(Mixed) Integer Nonlinear Programming

Adjustable Robust Nonlinear Network Design Without Controllable Elements under Load Scenario Uncertainties

  • Johannes Thürauf
  • Julia Grübel
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Energy, Robust Optimization Tags , , , ,

    We study network design problems for nonlinear and nonconvex flow models without controllable elements under load scenario uncertainties, i.e., under uncertain injections and withdrawals. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, load scenarios within a given … Read more

    Using Disjunctive Cuts in a Branch-and-Cut Method to Solve Convex Integer Nonlinear Bilevel Problems

  • Horländer Andreas
  • Ivana Ljubić
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Bilevel Optimization, Cutting Plane Approaches Tags , , ,

    We present a branch-and-cut method for solving convex integer nonlinear bilevel problems, i.e., bilevel models with nonlinear but jointly convex objective functions and constraints in both the upper and the lower level. To this end, we generalize the idea of using disjunctive cuts to cut off integer-feasible but bilevel-infeasible points. These cuts can be obtained … Read more

    Neur2BiLO: Neural Bilevel Optimization

  • Justin Dumouchelle
  • Esther Julien
  • Jannis Kurtz
  • Elias B. Khalil
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming Tags , ,

    Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness.  While exact solvers have been proposed for mixed-integer~linear bilevel optimization, they tend to scale poorly with problem … Read more

    Robust support vector machines via conic optimization

  • Shaoning Han
  • Andrés Gómez
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Optimization in Data Science Tags , , , ,

    We consider the problem of learning support vector machines robust to uncertainty. It has been established in the literature that typical loss functions, including the hinge loss, are sensible to data perturbations and outliers, thus performing poorly in the setting considered. In contrast, using the 0-1 loss or a suitable non-convex approximation results in robust … Read more

    On Sparse Canonical Correlation Analysis

  • Yongchun Li
  • Santanu S. Dey
  • Weijun Xie
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Integer Programming Tags , ,

    The classical Canonical Correlation Analysis (CCA) identifies the correlations between two sets of multivariate variables based on their covariance, which has been widely applied in diverse fields such as computer vision, natural language processing, and speech analysis. Despite its popularity, CCA can encounter challenges in explaining correlations between two variable sets within high-dimensional data contexts. … Read more

    An outer approximation method for solving mixed-integer convex quadratic programs with indicators

  • Linchuan Wei
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming Tags , ,

    Mixed-integer convex quadratic programs with indicator variables (MIQP) encompass a wide range of applications, from statistical learning to energy, finance, and logistics. The outer approximation (OA) algorithm has been proven efficient in solving MIQP, and the key to the success of an OA algorithm is the strength of the cutting planes employed. In this paper, … Read more

    Computing an approximation of the nondominated set of multi-objective mixed-integer nonlinear optimization problems

  • Leo Warnow
  • Gabriele Eichfelder
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Multi-Criteria Optimization Tags , , , ,

    In practical applications, one often has not only one, but several objectives that need to be optimized simultaneously. What is more, modeling such real world problems usually involves using both, continuous and integer variables. This then results in multi-objective mixed-integer optimization problems, which are in focus of this paper. We present an approximation concept, called … Read more

    Solving Nonconvex Optimization Problems using Outer Approximations of the Set-Copositive Cone

  • Kurt M. Anstreicher
  • MarkusGabl
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Global Optimization Theory Tags , ,

    We consider the solution of nonconvex quadratic optimization problems using an outer approximation of the set-copositive cone that is iteratively strengthened with conic constraints and cutting planes. Our methodology utilizes an MILP-based oracle for a generalization of the copositive cone that considers additional linear equality constraints. In numerical testing we evaluate our algorithm on a … Read more

    On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

  • Immanuel M. Bomze
  • Bo Peng
  • Yuzhou Qiu
  • E. Alper Yildirim
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming, Semi-definite Programming Tags , , , ,

    Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, … Read more

    Benchmarking Piecewise Linear Reformulations for MINLPs: A Computational Study Based on the Open-Source Framework PWL-T-Rex

  • Kristin Braun
  • Robert Burlacu
    • Categories (Mixed) Integer Nonlinear Programming Tags , , ,

    Solving mixed-integer nonlinear problems by means of piecewise linear relaxations has emerged as a reasonable alternative to the commonly used spatial branch-and-bound. These relaxations have been modeled by various mixed-integer models in recent decades. The idea is to exploit the availability of mature solvers for mixed-integer problems. In this work, we implement a framework that … Read more