraint qualification, which requires that the second-order linearized tangent set is included in the closed convex hull of the kernel of the parabolic subderivative of the penalty term. In particular, for a penalty term with order \(\frac{2}{3}\), by assuming the nonpositiveness of a sum of a second-order derivative and a third-order derivative of the original data and applying a third-order Taylor expansion, we obtain the second-order necessary condition.

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References

  1. Karush, W.: Minima of functions of several variables with inequalities as side constraints. Master’s thesis, Department of Mathematics, University of Chicago, Chicago, IL (1939)

  2. Kuhn, H.W., Tucker, A.W.: Nonlinear programming. Proceedings of 2nd Berkeley Symposium, pp. 481–492. University of California Press, Berkeley (1951)

  3. Ioffe, A.D.: Necessary and sufficient conditions for a local minimum. 3: Second order conditions and augmented duality. SIAM J. Control Optim. 17(2), 266–288 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Guignard, M.: Generalized Kuhn–Tucker conditions for mathematical programming problems in a Banach space. SIAM J. Control Optim. 7(2), 232–241 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gould, F.J., Tolle, Jon W.: A necessary and sufficient qualification for constrained optimization. SIAM J. Appl. Math. 20(2), 164–172 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  6. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  7. Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31(2), 143–165 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ben-Tal, A., Zowe, J.: A unified theory of first and second order conditions for extremum problems in topological vector spaces. Math. Program. Stud. 19, 39–76 (1982)

    Article  MathSciNet  Google Scholar 

  9. Rockafellar, R.T.: Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. Math. Oper. Res. 14, 462–484 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ioffe, A.D.: On some recent developments in the theory of second order optimality conditions. In: Dolecki, S. (ed.) Optimization. Lecture Notes in Mathematics, vol. 1405, pp. 55–68. Springer, Berlin (1989)

  11. Burke, J.V., Poliquin, R.A.: Optimality conditions for non-finite valued convex composite functions. Math. Program. 57, 103–120 (1992)

    Article  MathSciNet  Google Scholar 

  12. Mangasarian, O.L., Fromovitz, S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17(1), 37–47 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  13. Rockafellar, R.T.: First- and second-order epi-differentiability in nonlinear programming. Trans. Am. Math. Soc. 307(1), 75–108 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kawasaki, H.: Second-order necessary conditions of the Kuhn-Tucker type under new constraint qualifications. J. Optim. Theory Appl. 57(2), 253–264 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  16. Anitescu, M.: Degenerate nonlinear programming with a quadratic growth condition. SIAM J. Optim. 10, 1116–1135 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Baccari, A., Trad, A.: On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. Optim. 15, 394–408 (2005)

    Article  MathSciNet  Google Scholar 

  18. Andreani, R., Echagüe, C.E., Schuverdt, M.L.: Constant-rank condition and second-order constraint qualification. J. Optim. Theory Appl. 146, 255–266 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Arutyunov, A.V.: Perturbations of extremal problems with constraints and necessary optimality conditions. J. Math. Sci. 54, 1342–1400 (1991)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  21. Burke, J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29(4), 968–998 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Eremin, I.I.: The penalty method in convex programming. Cybernet. Syst. Anal. 3(4), 53–56 (1967)

    Article  MathSciNet  Google Scholar 

  23. Zangwill, W.I.: Non-linear programming via penalty functions. Manag. Sci. 13(5), 344–358 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  24. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  25. Burke, J.V.: Calmness and exact penalization. SIAM J. Control Optim. 29(2), 493–497 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rubinov, A.M., Yang, X.Q.: Lagrange-Type Functions in Constrained Non-Convex Optimization. Kluwer Academic Publishers, Dordrecht (2003)

    Book  MATH  Google Scholar 

  27. Yang, X.Q., Meng, Z.Q.: Lagrange multipliers and calmness conditions of order \(p\). Math. Oper. Res. 32(1), 95–101 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Meng, K.W., Yang, X.Q.: Optimality conditions via exact penalty functions. SIAM J. Optim. 20(6), 3208–3231 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pietrzykowski, T.: An exact potential method for constrained maxima. SIAM J. Numer. Anal. 6(2), 299–304 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  30. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  31. Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28(3), 533–552 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Chartrand, R., Staneva, V.: Restricted isometry properties and nonconvex compressive sensing. Inverse Probl. 24, 1–14 (2008)

    Article  MathSciNet  Google Scholar 

  33. Wright, S.J., Nowak, R.D., Figueiredo, M.A.T.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process 57(7), 2479–2493 (2009)

    Article  MathSciNet  Google Scholar 

  34. Lewis, A.S., Wright, S.J.: Identifying activity. SIAM J. Optim. 21(2), 597–614 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. Izmailov, A.F., Solodov, M.V.: Error bounds for 2-regular mappings with lipschitzian derivatives and their applications. Math. Program. 89, 413–435 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  36. Arutyunov, A.V., Avakov, E.R., Izmailov, A.F.: Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications. Math. Program. 114, 37–68 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Aubin, J.-P., Ekeland, L.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  38. Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)

    MATH  Google Scholar 

  39. Ng, K.F., Zheng, X.Y.: Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12(1), 1–17 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  40. Wilmott, P., Dewynne, J., Howison, S.: Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford, UK (1993)

    Google Scholar 

  41. Wang, S., Yang, X.Q., Teo, K.L.: A power penalty method for a linear complementarity problem arising from American option valuation. J. Optim. Theory Appl. 129, 227–254 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  42. Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math. 91, 199–218 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  43. Bensoussan, A., Lions, J.L.: Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam (1982)

    MATH  Google Scholar 

  44. Tian, B.S., Yang, X.Q., Meng, K.W.: An interior-point \(\ell _{\frac{1}{2}}\)-penalty method for inequality constrained nonlinear optimization (submitted)

  45. Curtis, F.E.: A penalty-interior-point algorithm for nonlinear constrained optimization. Math. Progr. Comput. 4(2), 181–209 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  46. Gould, N.I.M., Toint, P.L., Orban, D.: An interior-point \(\ell _1\)-penalty method for nonlinear optimization. Groupe d’études et de recherche en analyse des décisions (2010)

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Acknowledgments

The first author was supported by the National Science Foundation of China (Grants: 71090402, 11201383). The second author was supported by the Research Grants Council of Hong Kong (PolyU 5295/12E). The authors would like to thank the two anonymous reviewers for their valuable comments and suggestions, which have helped to improve the paper.

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Authors and Affiliations

  1. School of Economics and Management, Southwest Jiaotong University, Chengdu, 610031, China

    Kaiwen Meng

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    Xiaoqi Yang

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  1. Kaiwen Meng
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  2. Xiaoqi Yang
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Correspondence to Xiaoqi Yang.

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Meng, K., Yang, X. First- and Second-Order Necessary Conditions Via Exact Penalty Functions. J Optim Theory Appl 165, 720–752 (2015). https://doi.org/10.1007/s10957-014-0664-x

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