Joaquim Dias Garcia

Raphael Araujo Sampaio, Joaquim Dias Garcia, Marcus Poggi et al.

Due to their conceptual simplicity, k-means algorithm variants have been extensively used for unsupervised cluster analysis. However, one main shortcoming of these algorithms is that they essentially fit a mixture of identical spherical Gaussians to data that vastly deviates from such a distribution. In comparison, general Gaussian Mixture Models (GMMs) can fit richer structures but require estimating a quadratic number of parameters per cluster to represent the covariance matrices. This poses two main issues: (i) the underlying optimization problems are challenging due to their larger number of local minima, and (ii) their solutions can overfit the data. In this work, we design search strategies that circumvent both issues. We develop more effective optimization algorithms for general GMMs, and we combine these algorithms with regularization strategies that avoid overfitting. Through extensive computational analyses, we observe that optimization or regularization in isolation does not substantially improve cluster recovery. However, combining these techniques permits a completely new level of performance previously unachieved by k-means algorithm variants, unraveling vastly different cluster structures. These results shed new light on the current status quo between GMM and k-means methods and suggest the more frequent use of general GMMs for data exploration. To facilitate such applications, we provide open-source code as well as Julia packages (UnsupervisedClustering.jl and RegularizedCovarianceMatrices.jl) implementing the proposed techniques.

Mathieu Besançon, Joaquim Dias Garcia, Benoît Legat et al.

We introduce DiffOpt.jl, a Julia library to differentiate through the solution of optimization problems with respect to arbitrary parameters present in the objective and/or constraints. The library builds upon MathOptInterface, thus leveraging the rich ecosystem of solvers and composing well with modeling languages like JuMP. DiffOpt offers both forward and reverse differentiation modes, enabling multiple use cases from hyperparameter optimization to backpropagation and sensitivity analysis, bridging constrained optimization with end-to-end differentiable programming. DiffOpt is built on two known rules for differentiating quadratic programming and conic programming standard forms. However, thanks ability to differentiate through model transformation, the user is not limited to these forms and can differentiate with respect to the parameters of any model that can be reformulated into these standard forms. This notably includes programs mixing affine conic constraints and convex quadratic constraints or objective function.

Andrew W. Rosemberg, Joaquim Dias Garcia, François Pacaud et al.

Differentiating through constrained optimization problems is increasingly central to learning, control, and large-scale decision-making systems, yet practical integration remains challenging due to solver specialization and interface mismatches. This paper presents a general and streamlined framework-an updated DiffOpt.jl-that unifies modeling and differentiation within the Julia optimization stack. The framework computes forward - and reverse-mode solution and objective sensitivities for smooth, potentially nonconvex programs by differentiating the KKT system under standard regularity assumptions. A first-class, JuMP-native parameter-centric API allows users to declare named parameters and obtain derivatives directly with respect to them - even when a parameter appears in multiple constraints and objectives - eliminating brittle bookkeeping from coefficient-level interfaces. We illustrate these capabilities on convex and nonconvex models, including economic dispatch, mean-variance portfolio selection with conic risk constraints, and nonlinear robot inverse kinematics. Two companion studies further demonstrate impact at scale: gradient-based iterative methods for strategic bidding in energy markets and Sobolev-style training of end-to-end optimization proxies using solver-accurate sensitivities. Together, these results demonstrate that differentiable optimization can be deployed as a routine tool for experimentation, learning, calibration, and design-without deviating from standard JuMP modeling practices and while retaining access to a broad ecosystem of solvers.

Joaquim Dias Garcia, Alexandre Street, Tito Homem-de-Mello et al.

Forecasting and decision-making are generally modeled as two sequential steps with no feedback, following an open-loop approach. In this paper, we present application-driven learning, a new closed-loop framework in which the processes of forecasting and decision-making are merged and co-optimized through a bilevel optimization problem. We present our methodology in a general format and prove that the solution converges to the best estimator in terms of the expected cost of the selected application. Then, we propose two solution methods: an exact method based on the KKT conditions of the second-level problem and a scalable heuristic approach suitable for decomposition methods. The proposed methodology is applied to the relevant problem of defining dynamic reserve requirements and conditional load forecasts, offering an alternative approach to current ad hoc procedures implemented in industry practices. We benchmark our methodology with the standard sequential least-squares forecast and dispatch planning process. We apply the proposed methodology to an illustrative system and to a wide range of instances, from dozens of buses to large-scale realistic systems with thousands of buses. Our results show that the proposed methodology is scalable and yields consistently better performance than the standard open-loop approach.

Raphael Araujo Sampaio, Gerson Couto Oliveira, Luiz Carlos da Costa et al.

The increasing penetration of variable renewable energy (VRE) has brought significant challenges for power systems planning and operation. These highly variable sources are typically distributed in the grid; therefore, a detailed representation of transmission bottlenecks is fundamental to approximate the impact of the transmission network on the dispatch with VRE resources. The fine grain temporal scale of short term and day-ahead dispatch, taking into account the network constraints, also mandatory for mid-term planning studies, combined with the high variability of the VRE has brought the need to represent these uncertainties in stochastic optimization models while taking into account the transmission system. These requirements impose a computational burden to solve the planning and operation models. We propose a methodology based on community detection to aggregate the network representation, capable of preserving the locational marginal price (LMP) differences in multiple VRE scenarios, and describe a real-world operational planning study. The optimal expected cost solution considering aggregated networks is compared with the full network representation. Both representations were embedded in an operation model relying on Stochastic Dual Dynamic Programming (SDDP) to deal with the random variables in a multi-stage problem.