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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]

We consider risk-averse formulations of stochastic linear programs having a structure that is common in real-life applications. Specifically, the optimization problem corresponds to controlling over a certain horizon a system whose dynamics is given by a transition equation depending affinely on an interstage dependent stochastic process. We put in place a rolling-horizon time consistent policy. For each time step, a risk-averse problem with constraints that are deterministic for the current time step and uncertain for future times is solved. To each uncertain constraint corresponds both a chance and a Conditional Value-at-Risk constraint. We show that the resulting risk-averse problems are numerically tractable, being at worst conic quadratic programs. For the particular case in which uncertainty appears only on the right-hand side of the constraints, such risk-averse problems are linear programs. We show how to write dynamic programming equations for these problems and define robust recourse functions that can be approximated recursively by cutting planes. The methodology is assessed and favourably compared with Stochastic Dual Dynamic Programming on a real size water-resource planning problem. [ABSTRACT FROM AUTHOR]

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau envelopes), will have a natural extension from convex optimization to nonconvex optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish. In this paper we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proved for the class of nonconvex functions that are prox-bounded and lower- $${\mathcal{C}}^2$$ and encouraging preliminary numerical testing is reported. [ABSTRACT FROM AUTHOR]

For solving nonsmooth convex constrained optimization problems, we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods. On the other hand, the filter criterion for accepting a candidate point as the new iterate is sometimes easier to satisfy than the usual descent condition in bundle methods. Some encouraging preliminary computational results are also reported. [ABSTRACT FROM AUTHOR]

For convex minimization we introduce an algorithm based on -space decomposition. The method uses a bundle subroutine to generate a sequence of approximate proximal points. When a primal-dual track leading to a solution and zero subgradient pair exists, these points approximate the primal track points and give the algorithm's , or corrector, steps. The subroutine also approximates dual track points that are -gradients needed for the method's -Newton predictor steps. With the inclusion of a simple line search the resulting algorithm is proved to be globally convergent. The convergence is superlinear if the primal-dual track points and the objective's -Hessian are approximated well enough. [ABSTRACT FROM AUTHOR]

Branch-and-Cut algorithms for general 0-1 mixed integer programs can be successfully implemented by using Lift-and-Project (L&P) methods to generate cuts. L&P cuts are drawn from a cone of valid inequalities that is unbounded and, thus, needs to be truncated, or "normalized". We consider general normalizations defined by arbitrary closed convex sets and derive dual problems for generating L&P cuts. This unified theoretical framework generalizes and covers a wide group of already known normalizations. We also give conditions for proving finite convergence of the cutting plane procedure that results from using such general L&P cuts. [ABSTRACT FROM AUTHOR]