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We introduce a modification of Fenchel's conjugation which is a particular case of Moreau's conjugation. We obtain good properties such as convexity of the conjugate function even though the function is not convex. We also introduce the concept of conjugate dual space as a class of continuous operators, while in the Fenchel conjugation the conjugate dual space is the classical topological dual space. Finally, we present some examples for illustrating the difference between the Fenchel–Moreau conjugation and our modification. [ABSTRACT FROM PUBLISHER]

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646-669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties. [ABSTRACT FROM AUTHOR]