arXiv subjects

Yves Lucet

Publications and source records attributed to Yves Lucet.

13 recordsLinked to original sources

arXiv2025

A unified vertical alignment and earthwork model in road design with a new convex optimization model for road networks

The vertical alignment optimization problem in road design seeks the optimal vertical alignment of a road at minimal cost, taking into account earthwork while meeting all safety and design requirements. In recent years, modelling techniques have been advanced to incorporate: side slopes, multiple material types, multiple hauling types, and road networks. However, the advancements were created disjointly with implementations that only made a single advancement to the basic model. Herein, we present a mixed-integer linear programming optimization model that unifies all previous advancements. The model further improves on previous work by maintaining convexity even in the multi-material setting. We compare our new model to previous models, validate it numerically, and demonstrate its capability in approximating material volumes. Our new model performs particularly well for determining the optimal vertical alignment for large road networks.

math.OC↗

arXiv2025

A linear-time algorithm to compute the conjugate of nonconvex bivariate piecewise linear-quadratic functions

We propose the first linear-time algorithm to compute the conjugate of (nonconvex) bivariate piecewise linear-quadratic (PLQ) functions (bivariate quadratic functions defined on a polyhedral subdivision). Our algorithm starts with computing the convex envelope of each quadratic piece obtaining rational functions (quadratic over linear) defined over a polyhedral subdivision. Then we compute the conjugate of each resulting piece to obtain piecewise quadratic functions defined over a parabolic subdivision. Finally we compute the maximum of all those functions to obtain the conjugate as a piecewise quadratic function defined on a parabolic subdivision. The resulting algorithm runs in linear time if the initial subdivision is a triangulation (or has a uniform upper bound on the number of vertexes for each piece). Our open-source implementation in MATLAB uses symbolic computation and rational numbers to avoid floating-point errors, and merges pieces as soon as possible to minimize computation time.

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arXiv2025

Closest univariate convex linear-quadratic function approximation with minimal number of Pieces

We compute the closest convex piecewise linear-quadratic (PLQ) function with minimal number of pieces to a given univariate piecewise linear-quadratic function. The Euclidean norm is used to measure the distance between functions. First, we assume that the number and positions of the breakpoints of the output function are fixed, and solve a convex optimization problem. Next, we assume the number of breakpoints is fixed, but not their position, and solve a nonconvex optimization problem to determine optimal breakpoints placement. Finally, we propose an algorithm composed of a greedy search preprocessing and a dichotomic search that solves a logarithmic number of optimization problems to obtain an approximation of any PLQ function with minimal number of pieces thereby obtaining in two steps the closest convex function with minimal number of pieces. We illustrate our algorithms with multiple examples, compare our approach with a previous globally optimal univariate spline approximation algorithm, and apply our method to simplify vertical alignment curves in road design optimization. CPLEX, Gurobi, and BARON are used with the YALMIP library in MATLAB to effectively select the most efficient solver.

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arXiv2025

An Innovative Data-Driven and Adaptive Reinforcement Learning Approach for Context-Aware Prescriptive Process Monitoring

The application of artificial intelligence and machine learning in business process management has advanced significantly, however, the full potential of these technologies remains largely unexplored, primarily due to challenges related to data quality and availability. We present a novel framework called Fine-Tuned Offline Reinforcement Learning Augmented Process Sequence Optimization (FORLAPS), which aims to identify optimal execution paths in business processes by leveraging reinforcement learning enhanced with a state-dependent reward shaping mechanism, thereby enabling context-sensitive prescriptions. Additionally, to compare FORLAPS with the existing models (Permutation Feature Importance and multi-task Long Short Term Memory model), we experimented to evaluate its effectiveness in terms of resource savings and process time reduction. The experimental results on real-life event logs validate that FORLAPS achieves 31% savings in resource time spent and a 23% reduction in process time span. To further enhance learning, we introduce an innovative process-aware data augmentation technique that selectively increases the average estimated Q-values in sampled batches, enabling automatic fine-tuning of the reinforcement learning model. Robustness was assessed through both prefix-level and trace-level evaluations, using the Damerau-Levenshtein distance as the primary metric. Finally, the model's adaptability across industries was further validated through diverse case studies, including healthcare treatment pathways, financial services workflows, permit applications from regulatory bodies, and operations management. In each domain, the proposed model demonstrated exceptional performance, outperforming existing state-of-the-art approaches in prescriptive decision-making, demonstrating its capability to prescribe optimal next steps and predict the best next activities within a process trace.

cs.LG↗

arXiv2024

A Review of AI and Machine Learning Contribution in Predictive Business Process Management (Process Enhancement and Process Improvement Approaches)

Purpose- The significance of business processes has fostered a close collaboration between academia and industry. Moreover, the business landscape has witnessed continuous transformation, closely intertwined with technological advancements. Our main goal is to offer researchers and process analysts insights into the latest developments concerning Artificial Intelligence (AI) and Machine Learning (ML) to optimize their processes in an organization and identify research gaps and future directions in the field. Design/methodology/approach- In this study, we perform a systematic review of academic literature to investigate the integration of AI/ML in business process management (BPM). We categorize the literature according to the BPM life-cycle and employ bibliometric and objective-oriented methodology, to analyze related papers. Findings- In business process management and process map, AI/ML has made significant improvements using operational data on process metrics. These developments involve two distinct stages: (1) process enhancement, which emphasizes analyzing process information and adding descriptions to process models, and (2) process improvement, which focuses on redesigning processes based on insights derived from analysis. Research limitations/implications- While this review paper serves to provide an overview of different approaches for addressing process-related challenges, it does not delve deeply into the intricacies of fine-grained technical details of each method. This work focuses on recent papers conducted between 2010 and 2024. Originality/value- This paper adopts a pioneering approach by conducting an extensive examination of the integration of AI/ML techniques across the entire process management lifecycle. Additionally, it presents groundbreaking research and introduces AI/ML-enabled integrated tools, further enhancing the insights for future research.

