(Mixed) Integer Nonlinear Programming

A Finitely Convergent Cutting Plane, and a Bender’s Decomposition Algorithm for Mixed-Integer Convex and Two-Stage Convex Programs using Cutting Planes

  • Fengqiao Luo
  • Sanjay Mehrotra
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Stochastic Programming Tags , , ,

    We consider a general mixed-integer convex program. We first develop an algorithm for solving this problem, and show its nite convergence. We then develop a finitely convergent decomposition algorithm that separates binary variables from integer and continuous variables. The integer and continuous variables are treated as second stage variables. An oracle for generating a parametric … Read more

    Modeling Design and Control Problems Involving Neural Network Surrogates

  • Dominic Yang
  • Prasanna Balaprakash
  • Sven Leyffer
    • Categories (Mixed) Integer Nonlinear Programming, Control Applications, Nonsmooth Optimization Tags , , , ,

    We consider nonlinear optimization problems that involve surrogate models represented by neural net-works. We demonstrate first how to directly embed neural network evaluation into optimization models, highlight a difficulty with this approach that can prevent convergence, and then characterize stationarity of such models. We then present two alternative formulations of these problems in the specific … Read more

    Exact Methods for Discrete Γ-Robust Interdiction Problems with an Application to the Bilevel Knapsack Problem

  • Yasmine Beck
  • Ivana Ljubić
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Robust Optimization Tags , , , ,

    Developing solution methods for discrete bilevel problems is known to be a challenging task – even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty. We study discrete min-max problems with a follower who faces uncertainties regarding the parameters of the lower-level problem. Adopting a … Read more

    Compact extended formulations for low-rank functions with indicator variables

  • Shaoning Han
  • Andrés Gómez
    • Categories (Mixed) Integer Nonlinear Programming Tags , , ,

    We study the mixed-integer epigraph of a special class of convex functions with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions Extended formulations describing the convex hull of such … Read more

    A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

  • Peijing Liu
  • Salar Fattahi
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming, Statistics Tags , , , , , ,

    In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This … Read more

    Complexity of optimizing over the integers

  • Amitabh Basu
    • Categories (Mixed) Integer Nonlinear Programming, Convex Optimization Tags , ,

    In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like “input”, “size” and “complexity” in the context of general mathematical optimization, avoiding context dependent definitions which is one of the … Read more

    Feasible rounding approaches and diving strategies in branch-and-bound methods for mixed-integer optimization

  • Christoph Neumann
  • Stefan Schwarze
  • Oliver Stein
  • Benjamin Müller
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming Tags , , , , ,

    In this paper, we study the behavior of feasible rounding approaches for mixed-integer linear and nonlinear optimization problems (MILP and MINLP, respectively) when integrated into tree search of branch-and-bound. Our research addresses two important aspects. First, we develop insights into how an (enlarged) inner parallel set, which is the main component for feasible rounding approaches, … Read more

    An Improved Penalty Algorithm using Model Order Reduction for MIPDECO problems with partial observations

  • Dominik Garmatter
  • Margherita Porcelli
  • Francesco Rinaldi
  • Martin Stoll
    • Categories (Mixed) Integer Nonlinear Programming, Optimization of Systems modeled by PDEs Tags , , , ,

    This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra … Read more

    Presolving for Mixed-Integer Semidefinite Optimization

  • Frederic Matter
  • Marc E. Pfetsch
    • Categories (Mixed) Integer Nonlinear Programming, Semi-definite Programming Tags , , ,

    This paper provides a discussion and evaluation of presolving methods for mixed-integer semidefinite programs. We generalize methods from the mixed-integer linear case and introduce new methods that depend on the semidefinite condition. The considered methods include adding linear constraints, bounds relying on 2 × 2 minors of the semidefinite constraints, bound tightening based on solving … Read more

    Applications of stochastic mixed-integer second-order cone optimization

  • Baha Alzalg
  • Hadjer Alioui
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Second-Order Cone Programming Tags , , ,

    Second-order cone programming problems are a tractable subclass of convex optimization problems and there are known polynomial algorithms for solving them. Stochastic second-order cone programming problems have also been studied in the past decade and efficient algorithms for solving them exist. A new class of interest to optimization community and practitioners is the mixed-integer version … Read more