Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization

We consider Karush-Kuhn-Tucker second-order optimality conditions for nonsmooth set-valued optimization with attention to the envelope-like effect. To analyse the critical feasible directions, which produce this phenomenon, we use the contingent derivatives, the adjacent derivatives and the corresponding asymptotic derivatives, since directions are explicitly involved in these kinds of derivatives. To pursue strong multiplier rules, we impose cone-Aubin conditions to deal with the objective and constraint maps separately. In this way, we can invoke constraint qualifications of the Kurcyusz-Robinson-Zowe type. To our knowledge, some of the results are new; they will be indicated explicitly. The paper also discusses improvements or extensions of known results.