Some Applications Of Abstract Subdifferentials In Optimization

The primary aim of this article is introduce some applications of abstract subdifferential in optimization problems.On one hand, this article is to derive Lagrange multiplier rules for vector optimization problems using a non-convex separation technique and the concept of abstract subdifferential. Furthermore, we present a method of estimation of the norms of such multipliers in very general cases and for many particular subdifferentials.In section 2.1, we mainly introduce the concept of abstract subdifferential.In section 2.2, we present some results for'exact'case in order to point out the main ideas used in the sequel as well.Theorem 2.6 Let X,Y e z,f eF(XJ) and S be a closed subset ofX. If xQ eS is a weak minimum of/over S with respect to K then forTheorem 2.7 Let Y e%and C be a closed subset of Y. If y0 eC is a weak minimizer of C with respect to K then for every eeintA-, there existsProposition 2.8 Let 7 e%and C be a closed subset of Y. If y0 g C is a weak minimum point for C with respect to K then there exist 3v e^:*\{0}, s.t:-v'e^(C,y0).In section 2.3, In order to apply fuzzy sum rules for subderivatives We present first some results containing necessary optimality conditions for a weak minima of a set.Theorem 2.9 Let Y ex and C be a closed subset of Y. If y0 eC is a weak minimum point for C with respect to K then for everyeeintk and e> 0 there exist z e B(yo,e)f]C, u&K*, u (e)= 1, v e Ne(C,z), s.t:In order to provide comparison of our results with some results in [4] we formulate a corollary of Theorem 2.9 and its proof.Corollary2.11 Let Ye%, Cbe a closed subset of Y, v0 eCbea weak minimum point for C with respect to AT. Then: (i) for every ee(0,p),3zeeB{y0,s)f\C, u]zk\\<\u]\<{p-s-l) s.t: (ii) there exists 3u*e^*\{0}, s.t:-u eN?(C,y0), provided that the application C->.v!->■Ne(C,y) has norm-weak* closed graph.In section2.4, we mainly introduce three applications of fuzzy necessary optimality conditions for vector optimization problems.On the other hand, based on a study of a minimization problem, we present the following results applicable to possibly non-convex sets in Banach space:an opproximate projection result, an extended extremal principle, a non-convex separation theorem, a generalized Bishop-phelps theorem and a separable point result.Our analysis is based on the consideration of the following minimization problem: in conjunction with the inclusion problem find x∈A such that F(x)=0, (IP) where F is a mapping from a Banach space X to another Banach space Y and A is a closed subset of X.Definition 3.1 Let X, Y∈χ. Let F be a mapping from X to Y and let A be a closed subset of X. Consider the minimization problem (MP) and the inclusion problem (IP). We say that x∈A is anε-minimizer of (MP) provided thatε> 0 and‖F(x)‖0. Then the following assertions hold:(ⅰ) (?)u∈A∩BX(x,ε),v∈BX(x,ε) and y*∈Sy* such that(ⅱ) Suppose the abstract subdifferential (?)a is complete and that F is Lipschitz on X with rank L. (?)x∈A∩BX(?)(x,ε) and y*∈Sy*, such that andTheorem 3.3 Let Xi∈χ(i =1,···,m)and Y∈χ. Let X =(?)Xi and F:X→Y be strictly differentiable. Let A:(?)Ai where each Ai is a closed subset of Xi and let x=(x1,···, xm )∈A. Suppose that x is an outerε-minimizer for the corresponding minimizationprob|em Ih{『＼P.).Then the foflowing assertions hold:(i)3y.∈S,+,"=(.气,…,".,)∈彳nB.(x,占)and 1,:(V1,…,1_,)∈B.,(x,占)SUC『7 that嗡suppose that the abstrac{SUbdifferentiaI 8,iS comp|ete and that F IS L『psc厅itz on X with rank L.Thefjx=(xl,…,x,.)∈彳n B. (x,占)andTheorem 3.4￡-efx∈Z and／ef 4,…,4,6e c『osed subsefs 0厂x.Lef 7"hen the f0||owing assert|ons hok|:(i)There ex『Sf2佧彳i n瓦(i,s)andx'∈x',㈦I=1 SUC,7价af(ii)／厂we幻rtherassume maf a.／s comp『efe,价en价ere e州sf一∈.4 r_.瓦(亏.s) andx'∈Ⅳ',㈦l=1 suc厅所afThe fOllowing result iS estabIished in some speciaI case SUCh as when X js the class Of all Banach spaces and a.is the Clarke-RockafeIIa subdi仟erentIal 0c. Theorem 3.6 Let X∈χand I = {1,2,…,m}, let Ai(i∈I) be closed subsets of X with∩i∈I Ai= (?), Let x=(x1,···,xm)∈(?)Ai andε> 0 such that Then the following assertions hold:(ⅰ) There exist ui∈Ai∩BX(xi,ε) and xi*∈X* such that and(ⅱ) Suppose the abstract subdifferential (?)a is complete. Then there existTheorem 3.8 Let X∈χand suppose that X is reflexive. Let J be an arbitrary index set and {Ai:i∈J)be a family of weakly closed subsets of X with empty intersection. Let{Xi:i∈J} be elements in X such that whereεis a positive constant. Then there exist xi∈Ai∩BX(xi,ε) and xi*∈X* such thatTheorem 3.10 Let X∈χand let A be a nonempty proper closed subset of X. Then the following assertions hold:(ⅰ) The set P:={x∈bdA: Na(x,A)≠{0}} is dense in bdA.(ⅱ) Let K:=∪x∈p Na(x,A). Then K is dense in barr(A), i.e., for any x*∈barr(A) andε>0 there exists y*∈K such that‖x*-y*‖≤ε.Theorem 3.12 Let x∈Xm be anε-separable point of the system {A1,...,Am} for someε> 0, Then the following assertions hold:(ⅰ) There exist xi∈Ai∩BX(xi,ε) and x*∈X*,‖x*‖=1 such that(ⅱ) If we further assume that (?)a is complete, then (63) can be strengthened to the following form:...