Given a square matrix $A$ and a polynomial $p$, the Crouzeix ratio is the norm of the polynomial on the field of values of $A$ divided by the 2-norm of the matrix $p(A)$. Crouzeix’s conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, … Read more
In this paper, we consider a class of nonsmooth problem that is the sum of a Lipschitz differentiable function and a nonsmooth and proper lower semicontinuous function. We discuss here the convergence rate of the function values for a nonmonotone accelerated proximal gradient method, which proposed in “Huan Li and Zhouchen Lin: Accelerated proximal gradient … Read more
There are first-optimize-then-discretize (indirect) and first-discretize-then-optimize (direct) methods to deal with infinite dimensional optimal problems numerically by use of finite element methods. Generally, both discretization techniques lead to different structures. Regarding the indirect method, one derives optimality conditions of the considered infinite dimensional problems in appropriate function spaces firstly and then discretizes them into suitable … Read more
We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely ${\cal DD}_n^*$ (resp., ${\cal SDD}_n^*$), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. (2022), between a set ${\cal S}$ and the semidefinite cone has the same … Read more
Let F be a set defined by quadratic constraints. Understanding the structure of the closed convex hull cl(C(F)) := cl(conv{xx’ | x in F}) is crucial to solve quadratically constrained quadratic programs related to F. A set G with complicated structure can be constructed by intersecting simple sets. This paper discusses the relationship between cl(C(F)) … Read more
Binarized neural networks are an important class of neural network in deep learning due to their computational efficiency. This paper contributes towards a better understanding of the structure of binarized neural networks, specifically, ideal convex representations of the activation functions used. We describe the convex hull of the graph of the signum activation function associated … Read more
This work introduces a complex variant of the timetabling problem, which is motivated by the case of Italian schools. The new requirements enforce to (i) provide the same idle times for teachers, (ii) avoid consecutive \emph{heavy} days, (iii) limit daily multiple lessons for the same class, (iv) introduce shorter time units to differentiate entry and … Read more
This paper addresses the optimal design of resilient systems, in which components can fail. The system can react to failures and its behavior is described by general mixed integer nonlinear programs, which allows for applications to many (technical) systems. This then leads to a three-level optimization problem. The upper level designs the system minimizing a … Read more
This paper builds on the premise that the most vulnerable areas or groups of people should be protected from disasters by being given priority in humanitarian operations, particularly when there are limited resources available for disaster management. The basis and the development of the paper are strongly aligned with the United Nations’ Sustainable Development Goals … Read more
The relaxation complexity $\mathrm{rc}(X)$ of the set of integer points $X$ contained in a polyhedron is the smallest number of facets of any polyhedron $P$ such that the integer points in $P$ coincide with $X$. It is a useful tool to investigate the existence of compact linear descriptions of $X$. In this article, we derive … Read more