Tootfinder

Santal\'o Geometry of Convex Polytopes
Dmitrii Pavlov, Simon Telen
https://arxiv.org/abs/2402.18955 https://arxiv.org/pdf/2402.18…

Santaló Geometry of Convex Polytopes
The Santaló point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santaló point traces out a semi-algebraic set. We describe and compute this geometry using algebraic and numerical techniques. We exploit connections with statistics, optimi…

This https://arxiv.org/abs/2401.17360 has been replaced.
link: https://scholar.google.com/scholar?q=a

Bender--Knuth Billiards in Coxeter Groups
Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where $I$ is a finite index set. Fix a nonempty convex subset $\mathscr{L}$ of $W$. If $W$ is of type $A$, then $\mathscr{L}$ is the set of linear extensions of a poset, and there are important Bender--Knuth involutions $\mathrm{BK}_i\colon\mathscr{L}\to\mathscr{L}$ indexed by elements of $I$. For arbitrary $W$ and for each $i\in I$, we introduce an operator $τ_i\colon W\to W$ (depending on $\mathscr{L}$) that we call a noninvertib…

This https://arxiv.org/abs/2306.00710 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Set-Valued Analysis of Generalized Barycentric Coordinates and Their Geometric Properties
Letting $P$ be a convex polytope in $\mathbb{R}^d$ with $n>d$ vertices, we study geometric and analytical properties of the set of generalized barycentric coordinates relative to any point $p\in P$. We prove that such sets are polytopes in $\mathbb{R}^n$ with at most $n-d-1$ vertices, and provide results about continuity and differentiability for the corresponding set-valued maps.

This https://arxiv.org/abs/2209.11370 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the Convexity of General Inverse $σ_k$ Equations
We prove that if a level set of a degree $n$ general inverse $σ_k$ equation $f(λ_1, \cdots, λ_n) = λ_1 \cdots λ_n - \sum_{k = 0}^{n-1} c_k σ_k(λ) = 0$ is contained in $q + Γ_n$ for some $q \in \mathbb{R}^n$, where $c_k$ are real numbers not necessary to be non-negative and $Γ_n$ is the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the level set of all general inverse $σ_k$ type equations, for example, the Monge--Ampère equa…

This https://arxiv.org/abs/2312.11200 has been replaced.
link: https://scholar.google.com/scholar?q=a

Solving the Optimal Experiment Design Problem with Mixed-Integer Convex Methods
We tackle the Optimal Experiment Design Problem, which consists of choosing experiments to run or observations to select from a finite set to estimate the parameters of a system. The objective is to maximize some measure of information gained about the system from the observations, leading to a convex integer optimization problem. We leverage Boscia.jl , a recent algorithmic framework, which is based on a nonlinear branch-and-bound algorithm with node relaxations solved to approximate optimalit…

This https://arxiv.org/abs/2302.09529 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Duality for coalgebras for Vietoris and monadicity
We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over Set. We deliver an analogous result for the upper, lower and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jonsson-Tarski duality for modal algebras beyond the 0-dimensional setting.

Relationships between Global and Local Monotonicity of Operators
Pham Duy Khanh, Vu Vinh Huy Khoa, Juan Enrique Mart\'inez-Legaz, Boris S. Mordukhovich
https://arxiv.org/abs/2403.20227

Relationships between Global and Local Monotonicity of Operators
The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between finite-dimensional and infinite-dimensional spaces. We first show that for single-valued operators with convex domains in locally convex topological spaces, their continuity ensures that their global monotonicity agrees with the local one around any point of the graph. This also holds for set-valued mappings defined on …

Set-valued Star-Shaped Risk Measures
Bingchu Nie, Dejian Tian, Long Jiang
https://arxiv.org/abs/2402.18014 https://arxiv.org/pdf/2402…

Set-valued Star-Shaped Risk Measures
In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-valued risk measures can be represented as the union of a family of set-valued convex risk measures, and set-valued normalized star-shaped risk measures can be represented as the union of a family of se…

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Generalized product property
Bergur Snorrason
https://arxiv.org/abs/2404.18728 https://arxiv.org/pdf/2404.18728
arXiv:2404.18728v1 Announce Type: new
Abstract: A famous result of Siciak is how the Siciak-Zakharyuta functions, sometimes called global extremal functions or pluricomplex Green functions with a pole at infinity, of two sets relate to the Siciak-Zakharyuta function of their cartesian product. In this paper Siciak's result is generalized to the setting of Siciak-Zakharyuta functions with growth given by a compact convex set, along with discussing why this generalization does not work in the weighted setting.

