Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications (Q925264)

The paper contains optimality conditions for set-constrained optimization problems: \[ \text{minimize }f(x), \] \[ \text{subject to }x\in D= \{x\in X\mid F(x)\in Q\}, \] where \(X\) and \(Y\) are Banach spaces, \(f: X\to\mathbb{R}\) is a (real-valued) smooth function, \(F: X\to Y\) is a (vector-valued) smooth mapping and \(Q\subset Y\) is a fixed closed convex set. As a generalization of the known Robinson constraint qualification the so-called 2-regularity condition of the mapping \(F\) at the point \(x^0\in D\) with respect to the set \(Q\) in a direction \(h\in X\) is introduced according to \[ 0\in\text{int}(F(x^0)+ \text{im\,}F'(x^0)+ F''(x^0)[h, (F'(x^0))' (Q- F(x^0))]- Q). \] After the characterization of this condition and the suitable description of the contingent cone \(T_D(x^0)\) by the first and second derivatives of \(F\), the authors provide the main results of the paper, i.e. first-order (only the first derivation of the objective function is used) and second-order (the first two derivatives of the objective function are used) necessary optimality conditions for the given problem under 2-regularity assumptions. The results are supported by illustrative examples.