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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]

Equilibrium problems with equilibrium constraints are appropriate modeling formulations in a number of important areas, such as energy markets, transportation planning, and logistics. These models often correspond to bilevel games, in which certain dual variables, representing the equilibrium price, play a fundamental role. We consider multileader single-follower equilibrium problems having a linear program in the lower level. Because in this setting the lowerlevel response to the leaders' decisions may not be unique, the game formulation becomes ill-posed. We resolve possible ambiguities by considering a sequence of bilevel equilibrium problems, endowed with a special regularization term. We prove convergence of the approximating scheme. Our technique proves useful numerically over several instances related to energy markets. When using PATH to solve the corresponding mixed-complementarity formulations, we exhibit that, in the given context, the regularization approach computes a genuine equilibrium price almost always, while without regularization the outcome is quite the opposite. [ABSTRACT FROM AUTHOR]

The optimal expansion of a power system with reduced carbon footprint entails dealing with uncertainty about the distribution of the random variables involved in the decision process. Optimisation under ambiguity sets provides a mechanism to suitably deal with such a setting. For two-stage stochastic linear programs, we propose a new model that is between the optimistic and pessimistic paradigms in distributionally robust stochastic optimisation. When using Wasserstein balls as ambiguity sets, the resulting optimisation problem has nonsmooth convex constraints depending on the number of scenarios and a bilinear objective function. We propose a decomposition method along scenarios that converges to a solution, provided a global optimisation solver for bilinear programs with polyhedral feasible sets is available. The solution procedure is applied to a case study on expansion of energy generation that takes into account sustainability goals for 2050 in Europe, under uncertain future market conditions. [ABSTRACT FROM AUTHOR]

We present a framework for analyzing convergence and local rates of convergence of a class of descent algorithms, assuming the objective function is weakly convex. The framework is general, in the sense that it combines the possibility of explicit iterations (based on the gradient or a subgradient at the current iterate), implicit iterations (using a subgradient at the next iteration, like in the proximal schemes), as well as iterations when the associated subgradient is specially constructed and does not correspond either to the current or the next point (this is the case of descent steps in bundle methods). Under the subdifferential-based error bound on the distance to critical points, linear rates of convergence are established. Our analysis applies, among other techniques, to prox-descent for decomposable functions, the proximal-gradient method for a sum of functions, redistributed bundle methods, and a class of algorithms that can be cast in the feasible descent framework for constrained optimization. [ABSTRACT FROM AUTHOR]

During the early months of the current COVID-19 pandemic, social distancing measures effectively slowed disease transmission in many countries in Europe and Asia, but the same benefits have not been observed in some developing countries such as Brazil. In part, this is due to a failure to organise systematic testing campaigns at nationwide or even regional levels. To gain effective control of the pandemic, decision-makers in developing countries, particularly those with large populations, must overcome difficulties posed by an unequal distribution of wealth combined with low daily testing capacities. The economic infrastructure of these countries, often concentrated in a few cities, forces workers to travel from commuter cities and rural areas, which induces strong nonlinear effects on disease transmission. In the present study, we develop a smart testing strategy to identify geographic regions where COVID-19 testing could most effectively be deployed to limit further disease transmission. By smart testing we mean the testing protocol that is automatically designed by our optimization platform for a given time period, knowing the available number of tests, the current availability of ICU beds and the initial epidemiological situation. The strategy uses readily available anonymised mobility and demographic data integrated with intensive care unit (ICU) occupancy data and city-specific social distancing measures. Taking into account the heterogeneity of ICU bed occupancy in differing regions and the stages of disease evolution, we use a data-driven study of the Brazilian state of Sao Paulo as an example to show that smart testing strategies can rapidly limit transmission while reducing the need for social distancing measures, even when testing capacity is limited. [ABSTRACT FROM AUTHOR]

We examine how different pricing frameworks deal with non-convex features typical of day-ahead energy prices when the power system is hydro-dominated, like in Brazil. For the system operator, requirements of minimum generation translate into feasibility issues that are fundamental to carry the generated power through the network. When utilities are remunerated at a price depending on Lagrange multipliers computed for a system with fixed commitment, the corresponding values sometimes fail to capture a signal that recovers costs. Keeping in mind recent discussions for the Brazilian power system, we analyse mechanisms that provide a compromise between the needs of the generators and those of the system operator. After characterizing when a price supports a generation plan, we explain in simple terms dual prices and related concepts, such as minimal uplifts and bi-dual problems. We present a new pricing mechanism that guarantees cost recovery to all agents, without over-compensations. Instead of using Lagrange multipliers, the price is defined as the solution to an optimization problem. The behaviour of the new rule is compared to two other proposals in the literature on illustrative examples, including a small, yet representative, hydro-thermal system. [ABSTRACT FROM AUTHOR]

Intermittent sources of energy represent a challenge for electrical networks, particularly regarding demand satisfaction at peak times. Energy management tools such as load shaving or storage systems can be used to mitigate intermittency. In this work, the value of different mechanisms to move energy through time is examined through a multi-objective programming approach, that aims at minimizing operating costs as well as carbon emissions. Among main achievements, we mention a sensitivity result that provides a novel three-dimensional Pareto front. The new tool complements the traditional curves of indifference costs with a quantitative measure for assessing the relative value of the conflicting objectives. The methodology is assessed on three instances representing typical configurations in Brazil, Germany and France, respectively corresponding to a system that is hydro-dominated, thermo-dominated, and with a balanced mix of hydro and thermal power. The corresponding Pareto fronts show that lowering carbon emissions can also reduce generation costs if the power mix is sufficiently diversified. [ABSTRACT FROM AUTHOR]