This https://arxiv.org/abs/2304.08596 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Hidden convexity, optimization, and algorithms on rotation matrices
This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to the constraint $X \in \text{SO}(n)$. Nonetheless, we show that certain linear images of $\text{SO}(n)$ are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical contributions show that any two-dimensional image of $\text{SO}(n)$ is conve…

This https://arxiv.org/abs/2302.08937 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Orthogonal Projection of Convex Sets with a Differentiable Boundary
Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior) and the boundary of its orthogonal projection onto the linear subspaces of the Euclidean space. A system of equations for these orthogonal projections is derived from this topological link. This result is illustrated by the projection of the unit ball of norm $…

Communication-Constrained STL Task Decomposition through Convex Optimization
Gregorio Marchesini, Siyuan Liu, Lars Lindemann, Dimos V. Dimarogonas
https://arxiv.org/abs/2402.17585

Communication-Constrained STL Task Decomposition through Convex Optimization
In this work, we propose a method to decompose signal temporal logic (STL) tasks for multi-agent systems subject to constraints imposed by the communication graph. Specifically, we propose to decompose tasks defined over multiple agents which require multi-hop communication, by a set of sub-tasks defined over the states of agents with 1-hop distance over the communication graph. To this end, we parameterize the predicates of the tasks to be decomposed as suitable hyper-rectangles. Then, we show…

Capacity Provisioning Motivated Online Non-Convex Optimization Problem with Memory and Switching Cost
Rahul Vaze, Jayakrishnan Nair
https://arxiv.org/abs/2403.17480

Capacity Provisioning Motivated Online Non-Convex Optimization Problem with Memory and Switching Cost
An online non-convex optimization problem is considered where the goal is to minimize the flow time (total delay) of a set of jobs by modulating the number of active servers, but with a switching cost associated with changing the number of active servers over time. Each job can be processed by at most one fixed speed server at any time. Compared to the usual online convex optimization (OCO) problem with switching cost, the objective function considered is non-convex and more importantly, at eac…

Finitely generated dyadic convex sets
K. Matczak, A. Mu\'cka, A. B. Romanowska
https://arxiv.org/abs/2403.17028 https://arxiv.org…

Finitely generated dyadic convex sets
Dyadic rationals are rationals whose denominator is a power of $2$. We define dyadic $n$-dimensional convex sets as the intersections with $n$-dimensional dyadic space of an $n$-dimensional real convex set. Such a dyadic convex set is said to be a dyadic $n$-dimensional polytope if the real convex set is a polytope whose vertices lie in the dyadic space. Dyadic convex sets are described as subreducts (subalgebras of reducts) of certain faithful affine spaces over the ring of dyadic numbers, or …

Set-valued Star-Shaped Risk Measures
Bingchu Nie, Dejian Tian, Long Jiang
https://arxiv.org/abs/2402.18014 https://arxiv.org/pdf/2402…

Set-valued Star-Shaped Risk Measures
In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-valued risk measures can be represented as the union of a family of set-valued convex risk measures, and set-valued normalized star-shaped risk measures can be represented as the union of a family of se…

This https://arxiv.org/abs/2210.04363 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Monge-Ampere system: convex integration in arbitrary dimension and codimension
In this paper, we study flexibility of the weak solutions to the Monge-Ampère system (MA) via convex integration. This system of Pdes is an extension of the Monge-Ampère equation in $d=2$ dimensions, naturally arising from the prescribed curvature problem, and closely related to the classical problem of isometric immersions (II). Our main result achieves density, in the set of subsolutions, of the Hölder regular $\mathcal{C}^{1,α}$ solutions to the weak formulation (VK) of (MA), for all $…

