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We discuss the (first- and second-order) optimality conditions for nonlinear programming under the relaxed constant rank constraint qualification (RCRCQ). Although the optimality conditions are well established in the literature, the proofs presented here are based solely on the well-known inverse function theorem. This is the only prerequisite from real analysis used to establish two auxiliary results needed to prove the optimality conditions. To be precise, we provide a simple and alternative proof that RCRCQ is a constraint qualification that implies strong second-order optimality conditions. [ABSTRACT FROM AUTHOR]

We generalize the existing formulation and results on linear separability of sets. In order to characterize the solution of the generalized problem, we use the concepts of convex hulls. For finite sets, it is well known the Support Vector Machine technique for finding the optimal separating hyperplane. Here we consider arbitrary sets, allowing infinite, unbounded and nonclosed sets. The problem is formulated as an optimization problem with possibly infinitely many constraints. We prove existence and uniqueness of the solution. Besides, we present some examples and counterexamples to many properties discussed in the text and statements in the literature. [ABSTRACT FROM AUTHOR]