Global Optimization

Adaptive discretization-based algorithms for semi-infinite programs with unbounded variables

  • Daniel Jungen
  • Hatim Djelassi
  • Alexander Mitsos
    • Categories Global Optimization Tags , ,

    The proof of convergence of adaptive discretization-based algorithms for semi-infinite programs (SIPs) usually relies on compact host sets for the upper- and lower-level variables. This assumption is violated in some applications, and we show that indeed convergence problems can arise when discretization-based algorithms are applied to SIPs with unbounded variables. To mitigate these convergence problems, … Read more

    Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons

  • Christian Bingane
    • Categories Combinatorial Optimization, Global Optimization, Nonlinear Optimization Tags , , ,

    A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m \ge 7$. In this paper, we construct, for each $n=2m$ and $m\ge 3$, a small $n$-gon whose area is the maximal value of a one-variable function. We show that, for all … Read more

    Bounding the separable rank via polynomial optimization

  • Sander Gribling
  • Monique Laurent
  • Andries Steenkamp
    • Categories Convex and Nonsmooth Optimization, Global Optimization Tags , , , , ,

    We investigate questions related to the set $\mathcal{SEP}_d$ consisting of the linear maps $\rho$ acting on $\mathbb{C}^d\otimes \mathbb{C}^d$ that can be written as a convex combination of rank one matrices of the form $xx^*\otimes yy^*$. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In … Read more

    The equilateral small octagon of maximal width

  • Christian Bingane
  • Charles Audet
    • Categories Combinatorial Optimization, Global Optimization, Nonlinear Optimization Tags , , ,

    A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24\%$ larger than the width of the regular octagon: $\cos(\pi/8)$. … Read more

    On Piecewise Linear Approximations of Bilinear Terms: Structural Comparison of Univariate and Bivariate Mixed-Integer Programming Formulations

  • Lukas Hager
  • Robert Burlacu
  • Andreas Bärmann
  • Thomas Kleinert
    • Categories (Mixed) Integer Linear Programming, Global Optimization Theory, Quadratic Programming Tags , , , ,

    Bilinear terms naturally appear in many optimization problems. Their inherent nonconvexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of … Read more

    A Reformulation Technique to Solve Polynomial Optimization Problems with Separable Objective Functions of Bounded Integer Variables

  • Andrew C. Trapp
  • Pitchaya Wiratchotisatian
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Nonlinear Optimization Tags , , ,

    Real-world problems are often nonconvex and involve integer variables, representing vexing challenges to be tackled using state-of-the-art solvers. We introduce a mathematical identity-based reformulation of a class of polynomial integer nonlinear optimization (PINLO) problems using a technique that linearizes polynomial functions of separable and bounded integer variables of any degree. We also introduce an alternative … Read more

    Global Optimization for Nonconvex Programs via Convex Proximal Point Method

  • Wenxun Xing
  • Yuanyi Zhao
    • Categories Applications - OR and Management Sciences, Global Optimization

    The nonconvex program plays an important role in the field of optimization and has a lot of applications in practice. However, for general nonconvex programming problems, the lack of verifiable global optimal conditions and the multiple local minimizers make global optimization hard in computation. In this paper, a convex proximal point algorithm (CPPA) is considered … Read more

    Global optimization using random embeddings

  • Coralia Cartis
  • Estelle Massart
  • Adilet Otemissov
    • Categories Global Optimization, Nonlinear Optimization Tags , , , ,

    We propose a random-subspace algorithmic framework for global optimization of Lipschitz-continuous objectives, and analyse its convergence using novel tools from conic integral geometry. X-REGO randomly projects, in a sequential or simultaneous manner, the high- dimensional original problem into low-dimensional subproblems that can then be solved with any global, or even local, optimization solver. We estimate … Read more

    Exactness in SDP relaxations of QCQPs: Theory and applications

  • Fatma Kılınç-Karzan
  • Alex L. Wang
    • Categories Global Optimization Theory, Semi-definite Programming

    Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems. In a QCQP, we are asked to minimize a (possibly nonconvex) quadratic function subject to a number of (possibly nonconvex) quadratic constraints. Such problems arise naturally in many areas of operations research, computer science, and engineering. Although QCQPs are NP-hard to solve in … Read more

    The Promise of EV-Aware Multi-Period OPF Problem: Cost and Emission Benefits

  • Sezen Ece Kayacık
  • Burak Kocuk
  • Tuğçe Yüksel
    • Categories Applications - OR and Management Sciences, Global Optimization, Second-Order Cone Programming Tags , ,

    In this paper, we study the Multi-Period Optimal Power Flow problem (MOPF) with electric vehicles (EV) under emission considerations. We integrate three different real-world datasets: household electricity consumption, marginal emission factors, and EV driving profiles. We present a systematic solution approach based on second-order cone programming to find globally optimal solutions for the resulting nonconvex … Read more