Inexact subgradient methods for semialgebraic functions
J\'er\^ome Bolte (TSE-R), Tam Le (UGA, LJK), \'Eric Moulines (CMAP, MBZUAI), Edouard Pauwels (TSE-R, IUF)
https://arxiv.org/abs/2404.19517

Inexact subgradient methods for semialgebraic functions
Motivated by the widespread use of approximate derivatives in machine learning and optimization, we study inexact subgradient methods with non-vanishing additive errors and step sizes. In the nonconvex semialgebraic setting, under boundedness assumptions, we prove that the method provides points that eventually fluctuate close to the critical set at a distance proportional to $ε^ρ$ where $ε$ is the error in subgradient evaluation and $ρ$ relates to the geometry of the problem. In the conve…

Sibson's formula for higher order Voronoi diagrams
Merc\`e Claverol, Andrea de las Heras-Parrilla, Clemens Huemer, Dolores Lara
https://arxiv.org/abs/2404.17422

Sibson's formula for higher order Voronoi diagrams
Let $S$ be a set of $n$ points in general position in $\mathbb{R}^d$. The order-$k$ Voronoi diagram of $S$, $V_k(S)$, is a subdivision of $\mathbb{R}^d$ into cells whose points have the same $k$ nearest points of $S$. Sibson, in his seminal paper from 1980 (A vector identity for the Dirichlet tessellation), gives a formula to express a point $Q$ of $S$ as a convex combination of other points of $S$ by using ratios of volumes of the intersection of cells of $V_2(S)$ and the cell of $Q$ in $V_1…

Nearly Optimal Regret for Decentralized Online Convex Optimization
Yuanyu Wan, Tong Wei, Mingli Song, Lijun Zhang
https://arxiv.org/abs/2402.09173 https://…

Nearly Optimal Regret for Decentralized Online Convex Optimization
We investigate decentralized online convex optimization (D-OCO), in which a set of local learners are required to minimize a sequence of global loss functions using only local computations and communications. Previous studies have established $O(n^{5/4}ρ^{-1/2}\sqrt{T})$ and ${O}(n^{3/2}ρ^{-1}\log T)$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners, $ρ<1$ is the spectral gap of the communication matrix, and $T$ is the time horizo…

This https://arxiv.org/abs/2403.14761 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Quantitative Steinitz theorem and polarity
The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points in $S$ whose convex hull contains the origin within its interior. Bárány, Katchalski, and Pach established a quantitative version of Steinitz's theorem, showing that for a convex polytope $Q$ in $\mathbb{R}^d$ containing the standard Euclidean unit ball $\mathbf{B}^d$, there exist at most $2d$ vertices of $Q$ whose convex h…

This https://arxiv.org/abs/2201.01362 has been replaced.
link: https://scholar.google.com/scholar?q=a

Billiards in generic convex bodies have positive topological entropy
We show that there exists a $C^2$ open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits.

Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space
David Xu
https://arxiv.org/abs/2402.12294 https://

Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space
We construct convex-cocompact representations of fundamental groups of closed hyperbolic surfaces into the isometry group of the infinite-dimensional hyperbolic space using bendings. We prove that convex-cocompact representations of finitely generated groups in the group of isometries of the infinite-dimensional hyperbolic space form an open set in the space of representations and that the space of deformations (up to conjugation) obtained by bending an irreducible representation of a surface g…

Entanglement Detection by Approximate Entanglement Witnesses
Samuel Dai, Ning Bao
https://arxiv.org/abs/2402.14755 https://arxiv.org/…

Entanglement Detection by Approximate Entanglement Witnesses
The problem of determining whether a given quantum state is separable is known to be computationally difficult. We develop an approach to this problem based on approximations of convex polytopes in high dimensions. By showing that a convex polytope constructed from a polynomial number of hyperplanes approximates the Euclidean ball arbitrarily well in high dimensions, we find evidence that a polynomial-sized set of approximate entanglement witnesses is potentially sufficient to determine the ent…

