Abstract : Mixed-Integer Non-Linear Programming (MINLP) remains a computational bottleneck for many real-world applications arising in several fields. In this talk, we first review traditional exact methods for solving MINLP and their computational limitations on medium- to large-scale problems. We then explore recent advances in approximation techniques that leverage machine learning methods to construct surrogate models, enabling high-quality heuristic solutions with reduced computational effort. Building on these ideas, we introduce a novel methodology grounded in statistical learning theory for adaptive surrogate modeling, designed to balance approximation accuracy with tractability. We conclude with a computational comparison of these learning-based approaches, highlighting their strengths, limitations, and practical implications for tackling large-scale MINLPs in industries like energy and hydraulic engineering.

Abstract : The envy-free cake-cutting problem asks for a way to divide a cake (identified with the interval [0,1]) among players with different tastes so that each receives a connected piece and no one envies another's share. The Stromquist–Woodall theorem from 1980 guarantees the existence of such an envy-free division under mild assumptions. Recently, there has been a surge of interest in this problem from various perspectives — computer science, social choice theory, economics, and topological combinatorics. In this talk, we present several extensions, from these perspectives: the cake may be "poisoned," there may be multiple cakes with joint preferences, or the cake may be discrete (as in the necklace-splitting problem). We will also discuss several challenging open questions.

Abstract : In this talk I will briefly survey previous work about the application of random projections to mathematical programming, and then talk about recent advances, specifically about quadratically constrained quadratic programs, as well as on a specific MINLP problem, i.e. the minimum sum-of-squares clustering. Joint work with Benedetto Manca and Pierre-Louis Poirion.

Abstract : We present recent branch-and-cut methods for solving convex integer nonlinear bilevel optimization problems, that is, bilevel models with nonlinear yet jointly convex objective functions and constraints at both the upper and lower levels. We first consider problems in which the nonlinearities are represented by second-order cone constraints in the upper level and by a convex objective function in the lower level. The proposed branch-and-cut framework relies on disjunctive cuts generated through a second-order cone programming (SOCP) cut-generation procedure. We then extend this approach by generalizing the use of disjunctive cuts to eliminate points that are integer-feasible but bilevel-infeasible. We show that such cuts can be derived by solving a cut-generation problem that is itself formulated as a single-level, nonconvex integer nonlinear optimization problem. Finally, we compare the proposed classes of disjunctive cuts from both theoretical and computational perspectives.

Abstract : Decomposition methods play a central role in large-scale optimization by enabling complex problems to be separated into smaller, more tractable subproblems. By exploiting the underlying structure of mathematical models, these approaches significantly improve computational efficiency and scalability in both linear and integer programming. This presentation provides an overview of major decomposition techniques used in operations research, with a particular focus on Dantzig–Wolfe Decomposition, Column Generation, and Branch-and-Price. We discuss their theoretical foundations, algorithmic frameworks, and several advanced solution strategies developed in recent years. Throughout the presentation, practical applications in combinatorial optimization are used to illustrate the main theoretical concepts and demonstrate the effectiveness of these methods in practice. The objective of this talk is to show how decomposition techniques make it possible to solve real-world optimization problems that are otherwise computationally intractable using classical approaches.