complexity

Matching Algorithms and Complexity Results for Constrained Mixed-Integer Optimal Control with Switching Costs

  • Felix Bestehorn
  • Christian Kirches
    • Categories (Mixed) Integer Nonlinear Programming, Approximation Algorithms Tags , , , ,

    We extend recent work on the performance of the combinatorial integral approximation decomposition approach for Mixed-Integer Optimal Control Problems (MIOCPs) in the presence of combinatorial constraints or switching costs on an equidistant grid. For the time discretized problem, we reformulate the emerging rounding problem in the decomposition approach as a matching problem on a bipartite … Read more

    The Non-Stop Disjoint Trajectories Problem

  • Benno Hoch
  • Frauke Liers
  • Sarah Neumann
  • Francisco Javier Zaragoza Martínez
    • Categories Combinatorial Optimization, Network Optimization Tags , , , , ,

    Consider an undirected network with traversal times on its edges and a set of commodities with connection requests from sources to destinations and release dates. The non-stop disjoint trajectories problem is to find trajectories that fulfill all requests, such that the commodities never meet. In this extension to the \NP-complete disjoint paths problem, trajectories must … Read more

    Complexity Aspects of Fundamental Questions in Polynomial Optimization

  • Jeffrey Zhang
    • Categories Game Theory, Global Optimization Theory, Nonlinear Optimization Tags , , , ,

    In this thesis, we settle the computational complexity of some fundamental questions in polynomial optimization. These include the questions of (i) finding a local minimum, (ii) testing local minimality of a candidate point, and (iii) deciding attainment of the optimal value. Our results characterize the complexity of these three questions for all degrees of the … Read more

    On the Complexity of Finding a Local Minimizer of a Quadratic Function over a Polytope

  • Amir Ali Ahmadi
  • Jeffrey Zhang
    • Categories Quadratic Programming Tags , , ,

    We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $c^n$ (for any constant $c \ge 0$) of a local minimizer of an $n$-variate quadratic function over a polytope. This result (even with $c=0$) answers a question of Pardalos and Vavasis that appeared in 1992 on a … Read more

    Complexity Aspects of Local Minima and Related Notions

  • Amir Ali Ahmadi
  • Jeffrey Zhang
    • Categories Global Optimization Theory, Nonlinear Optimization, Semi-definite Programming Tags , , , , ,

    We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if … Read more

    Selective Maximum Coverage and Set Packing

  • Björn Tauer
  • Felix J. L. Willamowski
    • Categories Approximation Algorithms Tags , , , ,

    In this paper we introduce the selective maximum coverage and the selective maximum set packing problem and variants of them. Both problems are strongly related to well studied problems such as maximum coverage, set packing, and (bipartite) hypergraph matching. The two problems are given by a collection of subsets of a ground set and index … Read more

    The Multi-Stop Station Location Problem

  • Miriam Ganz
  • Erik Mühmer
  • Felix J. L. Willamowski
    • Categories Approximation Algorithms, Network Optimization, Transportation Tags , , , ,

    We introduce the (directed) multi-stop station location problem. The goal is to install stations such that ordered (multi-)sets of stops can be traversed with respect to range restrictions that are reset whenever a station is visited. Applications arise in telecommunications and transportation, e.g., charging station placement problems. The problem generalizes several network optimization problems such … Read more

    Complexity of cutting planes and branch-and-bound in mixed-integer optimization

  • Amitabh Basu
  • Michele Conforti
  • Marco Di Summa
  • Hongyi Jiang
    • Categories (Mixed) Integer Nonlinear Programming, Branch and Cut Algorithms, Cutting Plane Approaches Tags , ,

    We investigate the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization. In particular, we study the relative efficiency of BB and CP, when both are based on the same family of disjunctions. We extend a result of Dash to the nonlinear setting which shows that for convex 0/1 problems, CP … Read more

    Computing Technical Capacities in the European Entry-Exit Gas Market is NP-Hard

  • Lars Schewe
  • Martin Schmidt
  • Johannes Thürauf
    • Categories Applications - Science and Engineering, Network Optimization Tags , , , ,

    As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes … Read more

    Stochastic Dual Dynamic Programming for Multistage Stochastic Mixed-Integer Nonlinear Optimization

  • Shixuan Zhang
  • Xu Andy Sun
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Stochastic Programming Tags , , ,

    In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with \emph{non-Lipschitz-continuous} value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient … Read more