Learning control strategy in soft robotics through a set of configuration spaces
Etienne M\'enagerDEFROST, Christian DuriezDEFROST, CRIStAL
https://arxiv.org/abs/2402.13649

Learning control strategy in soft robotics through a set of configuration spaces
The ability of a soft robot to perform specific tasks is determined by its contact configuration, and transitioning between configurations is often necessary to reach a desired position or manipulate an object. Based on this observation, we propose a method for controlling soft robots that involves defining a graph of configuration spaces. Different agents, whether learned or not (convex optimization, expert trajectory, and collision detection), use the structure of the graph to solve the desir…

Functions on a convex set which are both $ \omega $-semiconvex and $ \omega $-semiconcave II
V\'aclav Kry\v{s}tof
https://arxiv.org/abs/2403.14901 http…

Functions on a convex set which are both $ ω$-semiconvex and $ ω$-semiconcave II
In a recent article (2022) we proved with L. Zajíček that if $ G\subset\R^n $ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $ f:G\to\R $ and a concave modulus $ ω$ such that $ \lim_{t\to\infty}ω(t)=\infty $, $ f $ is both semiconvex and semiconcave with modulus $ ω$ and $ f\notin C^{1,ω}(G) $. Here we improve the previous result as follows: If $ G $ is as above and $ ω(t)=t^α $ for some $ α\in(0,1) $, then…

Overfitting Reduction in Convex Regression
Zhiqiang Liao, Sheng Dai, Eunji Lim, Timo Kuosmanen
https://arxiv.org/abs/2404.09528 https://

Overfitting Reduction in Convex Regression
Convex regression is a method for estimating an unknown function $f_0$ from a data set of $n$ noisy observations when $f_0$ is known to be convex. This method has played an important role in operations research, economics, machine learning, and many other areas. It has been empirically observed that the convex regression estimator produces inconsistent estimates of $f_0$ and extremely large subgradients near the boundary of the domain of $f_0$ as $n$ increases. In this paper, we provide theoret…

Circular Distribution of Agents using Convex Layers
Gautam Kumar, Ashwini Ratnoo
https://arxiv.org/abs/2404.11351 https://arxiv.org/p…

Circular Distribution of Agents using Convex Layers
This paper considers the problem of conflict-free distribution of agents on a circular periphery encompassing all agents. The two key elements of the proposed policy include the construction of a set of convex layers (nested convex polygons) using the initial positions of the agents, and a novel search space region for each of the agents. The search space for an agent on a convex layer is defined as the region enclosed between the lines passing through the agent's position and normal to its sup…

Sibson's formula for higher order Voronoi diagrams
Merc\`e Claverol, Andrea de las Heras-Parrilla, Clemens Huemer, Dolores Lara
https://arxiv.org/abs/2404.17422

Sibson's formula for higher order Voronoi diagrams
Let $S$ be a set of $n$ points in general position in $\mathbb{R}^d$. The order-$k$ Voronoi diagram of $S$, $V_k(S)$, is a subdivision of $\mathbb{R}^d$ into cells whose points have the same $k$ nearest points of $S$. Sibson, in his seminal paper from 1980 (A vector identity for the Dirichlet tessellation), gives a formula to express a point $Q$ of $S$ as a convex combination of other points of $S$ by using ratios of volumes of the intersection of cells of $V_2(S)$ and the cell of $Q$ in $V_1…

Generalising realisability in statistical learning theory under epistemic uncertainty
Fabio Cuzzolin
https://arxiv.org/abs/2402.14759 https://

Generalising realisability in statistical learning theory under epistemic uncertainty
The purpose of this paper is to look into how central notions in statistical learning theory, such as realisability, generalise under the assumption that train and test distribution are issued from the same credal set, i.e., a convex set of probability distributions. This can be considered as a first step towards a more general treatment of statistical learning under epistemic uncertainty.

