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2 results

  • A NONNEGATIVITY CRITERION FOR ERDŐS MATRICES

  • Part of
  • FRÉDÉRIC MORNEAU-GUÉRIN
  • Journal:
    Bulletin of the Australian Mathematical Society , First View
    Published online by Cambridge University Press:
    02 October 2026, pp. 1-11
    • Article
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  • Let A be an $n\times n$ doubly stochastic matrix. The Marcus–Ree inequality asserts that $\|A\|_{\mathrm F}^2\leq \mathrm {maxtrace}(A)$. Matrices attaining equality are called Erdős matrices. Recent work of Karmakar et al. [‘Characterization of Erdős matrices by their zero entries’, Linear Algebra Appl. 739 (2026), 154–169] shows that every Erdős matrix is a restricted common diagonal sum matrix. By the structural theory of Brualdi and Dahl [‘Diagonal sums of doubly stochastic matrices’, Linear Multilinear Algebra 70 (2022), 4946–4972], a restricted common diagonal sum matrix with fully indecomposable skeleton $S=(s_{ij})$ admits additive potentials satisfying $a_{ij}=(u_i+v_j)s_{ij}$. Karmakar et al. observed that $\min _i u_i+\min _jv_j\geq 0$ is sufficient for Erdős-ness. We prove that it is also necessary. Thus, recognition reduces to solving one linear system per fully indecomposable block and checking signs, with no residual maxtrace computation.

  • 6 - Mixed-Integer Programming Models

  • Ignacio E. Grossmann, Carnegie Mellon University, Pennsylvania
  • Book:
    Advanced Optimization for Process Systems Engineering
    Published online:
    11 June 2021
    Print publication:
    25 March 2021, pp 53-61

  • Summary

    This chapter introduces basic models for mixed-integer linear programming. It starts with multiple-choice constraints, and implications that are formulated as inequalities with 0-1 variables. Next, constraints with continuous variables are introduced, such as discontinuous domains and cost functions with fixed charges, which are formulated as linear mixed-integer constraints. Finally, the chapter closes by introducing classic OR problems, the assignment problem, facility location problem, knapsack problem, set covering problem and the traveling salesman problem, all of which are formulated as linear integer and mixed-integer linear programming models.