## On second-order conditions in unconstrained optimization.(English)Zbl 1211.90276

The main aim of this paper is to generalize early obtained sufficient second-order optimality conditions which were introduced in terms of the Peano type and the Dini type of directional derivatives for the class of $$C^{1,1}$$ functions or for the class of stable functions.

### MSC:

 90C46 Optimality conditions and duality in mathematical programming 49K10 Optimality conditions for free problems in two or more independent variables 26B05 Continuity and differentiation questions
Full Text:

### References:

 [1] Auslender A. (1984). Stability in mathematical programming with nondifferentiable data. SIAM J. Control Optim. 22: 239–254 · Zbl 0538.49020 [2] Bednařík, D., Pastor, K.: A technical note concerning one example. Nonlinear Anal. 55, 187–189 (2003) · Zbl 1055.49507 [3] Bednařík D. and Pastor K. (2004). On characterization of convexity for regularly locally Lipschitz functions. Nonlinear Anal. 57: 85–97 · Zbl 1073.49008 [4] Bednařík D. and Pastor K. (2004). Elimination of strict convergence in optimization. SIAM J. Control Optim. 43(3): 1063–1077 · Zbl 1089.49023 [5] Ben-Tal A. and Zowe J. (1982). Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Math. Program. 24: 70–91 · Zbl 0488.90059 [6] Ben-Tal A. and Zowe J. (1985). Directional derivatives in nonsmooth optimization. J. Optim. Theory Appl. 47: 483–490 · Zbl 0556.90074 [7] Chan W.L., Huang L.R. and Ng K.F. (1994). On generalized second-order derivatives and Taylor expansions in nonsmooth optimization. SIAM J. Control Optim. 32: 591–611 · Zbl 0801.49016 [8] Chaney R.W. (1987). Second-order necessary conditions in constrained semismooth optimization. SIAM J. Control Optim. 25: 1072–1081 · Zbl 0635.49013 [9] Chaney, R.W.: Second-order sufficient conditions in nonsmooth optimization. Math. Oper. Res. 13, 660–673 (1988) · Zbl 0671.49013 [10] Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York · Zbl 0582.49001 [11] Cominetti R. and Correa R. (1990). A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28: 789–809 · Zbl 0714.49020 [12] Georgiev P.G. and Zlateva N.P. (1996). Second-order subdifferentials of C 1,1 functions and optimality conditions. Set-Valued Anal. 4: 101–117 · Zbl 0864.49012 [13] Ginchev I., Guerraggio A. and Rocca M. (2006). From scalar to vector optimization. Appl. Math. 51: 5–36 · Zbl 1164.90399 [14] Hiriart–Urruty J.B., Strodiot J.J. and Nguyen V.H. (1984). Generalized Hessian matrix and second-order optimality conditions for problems with C 1,1 data. Appl. Math. Optim. 11: 43–56 · Zbl 0542.49011 [15] Huang L.R. and Ng K.F. (1994). Second-order necessary and sufficient conditions in nonsmooth optimization. Math. Program. 66: 379–402 · Zbl 0824.90123 [16] Huang L.R. and Ng K.F. (1996). On some relations between Chaney’s generalized second-order directional derivative and that of Ben-Tal and Zowe. SIAM J. Control Optim. 34: 1220–1234 · Zbl 0877.49019 [17] Huang L.R. and Ng K.F. (1997). On lower bounds of the second-order directional derivatives of Ben-Tal-Zowe and Chaney. Math. Oper. Res. 22: 747–753 · Zbl 0886.49018 [18] Kawasaki H. (1988). An envelope-like effect of infinite many inequality constraints on second-order necessary conditions for minimization problems. Math. Program. 41: 73–96 · Zbl 0661.49012 [19] Klatte D. (2000). Upper Lipschitz behavior of solutions to perturbed C 1,1 programs. Math. Program. (Ser B) 88: 285–311 · Zbl 1017.90111 [20] Maruyama Y. (1990). Second-order necessary conditions for nonlinear optimization problems in Banach spaces and their application to an optimal control problem. Math. Oper. Res. 15: 467–482 · Zbl 0718.49024 [21] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330, 331. Springer, Berlin (2006) [22] Pastor K. (2001). On relations among the generalized second-order directional derivatives. Discuss. Math. Diff. Incl. Contr. Optim. 21: 235–247 · Zbl 1002.49021 [23] Pastor K. (2005). Convexity and generalized second-order derivatives for locally Lipschitz functions. Nonlinear Anal. 60: 547–555 · Zbl 1067.49014 [24] Peano G. (1891). Sulla formula di Taylor. Atti dell’ Accademia delle Science di Torino 27: 40–46 · JFM 23.0253.01 [25] Qi L. (1994). Superlinearly convergent approximate Newton methods for LC 1 optimization problem. Math. Program. 64: 277–294 · Zbl 0820.90102 [26] Rockafellar R.T. (1989). Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. Math. Oper. Res. 14: 462–484 · Zbl 0698.90070 [27] Rockafellar, R.T., Wets, J.B.: Variational Analysis. Springer, New York (1998) [28] Thompson, B.S.: Real Analysis. Springer, Berlin (1985) · Zbl 0586.03051 [29] Torre, D.L., Rocca, M.: Remarks on second order generalized derivatives for differentiable functions with Lipschitzian jacobian. Appl. Math. E-Notes 3, 130–137 (2003) · Zbl 1057.49016 [30] Yang X.Q. (1996). On second-order directional derivatives. Nonlinear Anal. 26: 55–66 · Zbl 0839.90138 [31] Yang X.Q. (1999). On relations and applications of generalized second-order directional derivatives. Nonlinear Anal. 36: 595–614 · Zbl 0990.49016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.