Second-Order Driectional Derivatives And Optimality Conditions

Second-order nonsmooth analysis by virtue of second-order directional derivatives have been extensively studied.In particular, second-order opti-mality conditions which be abtained by a variety of second-order directional derivatives play a crucial role in optimization theory. In the last few years, the useful connections between second-order directional derivatives for locally Lipschtiz and C1,1 functions have been deeply studied. Moreover, People also compared second-order optimality conditions obtained by using different types of second-order directional derivatives, and made a series of interesting results.This is a survey of obtained results of some important second-order di-rectional derivatives and optimality conditions.We present some results on second-order directional derivatives and opti-mality conditions researched by X. Q. Yang, K. Pastor, and D. Bednarik since 1996, including locally Lipschtiz and C1,1 functions, we establish the con-nections between second-order directional derivatives and compare optimality conditions obtained by them.Besides, we also give second-order sufficient optimality conditions for l-stable function by Peano derivatives.1.§3.1 We present some results on second-order directional derivatives and optimality conditions for locally Lipschtiz and C1,1 functions. In which,§3.1.1 we establish the connections between second-order directional deriva-tives;§3.1.2 Give the charaterization of convexity for functions with various properties;§3.1.2 We compare second-order optimality conditions obtained by using different types of second-order directional derivatives 1.1. Yang [9] es-tablish connections between the parabolic second-order directional derivatives introduced by Ben-Tal and Zowe [4] and the generalized second-order direc-tional derivatives proposed by Cominetti-Correa [2].Theorem 1 Let f:X→R be C1,1. Then we have for x, u, v∈X, If f(2)(x; u, v) exists, then for x, u, v∈X,1.2. Yang [10] introduced the new parabolic second-order directional derivatives for locally Lipschitz functions and establish connections with other parabolic second-order directional derivatives.Theorem 2 Letf:X→R be a locally Lipschitz function and let x, u∈X. ThenTheorem 3 Let f:X→R be a locally Lipschitz function and let x, u∈X. ThenTheorem 4 Let f:X→R be a locally Lipschitz function and let x, u∈X.Then Moreover, whereTheorem 5 Let f:X→R be a locally Lipschitz function and let x, u∈X. Then Moreover,1.3 Huang and Ng [22] studied some Relations between Chaney's Gen-eralized Second-order Directional Derivatives and that of Ben-Tal and Zowe.定理13 ([22]) Letx, u, x*∈Rn satisy the properties: (i) x*∈(?)uf(x) and x*(u)= f'(x; u). (ii) Both andγ= sup{D+(2)f(x; u, v) - x*(v):v∈S} are finite. (ii) fC(2)(x;x*,u) exists. Then, D(2) f(x;u,v) exists, and1.4 Pastor [13] Promotes the conclusion in[12] and established connec-tions among generalized second-order directional derivatives for regularly lo-cally Lipschitz functions. Theorem 6设f:X→R be a regularly locally Lipschitz function, x,h∈X. Then1.5 2005年, Pastor [15] compared the lower generalized second-order directional derivatives in the sense of Michel-Penot with the generalized second-order directional derivatives for C1,1.Theorem 7 Let f:X→R be a C1,1 function, x∈X, u, v∈X.. Then1.6 Yang [10] stated a second-order sufficient condition for the minimiza-tion problem (P)Theorem 8 Assume that f is C1,1 and▽f(x)=0.If either (ⅰ)f(2)(a;u,0)>0,u≠0;(assume f(2)(a; u,0) exists) (ⅱ)f∞(a;u, -u)<0,(?)∈X, u≠0. Then a is a strict local minimizer of(P).[13] stated a new second-order necessary condition for locally Lipschitz functions.Theorem 9 (necessary condition) Let f:X→R is a locally Lipschitz function and f attain a local minimum at x. Then Pastor [15] by means of the Clarke subdifferential state a new sufficient opti-mality condition for locally Lipschitz functions.Theorem 10 ([15]) Let f:X→R be Lipschitz on a neighbourhood U of x∈X and let f have the Frechet generalized second-order derivative at x. If 0∈(?)cf(x) and c>0 is such that for every h∈S we have f+(2)(x; h, h)≥c, then f attains a strict local minimum at x.2.§3.2 Bednarik,和Pastor [17] have attempted to weaken the C1,1 assumption and study so called l-stable function in order to giver second-order suifficient optimality conditions by Peano derivatives.定理14 Let f:Rn→R be continuous near x∈Rn and let f be l-stable at x.If fl(y; h)=0 for every h∈S, and then x is an isolated minimizer of order 2 for f. Conversely, each isolated minimizer of order 2 satisfies these sufficient conditions.