Second-order characterization of convex functions and its applications

• Autorzy: Nadi Mohammad Taghi , Yao Jen Chih , Zafarani Jafar

Identyfikatory

DOI

10.1515/jaa-2019-0006

• Języki publikacji: EN

Abstrakty

EN

Some developments of the second-order characterizations of convex functions are investigated by using the coderivative of the subdifferential mapping. Furthermore, some applications of the second-order subdifferentials in optimization problems are studied.

Słowa kluczowe

EN

convexity; second-order subdifferentials; positive definite; optimality conditions

PL

wypukłość; subróżniczka drugiego rzędu; dodatnio określony; warunki optymalności

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2019
• Tom: Vol. 25, nr 1
• Strony: 49--58
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

Nadi Mohammad Taghi

• University of Isfahan, Isfahan, 81745-163, Iran

autor

Yao Jen Chih

• Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

autor

Zafarani Jafar

• University of Isfahan and Sheikhbahaee University, Isfahan, 81745-163, Iran

Bibliografia

• [1] D. Bednařík and K. Pastor, On characterization of convexity for regularly locally Lipschitz functions, Nonlinear Anal. 57 (2004), no. 1, 85-97.
• [2] N. H. Chieu, T. D. Chuong, J.-C. Yao and N. D. Yen, Characterizing convexity of a function by its Fréchet and limiting second-order subdifferentials, Set-Valued Var. Anal. 19 (2011), no. 1, 75-96.
• [3] N. H. Chieu and N. Q. Huy, Second-order subdifferentials and convexity of real-valued functions, Nonlinear Anal. 74 (2011), no. 1, 154-160.
• [4] N. H. Chieu, G. M. Lee, B. S. Mordukhovich and T. T. A. Nghia, Coderivative characterizations of maximal monotonicity for set-valued mappings, J. Convex Anal. 23 (2016), no. 2, 461-480.
• [5] R. Cominetti and R. Correa, A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim. 28 (1990), no. 4, 789-809.
• [6] J.-P. Crouzeix, Criteria for generalized convexity and generalized monotonicity in the differentiable case, in: Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optim. Appl. 76, Springer, New York (2005), 89-119.
• [7] D. Drusvyatskiy and A. S. Lewis, Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential, SIAM J. Optim. 23 (2013), no. 1, 256-267.
• [8] I. Ginchev and V. I. Ivanov, Second-order characterizations of convex and pseudoconvex functions, J. Appl. Anal. 9 (2003), no. 2, 261-273.
• [9] T. Jabarootian and J. Zafarani, Generalized invariant monotonicity and invexity of non-differentiable functions, J. Global Optim. 36 (2006), no. 4, 537-564.
• [10] T. Jabarootian and J. Zafarani, Generalized vector variational-like inequalities, J. Optim. Theory Appl. 136 (2008), no. 1, 15-30.
• [11] T. Jabarootian and J. Zafarani, Characterizations of preinvex and prequasiinvex set-valued maps, Taiwanese J. Math. 13 (2009), no. 3, 871-898.
• [12] V. Jeyakumar and X. Q. Yang, Approximate generalized Hessians and Taylor’s expansions for continuously Gâteaux differentiable functions, Nonlinear Anal. 36 (1999), no. 3, 353-368.
• [13] A. B. Levy, R. A. Poliquin and R. T. Rockafellar, Stability of locally optimal solutions, SIAM J. Optim. 10 (2000), no. 2, 580-604.
• [14] B. S. Mordukhovich, Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR 254 (1980), no. 5, 1072-1076.
• [15] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I. Basic Theory, Grundlehren Math. Wiss. 330, Springer, Berlin, 2006.
• [16] B. S. Mordukhovich and T. T. A. Nghia, Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces, Nonlinear Anal. 86 (2013), 159-180.
• [17] B. S. Mordukhovich and J. V. Outrata, On second-order subdifferentials and their applications, SIAM J. Optim. 12 (2001), no. 1, 139-169.
• [18] B. S. Mordukhovich and R. T. Rockafellar, Second-order subdifferential calculus with applications to tilt stability in optimization, SIAM J. Optim. 22 (2012), no. 3, 953-986.
• [19] B. S. Mordukhovich, R. T. Rockafellar and M. E. Sarabi, Characterizations of full stability in constrained optimization, SIAM J. Optim. 23 (2013), no. 3, 1810-1849.
• [20] M. Oveisiha and J. Zafarani, Vector optimization problem and generalized convexity, J. Global Optim. 52 (2012), no. 1, 29-43.
• [21] M. Oveisiha and J. Zafarani, Super efficient solutions for set-valued maps, Optimization 62 (2013), no. 6, 817-834.
• [22] J.-P. Penot, Calculus Without Derivatives, Grad. Texts in Math. 266, Springer, New York, 2013.
• [23] R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1805-1838.
• [24] R. A. Poliquin and R. T. Rockafellar, Tilt stability of a local minimum, SIAM J. Optim. 8 (1998), no. 2, 287-299.
• [25] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216.
• [26] R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, New York, 1998.
• [27] X. Q. Yang and V. Jeyakumar, Generalized second-order directional derivatives and optimization with C1,1 functions, Optimization 26 (1992), no. 3-4, 165-185.
• [28] C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).