arXiv subjects

Julio González-Díaz

Publications and source records attributed to Julio González-Díaz.

11 recordsLinked to original sources

arXiv2026

Bound tightening in lifted formulations: (sub)solver-dependent impact on performance in RLT-based algorithms

In this paper we explore a relevant aspect of the interplay between two core elements of global optimization algorithms for nonconvex nonlinear programming problems. The first one is the reformulation of the original problem, which requires the introduction of auxiliary variables with the goal of defining convex relaxations that can be solved both reliably and efficiently on a node-by-node basis. The second one, bound tightening or, more generally, domain reduction, allows to reduce the search space to be explored by the branch-and-bound algorithm. We are interested in the performance implications of propagating the bounds of the original variables to the auxiliary ones in the lifted space: does this propagation reduce the overall size of the tree? does it improve the efficiency of solving the node relaxations? To better understand the above interplay, we focus on the reformulation-linearization technique for polynomial optimization. In this setting we present a theoretical result on the implicit bounds of the auxiliary variables in the RLT relaxations, which sets the stage for the ensuing computational study, whose goal is to assess to what extent the performance of an RLT-based algorithm may be affected by the decision to explicitly propagate the bounds on the original variables to the auxiliary ones.

math.OC↗

arXiv2026

A note on the convergence guarantees of RLT-based algorithms for polynomial optimization

This paper identifies and addresses a mathematical oversight in one of the foundational results on the Reformulation-Linearization Technique (RLT) for polynomial optimization. Correctness of the original result can be easily recovered by adding a minor and natural assumption. However, not being aware of this nuance may lead to the loss of convergence guarantees in RLT-based algorithms, especially if they rely on filtering strategies for the construction of the bound-factor constraints.

math.OC↗

arXiv2026

A new classification method based on Minimum Spanning Trees

Minimum Spanning Trees have been used in unsupervised learning, particularly in clustering tasks, due to their ability to recognize clusters by removing edges that are considered inconsistent in defining those clusters. This paper aims to study the use of Minimum Spanning Trees in supervised learning. Specifically, we propose a classification algorithm based on Minimum Spanning Trees. To improve its performance, we introduce a robust version of the method that is also computationally more efficient. We evaluate the effectiveness of our proposed method through an extensive simulation study. We also apply the proposed methodology to a real-world case study involving aircraft trajectories.

cs.LG↗

arXiv2026

Solving Sparse Mixed-Integer Quadratic Problems: Application to the Unit Commitment Problem with Optimal Power Flow

Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQP) arise in a variety of applications, particularly in energy, water, and gas systems, where discrete decisions interact with nonconvex quadratic constraints. These problems are computationally challenging due to the combination of combinatorial complexity and nonconvexity, often rendering traditional exact methods ineffective for large-scale instances. We propose a solution framework for sparse MIQCQPs that integrates semidefinite programming relaxations with chordal decomposition techniques to exploit sparsity. By leveraging problem structure, we significantly reduce the size of the semidefinite constraints into smaller, tractable blocks, improving the scalability of the relaxation and the overall branch-and-bound procedure. We evaluate our framework on the Unit Commitment problem with AC Optimal Power Flow constraints, a practically relevant and highly challenging problem that couples discrete generation decisions with nonlinear AC network physics. Computational results on standard IEEE test cases with up to 118 buses show that our approach provides strong bounds and high-quality solutions, while scaling significantly better than state-of-the-art global optimization solvers.

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arXiv2025

Impact of domain reduction techniques in polynomial optimization: A computational study

Domain reduction techniques are at the core of any global optimization solver for NLP or MINLP problems. In this paper, we delve into several of these techniques and assess the impact they may have in the performance of an RLT-based algorithm for polynomial optimization problems. These techniques include i) the use of (nonlinear) conic relaxations for optimality-based bound tightening, ii) the use of Lagrangian dual information to enhance feasibility-based bound tightening, and iii) different strategies for branching point selection. One of this paper's main contributions is providing insights into the relative impact of these techniques with respect to each other, which we hope will guide the efforts to develop and implement this type of enhancements in other solvers. We also explore how a solver equipped with these domain reduction enhancements can further improve its performance by using machine learning to better choose the best domain reduction approach to use on a given instance.