cs.AI↗

arXiv2021

Towards the Biconjugate of Bivariate Piecewise Quadratic Functions

Computing the closed convex envelope or biconjugate is the core operation that bridges the domain of nonconvex with convex analysis. We focus here on computing the conjugate of a bivariate piecewise quadratic function defined over a polytope. First, we compute the convex envelope of each piece, which is characterized by a polyhedral subdivision such that over each member of the subdivision, it has a rational form (square of a linear function over a linear function). Then we compute the conjugate of all such rational functions. It is observed that the conjugate has a parabolic subdivision such that over each member of its subdivision, it has a fractional form (linear function over square root of a linear function). This computation of the conjugate is performed with a worst-case linear time complexity algorithm. Our results are an important step toward computing the conjugate of a piecewise quadratic function, and further in obtaining explicit formulas for the convex envelope of piecewise rational functions.

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arXiv2018

Computation of the Epsilon-Subdifferential of Convex Piecewise-Defined Functions in Optimal Worst-Case Time

The $ε$-subdifferential of convex univariate piecewise linear-quadratic functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using dichotomic search, we show how the computation can be performed in logarithmic worst-case time. Furthermore, a new algorithm to compute the entire graph of the $ε$-subdifferential in linear time is presented. Both algorithms are not limited to convex PLQ functions but are also applicable to any convex piecewise-defined function with little restrictions.

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arXiv2017

Visualization of the ε-Subdifferential of Piecewise Linear-Quadratic Functions

Computing explicitly the ε-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class of (convex univariate) piecewise linear-quadratic functions for which existing numerical libraries allow practical computations. We visualize the results in a primal, dual, and subdifferential views through several numerical examples. We also provide a visualization of the Brøndsted-Rockafellar Theorem.

math.OC↗

arXiv2017

Best practices for comparing optimization algorithms

Comparing, or benchmarking, of optimization algorithms is a complicated task that involves many subtle considerations to yield a fair and unbiased evaluation. In this paper, we systematically review the benchmarking process of optimization algorithms, and discuss the challenges of fair comparison. We provide suggestions for each step of the comparison process and highlight the pitfalls to avoid when evaluating the performance of optimization algorithms. We also discuss various methods of reporting the benchmarking results. Finally, some suggestions for future research are presented to improve the current benchmarking process.

math.OC↗

arXiv2017

Multi-Haul Quasi Network Flow Model for Vertical Alignment Optimization

The vertical alignment optimization problem for road design aims to generate a vertical alignment of a new road with a minimum cost, while satisfying safety and design constraints. We present a new model called multi-haul quasi network flow (MH-QNF) for vertical alignment optimization that improves the accuracy and reliability of previous mixed integer linear programming models. We evaluate the performance of the new model compared to two state-of-the-art models in the field: the complete transportation graph (CTG) and the quasi network flow (QNF) models. The numerical results show that, within a 1% relative error, the proposed model is robust and solves more than 93% of test problems compared to 82% for the CTG and none for the QNF. Moreover, the MH-QNF model solves the problems approximately 8 times faster than the CTG model.

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arXiv2015

Multiple-Path Selection for new Highway Alignments using Discrete Algorithms

This paper addresses the problem of finding multiple near-optimal, spatially-dissimilar paths that can be considered as alternatives in the decision making process, for finding optimal corridors in which to construct a new road. We further consider combinations of techniques for reducing the costs associated with the computation and increasing the accuracy of the cost formulation. Numerical results for five algorithms to solve the dissimilar multipath problem show that a "bidirectional approach" yields the fastest running times and the most robust algorithm. Further modifications of the algorithms to reduce the running time were tested and it is shown that running time can be reduced by an average of 56 percent without compromising the quality of the results.

cs.DS↗

arXiv2015

Optimizing horizontal alignment of roads in a specified corridor

Finding an optimal alignment connecting two end-points in a specified corridor is a complex problem that requires solving three interrelated sub-problems, namely the horizontal alignment, vertical alignment and earthwork optimization problems. In this research, we developed a novel bi-level optimization model combining those three problems. In the outer level of the model, we optimize the horizontal alignment and in the inner level of the model a vertical alignment optimization problem considering earthwork allocation is solved for a fixed horizontal alignment. Derivative-free optimization algorithms are used to solve the outer problem. The result of our model gives an optimal horizontal alignment in the form of a linear-circular curve and an optimal vertical alignment in the form of a quadratic spline. Our model is tested on real-life data. The numerical results show that our approach improves the road alignment designed by civil engineers by 27\% on average, resulting in potentially millions of dollars of savings.

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arXiv2014

On the convexity of piecewise-defined functions

Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we conclude that the entire function is convex? In this paper we provide several convenient, verifiable conditions guaranteeing convexity (or the lack thereof). Several examples are presented to illustrate our results.

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