A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ sides are unknown when $s \ge 4$. In this paper, we construct a family of convex small $n$-gons, $n=2^s$ with $s\ge 4$, and show that their perimeters and their widths are within … Read more
Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of … Read more
In bike sharing systems, the spatiotemporal imbalance of bike flows leads to shortages of bikes in some areas and overages in some others, depending on the time of the day, resulting in user dissatisfaction. Repositioning needs to be performed timely to deal with the spatiotemporal imbalance and to meet customer demand in time. In this … Read more
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
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
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
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