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Clóvis Caesar Gonzaga, a renowned figure in interior point methods, made significant contributions to the field of operational research. His analysis of path following methods in linear programming, presented in the late 1980s, remains the best complexity bound in the field. Born in Brazil in 1944, Gonzaga obtained degrees in electronic engineering and systems engineering and computer science. He held various academic positions, including professorships at COPPE UFRJ and UFSC. Gonzaga's work revolutionized the optimization community worldwide, and his legacy continues to inspire researchers in Brazil and beyond. [Extracted from the article]

In this work, we propose integral global optimality conditions for multiobjective problems not necessarily differentiable. The integral characterization, already known for single objective problems, are extended to multiobjective problems by weighted sum and Chebyshev weighted scalarizations. Using this last scalarization, we propose an algorithm for obtaining an approximation of the weak Pareto front whose effectiveness is illustrated by solving a collection of multiobjective test problems. [ABSTRACT FROM AUTHOR]

We describe two algorithms for solving differentiable convex optimization problems constrained to simple sets in ℝn, i.e., sets on which it is easy to project an arbitrary point. The algorithms are optimal in the sense that they achieve an absolute precision of ε in relation to the optimal value in O (1√ε) iterations using only first-order information. This complexity depends on an (unknown) Lipschitz constant L* for the function derivatives and on a (known) strong convexity constant μ* ≥ 0 . The algorithms are extensions of well-known methods devised by Nesterov [Introductory Lectures on Convex Optimization, Kluwer Academic, Boston, 2004], without the need for estimates of L* and including (in the second algorithm) line searches and an adaptive procedure for estimating a strong convexity constant. The algorithms are such that all computed points are feasible, and the complexity analysis follows a simple geometric approach. Numerical tests for box-constrained quadratic problems are presented. [ABSTRACT FROM AUTHOR]

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]

We present a general filter algorithm that allows a great deal of freedom in the step computation. Each iteration of the algorithm consists basically in computing a point which is not forbidden by the filter, from the current point. We prove its global convergence, assuming that the step must be efficient, in the sense that, near a feasible nonstationary point, the reduction of the objective function is "large." We show that this condition is reasonable, by presenting two classical ways of performing the step which satisfy it. In the first one, the step is obtained by the inexact restoration method of Martínez and Pilotta. In the second, the step is computed by sequential quadratic programming. [ABSTRACT FROM AUTHOR]

This paper presents some examples of ill-behaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data. [ABSTRACT FROM AUTHOR]

In this paper we present a filter algorithm for nonlinear programming and prove its global convergence to stationary points. Each iteration is composed of a feasibility phase, which reduces a measure of infeasibility and an optimality phase, which reduces the objective function in a tangential approximation of the feasible set. These two phases are totally independent,and the only coupling between them is provided by the filter. The method is independent of the internal algorithms used in each iteration, as long as these algorithms satisfy reasonable assumptions on their efficiency. Under standard hypotheses, we show two results: for a filter with minimum size, the algorithm generates a stationary accumulation point; for a slightly larger filter, all accumulation points are stationary. [ABSTRACT FROM AUTHOR]