Necessary Optimality Conditions In Terms Of Convexificators For Nonsmooth Optimization

The idea of convexificator, which is introduced in 1994 as a weaker version of the notion of subdifferentials of nonsmooth functions, has been used to extend,unify, and sharpen various results in nonsmooth analysis and optimization.On one hand, noncompactness of convexificators can provide a useful tool for analyzing continuous function, not necessarily Lipschitz. For a Lipschitz function,on the other hand, most known subdifferentials, such as the subdifierentials of Clarke, Michel-Penot, Ioffe-Morduchovich, and Treiman are convexificators, and these known subdifferentials may contain the convex hull of a convexificator.Therefore, from the viewpoint of optimization and applications, the descriptions of the necessary optimality conditions in terms of convexificators provide sharper results. Various important results have so far been obtained concerning calculusrules for convexificators, characterization of generalized convexity of functions through convexificators, and F-J type, K-T type, and stronger K-T type necessary optimality conditions in terms of convexificators for nonsmooth optimization, etc.The purpose of this thesis is (i) to study and to make a survey of the calculusrules for convexificators, the problem of hunting for a smaller sub differential, and, in particular, necessary optimality conditions and constraint qualifications in terms of convexificators for nonsmooth optimization; (ii) to investigate for Lipschitz functions relationship of convexificators and another important subdif- ferential called exhausters.The thesis contains four parts.In the second part, we give the notion of convexificator, present the main calculus properties of convexificators, including generalized mean-value theorems and chain rules for composite functions, and discuss the problem of hunting for a smaller subdifferential. These are helpful for the study of necessary optimality conditions. As follows, we introduce the notion of convexificator.Let X be a real Banach space.Definition 1 (1) The function f : X→(?) = R∪{+∞} is said to have an upper convexificator (?)^*f(x) at x if (?)^*f(x) (?) X^* is weak^* closed and, for each v∈X,(2) The function f : X→(?)= R∪{+∞} is said to have a lower convexificator(?)^*f(x) at x if (?)_*f(x)(?)X^* is weak^* closed and, for each v∈X,(3) The function f : X→(?) = R∪{+∞} is said to have a convexificator (?)^*f(x) at x if it is both upper and lower convexificator of f at x.In the third part, for nonsmooth optimization problems, various important results concerning necessary optimality conditions expressed in terms of convexificatorsare presented. As follows, we stated the main results.1. Minimization with inequality constraint.Theorem 1 Let x₀ be a minimizer of problem (P). Assume that f, g₁,…, g_m are continuous and admit bounded convexificators (?)^*f(x₀), (?)^*g₁(x₀),…, (?)^*g_m(x₀) respectively and that (?)^*f, (?)^*g₁,…, (?)^*g_m are upper semicontinuous at x₀. Then, there exist scalarsλ₀≥0,λ₁≥0,…,λ_m≥0, with∑_i=0^mλ_i=1,such thatλ_igi(x₀) = 0.Theorem 2 Let x₀∈X be a local minimizer of problem (PI). Suppose that at x₀, f admits an upper semiregular convexificator (?)^*f(x₀) and g_j admits an upper convexificator (?)^*g_j(x₀) for each j∈J(x₀). If (ACQ) holds at x₀, then,If, in addition, (?)^*f(x₀) is bounded, thenTheorem 3 Let x₀∈X be an efficient solution to problem (VP). Suppose that, at x₀,(a) f_i and g_j admit, respectively, upper convexificator (?)^*f_i(x₀) and (?)^*g_j(x₀), i∈I and j∈J(x₀);(b) f_i is directionally differentiable, i∈I, with f′_i0(x₀,·) being linear for some i₀∈I and f′_k(x₀,·) sublinear for all k∈I\{i₀};(c) g_j^↓(x₀,·) is sublinear j∈J(x₀).If (GGCQ) holds at x₀, then there exist real numbersα_i > 0 andβ_j≥0, i∈I and j∈J(x₀), such that2. Minimization with a set constraint.Theorem 4 Let us consider the problem (GVP) where each f_j is a locally Lipschitz function and C a closed convex set. Let x₀∈C be a weak efficient point for (GVP). Let us consider that for each j = 1,2,…, k, the functions f_j have a bounded upper semiregular covexificator (?)^*f_j(x₀) at x₀. Then there existsÏ„_j≥0, not all zero such thatwhere N(C, x₀) = {v∈Rⁿ| < v, z - x₀ >≤0, (?)z∈C}.Theorem 5 Let x₀∈X be a local minimizer of problem (PS). Suppose that f admits an upper convexificator (?)^*f(x₀) at x₀. Then for every closed cone A(?)A(X,x₀),0∈cl(co(?)^*f(x₀) + A^-).If, in addition, (?)^*f(x₀) is bounded, then0∈co(?)^*f(x₀) + A^-.3. Minimization with equality constraint and inequality constraint.Theorem 6 Assume that (?)^*H is an approximate Jacobian map of H which is upper semicontinuous at x₀. If x₀ is a local weakly efficient solution of (VP₁), then there is a vectorλ₀ = (ξ₀,θ₀,γ₀)∈T such that0∈λ₀(clco(?)^*H(x₀)∪co(((?)^*H(x₀))_∞\{0})),θ_0g(x₀) = 0.4. Minimization with multi-constraint.Theorem 7 For the problem (PE), let F(x) = (f₀(x),…, f_m(x)), Assume that F admits a locally bounded approximate Jacobian at (?)∈Rⁿ. If (?) is a minimizer of (PE), then there exist Lagrange multipliersλ₀≥0,…,λ_p≥0,λ_p+1,…,λ_m, not all zero, such thatλ_if_i(?) = 0, i = 1,…,m, where e_i = [0,…, 0,1,0,…, 0]^T∈R^m+1 is a unit vector.In the forth part, we prove a theorem about relationship of convexificators and exhausters.Theorem 8 Let f : Rⁿ→R be Dini directionally differentiable, let f′(x, g) be the Dini derivative of f at x in the direction g, and let h(g) = f′(x, g). Assume h(g) is a positively homogeneous function and f(x), h(g) are locally Lipschitz.Then, there exist an upper exhauster E^*h of h and an upper semiregular convex-ificator (?)^*f(x) of f(x), such that(?)^*f(x)∈E^*h.