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This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient conditions for a critical point to be a local minimizer, we also present and discuss counterexamples for some statements that can be found in the literature of undergraduate Calculus related to Lagrange Multipliers, such as 'between the critical points, the ones which have the smallest image (under the function) are minimizers' or 'a single critical point (which is a local minimizer) is a global minimizer'.

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]

This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient conditions for a critical point to be a local minimizer, we also present and discuss counterexamples for some statements that can be found in the literature of undergraduate Calculus related to Lagrange Multipliers, such as 'between the critical points, the ones which have the smallest image (under the function) are minimizers' or 'a single critical point (which is a local minimizer) is a global minimizer'. [ABSTRACT FROM AUTHOR]

Augmented Lagrangian (AL) algorithms are very popular and successful methods for solving constrained optimization problems. Recently, global convergence analysis of these methods has been dramatically improved by using the notion of sequential optimality conditions. Such conditions are necessary for optimality, regardless of the fulfillment of any constraint qualifications, and provide theoretical tools to justify stopping criteria of several numerical optimization methods. Here, we introduce a new sequential optimality condition stronger than previously stated in the literature. We show that a well-established safeguarded Powell–Hestenes–Rockafellar (PHR) AL algorithm generates points that satisfy the new condition under a Lojasiewicz-type assumption, improving and unifying all the previous convergence results. Furthermore, we introduce a new primal–dual AL method capable of achieving such points without the Lojasiewicz hypothesis. We then propose a hybrid method in which the new strategy acts to help the safeguarded PHR method when it tends to fail. We show by preliminary numerical tests that all the problems already successfully solved by the safeguarded PHR method remain unchanged, while others where the PHR method failed are now solved with an acceptable additional computational cost. [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]