Notes: This session presents practical exercises to be performed by the participants. A computer and an Internet connection are required to participate.

The challenges faced by operational research (OR) practitioners, whether in operational scenarios or during techno-economic analyses, are typically intricate and inherently non-linear. A skilled modeler's task is to simplify this complexity and identify the core issue whose resolution untangles the rest.
A common strategy is to linearize the problem to leverage the wealth of theoretical and practical tools provided by linear programming, such as duality, sensitivity analysis, and efficient algorithms. While this approach is often effective, it's important to be acquainted with non-linear optimization methods to avoid overly relying on linearization out of habit. In many instances, problems can be efficiently solved directly without linearization, utilizing appropriate non-linear methods and tools.
This tutorial will showcase several instances of non-linear problems, typically addressed through linear relaxation, yet warranting a direct approach. It will delve into the methodologies and tools employed to tackle these challenges. This tutorial aims at an audience familiar with combinatorial and linear programming, focusing on practical examples and discussions of both linear and non-linear approaches. The focus will be on practical application, with participants implementing various examples in a Python notebook using the Artelys Knitro solver. This session aims to empower participants to recognize the type of non-linear problem they encounter, determine appropriate methods and tools for solving it, and learn how to effectively utilize them.

Consider a hypergraph $H=(V,E)$, with a weights vector $w\in \mathbb{R}^E_+$, and a fixed number $k$. The $k$-cut problem consists of finding a partition $\{S_1,\dots,S_k\}$ of $V$ that minimizes $w(\delta(S_1,\dots,S_k))$, where $\delta(S_1,\dots,S_k)$ is the set of hyperedges having at least two elements into two different sets of the partition. Based on the lower bound of Narayanan (1996) we develop a combinatorial branch and bound algorithm for computing the optimal solution for the $k$-cut problem. We also show that a direct extension of Naor's (2001) linear integer program for the $k$-cut problem in graphs will not holds in the case of hypergraphs. For this, we produce a new linear integer program that we use to develop a branch and cut algorithm using Cplex. We also consider several heuristics for computing an upper bound. Finally, we give experimental results comparing all these approachs.

Submodular maximization over a matroid constraint is a fundamental problem with various applications in machine learning. Some of these applications involve decision-making over datapoints with sensitive attributes such as gender or race. In such settings, it is crucial to guarantee that the selected solution is fairly distributed with respect to this attribute. Recently, fairness has been investigated in submodular maximization under a cardinality constraint in both the streaming and offline settings, however the more general problem with matroid constraint has only been considered in the streaming setting and only for monotone objectives. This work fills this gap. We propose various algorithms and impossibility results offering different trade-offs between quality, fairness, and generality.

In this talk, we investigate nonlinear optimization problems using algebraic topology. Specifically, we utilize Borsuk-Ulam's theorem, which states that for any continuous mapping from the n-sphere to n-dimensional Euclidean space, there exists a pair of antipodal points that are mapped to the same point. We apply this theorem to the optimal-value functions of an n-tuple of parametric optimization problems and present an antipodal theorem for them. Furthermore, by considering the support function of an n-tuple of convex sets as the objective functions, we introduce a ham-sandwich-type theorem.