Higher-Order Karush--Kuhn--Tucker Conditions in Nonsmooth Optimization
Phan Quốc Khánh, Nguyễn Minh Tùng · SIAM Journal on Optimization · 2018
For a general set-valued problem with set and inclusion constraints we establish higher-order Karush--Kuhn--Tucker multiplier rules with additional complementarity slackness conditions in terms of contingent-type derivatives of index $\gamma \in \{0,1\}$ for weak and firm solutions. Our multiplier rules are in a nonclassical form with a supremum expression on the right-hand side (instead of zero). An example is provided to show a case in which the fourth-order envelope-like effect occurs. We also prove Karush--Kuhn--Tucker rules in terms of Studniarski's derivatives under directional higher-order Hölder metric subregularity. The results are novel even in classical finite-dimensional scalar problems of nonlinear programming. As applications, we consider vector nonlinear programming to have detailed comparisons with known results.