The V U -algorithm is a superlinearly convergent method for minimizing nonsmooth, convex functions. At each iteration, the algorithm works with a certain V -space and its orthogonal U -space, such that the nonsmoothness of the objective function is concentrated on its projection onto the V -space, and on the U -space the projection is smooth. This structure allows for an alternation between a Newton-like step where the function is smooth, and a proximal-point step that is used to find iterates with promising V U -decompositions. We establish a derivative-free variant of the V U -algorithm for convex finite-max objective functions. We show global convergence and provide numerical results from a proof-of-concept implementation, which demonstrates the feasibility and practical value of the approach. We also carry out some tests using nonconvex functions and discuss the results. [ABSTRACT FROM AUTHOR]

To describe the joint dynamics of prices of crude oil and refined products we extend two-factor models to a multidimensional setting. The new model captures directly the general correlation structure between the different commodities in the form of certain covariance matrix. Since the associated state-space formulation makes use of such correlations, the feasible set of the resulting estimation problem includes the cone of positive semidefinite matrices. Tractability is ensured by means of an interior point method, specially tailored for nonlinear semidefinite programming problems. For different sets of historical prices of crude oil, heating oil, and gasoline, the empirical out-of-sample forecasts obtained with the approach proposed in this work systematically provide an excellent fit to data. [ABSTRACT FROM AUTHOR]

Joint chance constrained problems give rise to many algorithmic challenges. Even in the convex case, i.e., when an appropriate transformation of the probabilistic constraint is a convex function, its cutting-plane linearization is just an approximation, produced by an oracle providing subgradient and function values that can only be evaluated inexactly. As a result, the cutting-plane model may lie above the true constraint. For dealing with such upper inexact oracles, and still solving the problem up to certain precision, a special numerical algorithm must be put in place. We introduce a family of constrained bundle methods, based on the so-called improvement functions, that is shown to be convergent and encompasses many previous approaches as well as new algorithms. Depending on the oracle accuracy, we analyze to what extent the considered methods solve the joint chance constrained program. The approach is assessed on real-life energy problems, arising when dealing with stochastic hydroreservoir management. [ABSTRACT FROM AUTHOR]

Stochastic programming problems arise in many practical situations, hi general, the deterministic equivalents of these problems can be very large and may not be solvable directly by general-purpose optimization approaches. For the particular case of two-stage stochastic programs, we consider decomposition approaches akin to a regularized L-shaped method that can handle inexactness ill the subproblem solution. From a nonsmooth optimization perspective, these variants amount to applying a proximal bundle method to an oracle that gives inaccurate values for the objective function and a subgradient. Rather than forcing early termination of the subproblem optimization to define inexact oracles, we select a small subset of scenarios for which the subproblem solution is exact, and we replace the information for the remaining scenarios by a fast procedure that does not involve solving an optimization problem. The inaccurate oracle information creates inexact cuts in the master program, which are well handled by the recently introduced inexact bundle methods. The proposed approaches are validated by encouraging numerical results on several two-stage stochastic linear programs found in the literature. [ABSTRACT FROM AUTHOR]

The mid-term operation planning of hydro-thermal power systems needs a large number of synthetic sequences to represent accurately stochastic streamflows. These sequences are generated by a periodic autoregressive model. If the number of synthetic sequences is too big, the optimization planning problem may be too difficult to solve. To select a small set of sequences representing the stochastic process well enough, this work employs two variants of the Scenario Optimal Reduction technique. The first variant applies such a technique at the last stage of a tree defined a priori for the whole planning horizon while the second variant combines a stage-wise reduction, preserving the periodic autoregressive structure, with resampling. Both approaches are assessed numerically on hydrological sequences generated for real configurations of the Brazilian power system. [ABSTRACT FROM AUTHOR]

We consider the problem of minimizing nonsmooth convex functions, defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite Hessian on the set of interest. This is a particular case of functions with primal-dual gradient structure, a notion closely related to the so-called VU space decomposition: At a given point, nonsmoothness is locally restricted to the directions of the subspace V, while along the subspace U the behavior of the function is twice differentiable. Constructive identification of the two subspaces is important, because it opens the way to devising fast algorithms for nonsmooth optimization (by following iteratively the manifold of smoothness on which superlinear U-Newton steps can be computed). In this work we show that, for the class of functions in consideration, the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points, provided a minimizer satisfies the nondegeneracy and strong transversality conditions. [ABSTRACT FROM AUTHOR]

Generation expansion planning problems should take into account not only satisfaction of future demand but also environmental constraints, such as government imposed limits on pollution produced by coal-based power plants. For a vertically integrated system of power plants, we model those features as a two-stage stochastic programming problem with recourse that can be solved by decomposition methods. We present results for different demand scenarios, corresponding to low, medium, and high economic future growth, as well as different environmental cases, with stringent and non-stringent emission limits for the thermal plants composing the electric mix. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H , we consider the problem of finding ( x , y ? T ( x )) satisfying x ? A and y ? A ? . An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn. The resulting generalized equation can be solved by using the proximal point algorithm. We consider instead the use of hybrid proximal methods. Hybrid methods use enlargements of operators, close in spirit to the concept of e-subdifferentials. We characterize the enlargement of the partial inverse operator in terms of the enlargement of T itself. We present a new algorithm of resolution that combines Spingarn and hybrid methods, we prove for this method global convergence only assuming existence of solutions and maximal monotonicity of T . We also show that, under standard assumptions, the method has a linear rate of convergence. For the important problem of finding a zero of a sum of maximal monotone operators T 1 , ..., T m , we present a highly parallelizable scheme. Finally, we derive a parallel bundle method for minimizing the sum of polyhedral functions. [ABSTRACT FROM AUTHOR]