Fast and Simple Multiclass Data Segmentation: An Eigendecomposition and Projection-Free Approach

  • Chiara Faccio
  • Margherita Porcelli
  • Francesco Rinaldi
  • Martin Stoll
    • Categories Data Science Algorithms, Data Science Applications, Nonlinear Optimization

    Graph-based machine learning has seen an increased interest over the last decade with many connections to other fields of applied mathematics. Learning based on partial differential equations, such as the phase-field Allen-Cahn equation, allows efficient handling of semi-supervised learning approaches on graphs. The numerical solution of the graph Allen-Cahn equation via a convexity splitting or … Read more

    Density, Determinacy, Duality and a Regularized Moment-SOS Hierarchy

  • Didier Henrion
    • Categories Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags ,

    The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar’s Positivstellensatz, which requires the feasible set to satisfy the algebraic Archimedean property. In this paper, we introduce a regularized moment-SOS hierarchy capable of handling problems on unbounded sets or bounded sets violating the Archimedean … Read more

    On constraint qualifications for lower-level sets and an augmented Lagrangian method

  • Roberto Andreani
  • Gabriel Haeser
  • Mariana da Rosa
  • Daiana O. Santos
    • Categories Nonlinear Optimization Tags , , ,

    In this paper we consider an augmented Lagrangian method with general lower-level constraints, that is, where some of the constraints are penalized while others are kept as subproblem constraints. Motivated by some recent results on optimization problems on manifolds, we present a general theory of global convergence when a feasible approximate KKT point is found … Read more

    Modeling Bloons Tower Defense as a temporal two-dimensional knapsack problem with irregular shapes and side constraints: integer programming-based approaches

  • Maxence Delorme
    • Categories Combinatorial Optimization, Integer Programming Tags , , , ,

    In Tower Defense (TD) games, the objective is to defend a specific point on the game map from mobile units by constructing towers with offensive capabilities. In this work, we focus on Bloons Tower Defense (Bloons TD), one of the earliest and most prominent TD games. We show that the problem of finding tower configurations … Read more

    The Fulfillment Regionalization Problem

  • Nidhima Grover
  • Xiaoyan Si
  • Alejandro Toriello
    • Categories (Mixed) Integer Nonlinear Programming, Supply Chain Management, Transportation Tags , , , ,

    In many retail industries, the retailer can choose the inventory location or fulfillment center (FC) that fulfills an order, yielding opportunities for inventory pooling and product selection expansion. However, fulfillment decisions are complex and must consider cost and speed, among various factors. With the unprecedented growth of the retail industry, companies must look for strategies … Read more

    Solving the Heilbronn Triangle Problem using Global Optimization Methods

  • Burak Kocuk
  • Amirali Modir
  • Amirhossein Monji
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Applications Tags , , , ,

    We study the Heilbronn triangle problem, which involves placing \(n\) points in the unit square such that the minimum area of any triangle formed by these points is maximized. A straightforward maximin formulation of this problem is highly non-linear and non-convex due to the existence of bilinear terms and absolute value equations. We propose two … Read more

    Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization

  • Jie Wang
    • Categories Infinite Dimensional Optimization, Optimization in Data Science, Robust Optimization

    Distributionally robust optimization (DRO) has emerged as a powerful paradigm for reliable decision-making under uncertainty. This paper focuses on DRO with ambiguity sets defined via the Sinkhorn discrepancy: an entropy-regularized Wasserstein distance, referred to as Sinkhorn DRO. Existing work primarily addresses Sinkhorn DRO from a dual perspective, leveraging its formulation as a conditional stochastic optimization … Read more

    An Elementary Proof of the Near Optimality of LogSumExp Smoothing

  • Thabo Samakhoana
  • Benjamin Grimmer
    • Categories Convex Optimization, Data Science Algorithms, Nonsmooth Optimization Tags , , , ,

    We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value by at most $\ln(d)$. We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must … Read more

    Combinatorial Benders Decomposition and Column Generation for Optimal Box Selection

  • Christoph Buchheim
  • Alice Kirchheim
  • Pia Schreynemackers
    • Categories Combinatorial Optimization, Cutting Plane Approaches, Transportation Tags , ,

    We consider a two-stage optimization problem with sparsity constraints, motivated by a common challenge in packaging logistics: minimizing the volume of transported air by optimizing the size and number of available packaging boxes, given the demand for order items. In the first stage, we select the optimal dimensions of the boxes, while in the second … Read more

    Robust optimality for nonsmooth mathematical programs with equilibrium constraints under data uncertainty

  • Vivek Laha
    • Categories Applications - OR and Management Sciences, Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization, Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization, Robust Optimization Tags

    We develop a unified framework for robust nonsmooth optimization problems with equilibrium constraints (UNMPEC). As a foundation, we study a robust nonsmooth nonlinear program with uncertainty in both the objective function and the inequality constraints (UNP). Using Clarke subdifferentials, we establish Karush–Kuhn–Tucker (KKT)–type necessary optimality conditions under an extended no–nonzero–abnormal–multiplier constraint qualification (ENNAMCQ). When the … Read more