Subset Of Non-convex Second-order Necessary Optimality Conditions And Lagrange Multiplier

In this paper, the active rank condition andΛ₁(x^*) are defined. When the setof lagrange multipliersΛ(x^*) is the bounded line segment and x^* satisfies the gen-eralized strict complementary slackness condition, the classical necessary second-order optimality condition is proved on the proper set. Two examples are given thatshowed that the classical necessary second-order optimality condition still holdswithout the conditions that the set of lagrange multipliersΛ(x^*) is the bounded linesegment and the generalized strict complementary slackness condition. The theorythat the set of lagrange multipliersΛ(x^*) is the bounded line segment is extended.