Upper estimates for the Hausdorff dimension of the temporal singular set in chemotaxis-fluid systems
Mario Fuest
https://arxiv.org/abs/2402.16582 https://<…

Upper estimates for the Hausdorff dimension of the temporal singular set in chemotaxis-fluid systems
The chemotaxis-fluid system \begin{align}\tag{$\star$}\label{prob:star} \begin{cases} n_t + u \cdot \nabla n = Δn - \nabla \cdot (n \nabla c), \\ c_t + u \cdot \nabla c = Δc - nc, \\ u_t + (u \cdot \nabla) u = Δu + \nabla P + n \nabla Φ, \quad \nabla \cdot u = 0, \end{cases} \end{align} models aerobic bacteria interacting with a fluid via transportation and buoyancy. When posed on a three-dimensional, smoothly bounded, convex domain $Ω$, \eqref{prob:star} complemented with suitable initial…

This https://arxiv.org/abs/2310.04176 has been replaced.
link: https://scholar.google.com/scholar?q=a

Approximating the set of Nash equilibria for convex games
In Feinstein and Rudloff (2023), it was shown that the set of Nash equilibria for any non-cooperative $N$ player game coincides with the set of Pareto optimal points of a certain vector optimization problem with non-convex ordering cone. To avoid dealing with a non-convex ordering cone, an equivalent characterization of the set of Nash equilibria as the intersection of the Pareto optimal points of $N$ multi-objective problems (i.e.\ with the natural ordering cone) is proven. So far, algorithms …

The Minkowski problem for the non-compact convex set with an asymptotic boundary condition
Ning Zhang
https://arxiv.org/abs/2402.12802 https://

The Minkowski problem for the non-compact convex set with an asymptotic boundary condition
In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric interpretation is obtained by the Hadamard variational formula. The Brunn-Minkowski and Minkowski inequalities for covolume are established, and the equivalence of these two inequalities are discussed as well. The Minkowski problem for non-compact convex set is p…

Convergence of Some Convex Message Passing Algorithms to a Fixed Point
Vaclav Voracek, Tomas Werner
https://arxiv.org/abs/2403.07004 https://

Convergence of Some Convex Message Passing Algorithms to a Fixed Point
A popular approach to the MAP inference problem in graphical models is to minimize an upper bound obtained from a dual linear programming or Lagrangian relaxation by (block-)coordinate descent. Examples of such algorithms are max-sum diffusion and sequential tree-reweighted message passing. Convergence properties of these methods are currently not fully understood. They have been proved to converge to the set characterized by local consistency of active constraints, with unknown convergence rat…

Spline Trajectory Tracking and Obstacle Avoidance for Mobile Agents via Convex Optimization
Akua Dickson, Christos G. Cassandras, Roberto Tron
https://arxiv.org/abs/2403.16900

Spline Trajectory Tracking and Obstacle Avoidance for Mobile Agents via Convex Optimization
We propose an output feedback control-based motion planning technique for agents to enable them to converge to a specified polynomial trajectory while imposing a set of safety constraints on our controller to avoid collisions within the free configuration space (polygonal environment). To achieve this, we 1) decompose our polygonal environment into different overlapping cells 2) write out our polynomial trajectories as the output of a reference dynamical system with given initial conditions 3) …

On 1-skeleton of the cut polytopes
Andrei V. Nikolaev
https://arxiv.org/abs/2402.13591 https://arxiv.org/pdf/2402.13591

On 1-skeleton of the cut polytopes
Given an undirected graph $G = (V,E)$, the cut polytope $\mathrm{CUT}(G)$ is defined as the convex hull of the incidence vectors of all cuts in $G$. The 1-skeleton of $\mathrm{CUT}(G)$ is a graph whose vertex set is the vertex set of the polytope, and the edge set is the set of geometric edges or one-dimensional faces of the polytope. We study the diameter and the clique number of 1-skeleton of cut polytopes for several classes of graphs. These characteristics are of interest since they estimat…

