Global Optimization

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

    Exact Matrix Completion via High-Rank Matrices in Sum-of-Squares Relaxations

  • Sunyoung Kim
  • Godai Azuma
  • Makoto Yamashita
    • Categories Convex Optimization, Global Optimization Theory, Semi-definite Programming Tags , , , ,

    We study exact matrix completion from partially available data with hidden connectivity patterns. Exact matrix completion was shown to be possible recently by Cosse and Demanet in 2021 with Lasserre’s relaxation using the trace of the variable matrix as the objective function with given data structured in a chain format. In this study, we introduce … Read more

    The limitation of neural nets for approximation and optimization

  • Tommaso Giovannelli
  • Oumaima Sohab
  • Luis Nunes Vicente
    • Categories Global Optimization, Nonlinear Optimization, Optimization in Data Science Tags , , ,

    We are interested in assessing the use of neural networks as surrogate models to approximate and minimize objective functions in optimization problems. While neural networks are widely used for machine learning tasks such as classification and regression, their application in solving optimization problems has been limited. Our study begins by determining the best activation function … 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

    Cone product reformulation for global optimization

  • Danique de Moor
  • Dick den Hertog
  • Dimitris Bertsimas
  • Jianzhe Zhen
  • Thodoris Koukouvinos
    • Categories Cone Programming, Global Optimization, Nonlinear Optimization

    In this paper, we study nonconvex optimization problems involving sum of linear times convex (SLC) functions as well as conic constraints belonging to one of the five basic cones, that is, linear cone, second order cone, power cone, exponential cone, and semidefinite cone. By using the Reformulation Perspectification Technique, we can obtain a convex relaxation … Read more

    A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

  • Jie Wang
  • Victor Magron
    • Categories Global Optimization, Global Optimization Theory, Semi-definite Programming Tags , , , ,

    This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to solve. In addition, we prove that global optimality is achieved when the ranks of the moment matrix and certain … Read more

    Convex envelopes of bounded monomials on two-variable cones

  • Pietro Belotti
    • Categories Global Optimization Theory Tags , ,

    \(\) We consider an \(n\)-variate monomial function that is restricted both in value by lower and upper bounds and in domain by two homogeneous linear inequalities. Such functions are building blocks of several problems found in practical applications, and that fall under the class of Mixed Integer Nonlinear Optimization. We show that the upper envelope … Read more

    AI Hilbert: A New Paradigm for Scientific Discovery by Unifying Data and Background Knowledge

  • Ryan Cory-Wright
  • Bachir El Khadir
  • Cristina Cornelio
  • Sanjeeb Dash
  • Lior Horesh
    • Categories Basic Sciences Applications, Global Optimization Applications, Semi-definite Programming Tags , , ,

    The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in settings with … Read more

    Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
    • Categories Global Optimization, Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , , , ,

    \(\) A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh asa unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone K, a convex subcone J of the … Read more

    On Common-Random-Numbers and the Complexity of Adaptive Sampling Trust-Region Methods

  • Sara Shashaani
    • Categories Optimization of Simulated Systems, Stochastic Approaches, Unconstrained Optimization

    \(\) In the context of simulation optimization (SO), Common Random Numbers (CRN) is the practice of querying the simulation-based oracle with the same random number stream at each point visited by an SO algorithm. This practice is widely believed to facilitate SO algorithm efficiency by preserving structure inherent to the objective function and gradient sample-paths. … Read more