Weak lower Studniarski derivative in set-valued optimization (Q2888009)

Let \(X,Y,Z\) be real normed spaces and \(F:X\to 2^Y,\, G:X\to 2^Z\) two set-valued maps. Suppose that \(Y,Z\) are ordered by the cones \(S\subset Y\) and \(Q\subset Z\) and let \(F_+(x)=F(x)+S,\, G_+(y)=G(y)+Q,\, ~x\in X,\, ~y\in Y\). The authors consider the constrained set-valued optimization problem (SOP): min \(F(x)\) for \(x\in M,\) where \(M:=\{x'\in X: G(x')\cap (-Q)\neq\emptyset\}.\) A point \(y_0\) in a subset \( D\) of \(Y\) is called an efficient point of \(D,\) denoted by \(y_0\in \mathrm{Min}_SD,\) if \((D-y_0)\cap(-S)=\{0\}.\) A pair \((x_0,y_0)\in M\times F(x_0)\) is called a strict local minimizer of order \(m\) (\(m\in\mathbb N\)) for (SOP), denoted by \((x_0,y_0)\in \mathrm{Strl}_S(F,M;m),\) if there exist a constant \(\alpha > 0\) and a neighborhood \(U\) of \(x_0\) such that \(\,(F(x)+S)\cap B(y_0,\alpha\|x-x_0\|^m)=\emptyset\,\) for all \(x\in M\cap U\setminus\{x_0\}.\) Finally, the weak lower Studniarski derivative of order \(m\) is defined by NEWLINE\[NEWLINE\begin{multlined} \underline{d}^mF(x_0,y_0)(u)=\\ \{v\in Y : \forall h_n\to 0+,\, \exists (u_n,v_n)\to (u,v) \; s.t. \;\forall n,\, y_0+h_n^mv_n\in F(x_0+h_nu_n).\end{multlined}NEWLINE\]NEWLINE This extends to the set-valued vector setting the lower Studniarski derivative of an extended real-valued function \(f\), defined by NEWLINE\[NEWLINE\underline{d}^m f(x,h)=\liminf_{t\to 0+,\, v\to h} t^{-m}\left(f(x+th)-f(x)\right),NEWLINE\]NEWLINE the higher-order analogue of Dini derivative, see \textit{M. Studniarski} [SIAM J. Control Optimization 24, 1044--1049 (1986; Zbl 0604.49017)]. The authors give higher-order necessary conditions for strict local minimizers of the problem (SOP) expressed in terms of weak lower Studniarski derivative. For instance, in Theorem 4.1, one proves that if \((x_0,y_0)\in \mathrm{Strl}_S(F,M;m),\) then NEWLINE\[NEWLINE [d^m_w(F_+,G_+)((x_0,y_0),z_0)(x)+(\theta_Y,z_0)\cap((-\mathrm{int} S) \times (-\mathrm {int} Q))=\emptyset,NEWLINE\]NEWLINE for all \(x\in \mathrm{dom} [(F_+,G_+)((x_0,y_0)z_0)]\), where \((F_+,G_+)(x',y')=F_+(x')\times G_+(y'),\, ~x'\in X,\,~ y'\in Y\). This extends a similar result obtained by Studniarski [loc. cit.] for nonsmooth real-valued functions.NEWLINENEWLINEIn the last section of the paper (Section 5), higher-order Fritz-John type optimality conditions for strict local minimizers of this problem are investigated by similar methods.