This https://arxiv.org/abs/2310.09154 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Every quantum helps: Operational advantage of quantum resources beyond convexity
Identifying what quantum-mechanical properties are useful to untap a superior performance in quantum technologies is a pivotal question. Quantum resource theories provide a unified framework to analyze and understand such properties, as successfully demonstrated for entanglement and coherence. While these are examples of convex resources, for which quantum advantages can always be identified, many physical resources are described by a nonconvex set of free states and their interpretation has so…

Quantitative Steinitz theorem and polarity
Grigory Ivanov
https://arxiv.org/abs/2403.14761 https://arxiv.org/pdf/2403.14761

Quantitative Steinitz theorem and polarity
The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points in $S$ whose convex hull contains the origin within its interior. Bárány, Katchalski, and Pach established a quantitative version of Steinitz's theorem, showing that for a convex polytope $Q$ in $\mathbb{R}^d$ containing the standard Euclidean unit ball $\mathbf{B}^d$, there exist at most $2d$ vertices of $Q$ whose convex h…

Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions
Fabrizio Grandoni, Edin Husi\'c, Mathieu Mari, Antoine Tinguely
https://arxiv.org/abs/2402.07666

Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions
In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the problem where the edges of the polygons can take at most $d$ fixed directions. We present an $8d/3$-approximation algorithm for this problem running in time $O((nd)^{O(d4^d)})$. The previous-best polynomial-time approximation (for constant $d$) was a classica…

This https://arxiv.org/abs/2308.01753 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Minimal Convex Environmental Contours
We develop a numerical method for the computation of a minimal convex and compact set, $\mathcal{B}\subset\mathbb{R}^N$, in the sense of mean width. This minimisation is constrained by the requirement that $\max_{b\in\mathcal{B}}\langle b , u\rangle\geq C(u)$ for all unit vectors $u\in S^{N-1}$ given some Lipschitz function $C$. This problem arises in the construction of environmental contours under the assumption of convex failure sets. Environmental contours offer descriptions of extreme en…

Convergence of Some Convex Message Passing Algorithms to a Fixed Point
Vaclav Voracek, Tomas Werner
https://arxiv.org/abs/2403.07004 https://

Convergence of Some Convex Message Passing Algorithms to a Fixed Point
A popular approach to the MAP inference problem in graphical models is to minimize an upper bound obtained from a dual linear programming or Lagrangian relaxation by (block-)coordinate descent. Examples of such algorithms are max-sum diffusion and sequential tree-reweighted message passing. Convergence properties of these methods are currently not fully understood. They have been proved to converge to the set characterized by local consistency of active constraints, with unknown convergence rat…

On some relations between the Perimeter, the Area and the Visual Angle of a Convex Set
Joaquim Bruna, Juli\`a Cuf\'i, Agust\'i Revent\'os
https://arxiv.org/abs/2404.08349

On some relations between the Perimeter, the Area and the Visual Angle of a Convex Set
We establish some relations between the perimeter, the area and the visual angle of a planar compact convex set. Our first result states that Crofton's formula is the unique universal formula relating the visual angle, length and area. After that we give a characterization of convex sets of constant width by means of the behaviour of its isotopic sets at infinity. Also for this class of convex sets we prove that the existence of an isotopic circle is enough to ensure that the considered set is …

This https://arxiv.org/abs/2307.09167 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On linearisation and uniqueness of preduals
We study strong linearisations and the uniqueness of preduals of locally convex Hausdorff spaces of scalar-valued functions. Strong linearisations are special preduals. A locally convex Hausdorff space $\mathcal{F}(Ω)$ of scalar-valued functions on a non-empty set $Ω$ is said to admit a strong linearisation if there are a locally convex Hausdorff space $Y$, a map $δ\colonΩ\to Y$ and a topological isomorphism $T\colon\mathcal{F}(Ω)\to Y_{b}'$ such that $T(f)\circ δ= f$ for all $f\in\mathca…

This https://arxiv.org/abs/2204.06953 has been replaced.
link: https://scholar.google.com/scholar?q=a