math.OC↗

arXiv2025

Parameter estimation in ODEs: assessing the potential of local and global solvers

We consider the problem of parameter estimation in dynamic systems described by ordinary differential equations. A review of the existing literature emphasizes the need for deterministic global optimization methods due to the nonconvex nature of these problems. Recent works have focused on expanding the capabilities of specialized deterministic global optimization algorithms to handle more complex problems. Despite advancements, current deterministic methods are limited to problems with a maximum of around five state and five decision variables, prompting ongoing efforts to enhance their applicability to practical problems. Our study seeks to assess the effectiveness of state-of-the-art general-purpose global and local solvers in handling realistic-sized problems efficiently, and evaluating their capabilities to cope with the nonconvex nature of the underlying estimation problems.

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arXiv2024

Optimal policies for stock redistribution in a retail network: Mathematical modeling and algorithmic solution

We study the problem of stock replenishment and transshipment in the retail industry. We develop a model that can accommodate different policies, including centralized redistribution (replenishment) and decentralized redistribution (lateral transshipments), allowing for direct comparisons between them. We present a numeric analysis in which the benchmark instances stem from the collaboration with a high-end clothing retail company. The underlying model, as usually in the field, is a large-scale mixed-integer linear programming problem. We develop a specific algorithmic procedure to solve this MILP problem and compare its performance with the direct solution via state-of-the-art solvers.

math.OC↗

arXiv2024

An extension of an RLT-based solver to MINLP polynomial problems

In this paper we extend the core branch-and-bound algorithm of an RLT-based solver for continuous polynomial optimization, RAPOSa, to handle mixed-integer problems. We do so by a direct adaptation, in which LP relaxations are replaced with MILP ones and, therefore, the additional burden caused by the discrete variables is taken care of by the auxiliary MILP solver. The analysis then focuses on the impact of a number of choices that must be made for the resulting algorithm to be effective at providing good lower and upper bounds upon termination.

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arXiv2024

Learning in Spatial Branching: Limitations of Strong Branching Imitation

Over the last few years, there has been a surge in the use of learning techniques to improve the performance of optimization algorithms. In particular, the learning of branching rules in mixed integer linear programming has received a lot of attention, with most methodologies based on strong branching imitation. Recently, some advances have been made as well in the context of nonlinear programming, with some methodologies focusing on learning to select the best branching rule among a predefined set of rules leading to promising results. In this paper we explore, in the nonlinear setting, the limits on the improvements that might be achieved by the above two approaches: learning to select the best variable (strong branching) and learning to select the best rule (rule selection).

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arXiv2022

Polynomial Optimization: Enhancing RLT relaxations with Conic Constraints

Conic optimization has recently emerged as a powerful tool for designing tractable and guaranteed algorithms for non-convex polynomial optimization problems. On the one hand, tractability is crucial for efficiently solving large-scale problems and, on the other hand, strong bounds are needed to ensure high quality solutions. In this research, we investigate the strengthening of RLT relaxations of polynomial optimization problems through the addition of nine different types of constraints that are based on linear, second-order cone, and semidefinite programming to solve to optimality the instances of well established test sets of polynomial optimization problems. We describe how to design these conic constraints and their performance with respect to each other and with respect to the standard RLT relaxations. Our first finding is that the different variants of nonlinear constraints (second-order cone and semidefinite) are the best performing ones in around $50\%$ of the instances. Additionally, we present a machine learning approach to decide on the most suitable constraints to add for a given instance. The computational results show that the machine learning approach significantly outperforms each and every one of the nine individual approaches.

math.OC↗

arXiv2022

Learning for Spatial Branching: An Algorithm Selection Approach

The use of machine learning techniques to improve the performance of branch-and-bound optimization algorithms is a very active area in the context of mixed integer linear problems, but little has been done for non-linear optimization. To bridge this gap, we develop a learning framework for spatial branching and show its efficacy in the context of the Reformulation-Linearization Technique for polynomial optimization problems. The proposed learning is performed offline, based on instance-specific features and with no computational overhead when solving new instances. Novel graph-based features are introduced, which turn out to play an important role for the learning. Experiments on different benchmark instances from the literature show that the learning-based branching rule significantly outperforms the standard rules.

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