spatial branch and bound

A graphical framework for global optimization of mixed-integer nonlinear programs

  • Danial Davarnia
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Applications, Global Optimization Theory Tags , , , , ,

    While mixed-integer linear programming and convex programming solvers have advanced significantly over the past several decades, solution technologies for general mixed-integer nonlinear programs (MINLPs) have yet to reach the same level of maturity. Various problem structures across different application domains remain challenging to model and solve using modern global solvers, primarily due to the lack … Read more

    Nonconvex optimization problems involving the Euclidean norm: Challenges, progress, and opportunities

  • Anatoliy Kuznetsov
  • Nikolaos Sahinidis
    • Categories Global Optimization, Optimization Software Benchmark Tags , , ,

    The field of global optimization has advanced significantly over the past three decades. Yet, the solution of even small instances of many nonconvex optimization problems involving the Euclidean norm to global optimality remains beyond the reach of modern global optimization methods. These problems include numerous well-known and high-impact open research questions from a diverse collection … Read more

    Spatial branch-and-bound for nonconvex separable piecewise linear optimization

  • Thomas Hübner
  • Akshay Gupte
  • Steffen Rebennack
    • Categories (Mixed) Integer Linear Programming, Global Optimization Tags , , , ,

    Nonconvex separable piecewise linear functions (PLFs) frequently appear in applications and to approximate nonlinearitites. The standard practice to formulate nonconvex PLFs is from the perspective of discrete optimisation, using special ordered sets and mixed integer linear programs (MILPs). In contrast, we take the viewpoint of global continuous optimization and present a spatial branch-and-bound algorithm (sBB) … Read more

    Strengthening SONC Relaxations with Constraints Derived from Variable Bounds

  • Ksenia Bestuzheva
  • Ambros Gleixner
    • Categories Constrained Nonlinear Optimization, Global Optimization, Optimization Software Design Principles Tags , , ,

    Nonnegativity certificates can be used to obtain tight dual bounds for polynomial optimization problems. Hierarchies of certificate-based relaxations ensure convergence to the global optimum, but higher levels of such hierarchies can become very computationally expensive, and the well-known sums of squares hierarchies scale poorly with the degree of the polynomials. This has motivated research into … Read more

    A branch-and-bound algorithm for non-convex Nash equilibrium problems

  • Peter Kirst
  • Stefan Schwarze
  • Oliver Stein
    • Categories Game Theory Tags , , , ,

    This paper introduces a spatial branch-and-bound method for the approximate computation of the set of all epsilon-Nash equilibria of continuous box-constrained non-convex Nash equilibrium problems. We explain appropriate discarding and fathoming techniques, provide a termination proof for a prescribed approximation tolerance, and report our computational experience. ArticleDownload View PDF

    Chance Constrained Programs with Gaussian Mixture Models

  • Zhaolin Hu
  • Wenjie Sun
  • Shushang Zhu
    • Categories Nonlinear Optimization, Stochastic Programming Tags , , , ,

    In this paper, we discuss input modeling and solution techniques for several classes of chance constrained programs (CCPs). We propose to use Gaussian mixture models (GMM), a semi-parametric approach, to fit the data available and to model the randomness. We demonstrate the merits of using GMM. We consider several scenarios that arise from practical applications … Read more

    Quantifying Double McCormick

  • Jon Lee
  • Emily Speakman
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory Tags , , , , , ,

    When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the … Read more

    On a nonconvex MINLP formulation of the Euclidean Steiner tree problems in n-space

  • Claudia D'Ambrosio
  • Marcia Fampa
  • Jon Lee
  • Stefan Vigerske
    • Categories (Mixed) Integer Nonlinear Programming, Combinatorial Optimization, Global Optimization Tags , ,

    The Euclidean Steiner Tree Problem in dimension greater than two is notoriously difficult. The successful methods for exact solution are not based on mathematical-optimization methods — rather, they involve very sophisticated enumeration. There are two types of mathematical-optimization formulations in the literature, and it is an understatement to say that neither scales well enough to … Read more

    On feasibility based bounds tightening

  • Pietro Belotti
  • Sonia Cafieri
  • Jon Lee
  • Leo Liberti
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory Tags , , ,

    Mathematical programming problems involving nonconvexities are usually solved to optimality using a (spatial) Branch-and-Bound algorithm. Algorithmic efficiency depends on many factors, among which the widths of the bounding box for the problem variables at each Branch-and-Bound node naturally plays a critical role. The practically fastest box-tightening algorithm is known as FBBT (Feasibility-Based Bounds Tightening): an … Read more

    An algorithmic framework for MINLP with separable non-convexity

  • Claudia D'Ambrosio
  • Jon Lee
  • Andreas Wächter
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Tags , , , ,

    Global optimization algorithms, e.g., spatial branch-and-bound approaches like those implemented in codes such as BARON and COUENNE, have had substantial success in tackling complicated, but generally small scale, non-convex MINLPs (i.e., mixed-integer nonlinear programs having non-convex continuous relaxations). Because they are aimed at a rather general class of problems, the possibility remains that larger instances … Read more