Convex Hulls of Grassmannians and Combinatorics of Symmetric Hypermatrices
It is known that the complex Grassmannian of $k$-dimensional subspaces can be identified with the set of projection matrices of rank $k$. It is also classically known that the convex hull of this set is the set of Hermitian matrices with eigenvalues between $0$ and $1$ and summing to $k$. We give a new proof of this fact. We also give an existence theorem for a certain combinatorial class of hypermatrices by a similar argument. This existence theorem can be rewritten into an existence theorem f…

This https://arxiv.org/abs/2302.07382 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Free Extreme points of free spectrahedrops and generalized free spectrahedra
Matrix convexity generalizes convexity to the dimension free setting and has connections to many mathematical and applied pursuits including operator theory, quantum information, noncommutative optimization, and linear control systems. In the setting of classical convex sets, extreme points are central objects which exhibit many important properties. For example, the Minkowski theorem shows that any element of a closed bounded convex set can be expressed as a convex combination of extreme point…

Computing the EHZ capacity is NP-hard
Karla Leipold, Frank Vallentin
https://arxiv.org/abs/2402.09914 https://arxiv.org/pdf/2402.0991…

Computing the EHZ capacity is NP-hard
The Ekeland-Hofer-Zehnder capacity (EHZ capacity) is a fundamental symplectic invariant of convex bodies. We show that computing the EHZ capacity of polytopes is NP-hard. For this we reduce the feedback arc set problem in bipartite tournaments to computing the EHZ capacity of simplices.

Formal Verification of the Empty Hexagon Number
Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, Marijn J. H. Heule
https://arxiv.org/abs/2403.17370

Formal Verification of the Empty Hexagon Number
A recent breakthrough in computer-assisted mathematics showed that every set of $30$ points in the plane in general position (i.e., without three on a common line) contains an empty convex hexagon, thus closing a line of research dating back to the 1930s. Through a combination of geometric insights and automated reasoning techniques, Heule and Scheucher constructed a CNF formula $ϕ_n$, with $O(n^4)$ clauses, whose unsatisfiability implies that no set of $n$ points in general position can avoid…

This https://arxiv.org/abs/2301.05932 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On continuation and convex Lyapunov functions
Suppose that the origin is globally asymptotically stable under a set of continuous vector fields on Euclidean space and suppose that all those vector fields come equipped with -- possibly different -- convex Lyapunov functions. We show that this implies there is a homotopy between any two of those vector fields such that the origin remains globally asymptotically stable along the homotopy. Relaxing the assumption on the origin to any compact convex set or relaxing convexity to geodesic convexi…

This https://arxiv.org/abs/2404.02341 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Fully chaotic conservative models for some torus homeomorphisms
We study homotopic-to-the-identity torus homeomorphisms, whose rotation set has nonempty interior. We prove that any such map is monotonically semiconjugate to a homeomorphism that preserves the Lebesgue measure, and that has the same rotation set. Furthermore, the dynamics of the quotient map has several interesting chaotic traits: for instance, it is topologically mixing, has a dense set of periodic points and is continuum-wise expansive. In particular, this shows that a convex compact set of…

This https://arxiv.org/abs/2403.04917 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csRO_…

A Mixed-Integer Conic Program for the Moving-Target Traveling Salesman Problem based on a Graph of Convex Sets
This paper introduces a new formulation that finds the optimum for the Moving-Target Traveling Salesman Problem (MT-TSP), which seeks to find a shortest path for an agent, that starts at a depot, visits a set of moving targets exactly once within their assigned time-windows, and returns to the depot. The formulation relies on the key idea that when the targets move along lines, their trajectories become convex sets within the space-time coordinate system. The problem then reduces to finding the…

When are Lossy Energy Storage Optimization Models Convex?
Feras Al Taha, Eilyan Bitar
https://arxiv.org/abs/2403.14010 https://arxiv.…

When are Lossy Energy Storage Optimization Models Convex?
We consider a class of optimization problems involving the optimal operation of a single lossy energy storage system that incurs energy loss when charging or discharging. Such inefficiencies in the energy storage dynamics are known to result in a nonconvex set of feasible charging and discharging power profiles. In this letter, we provide an equivalent reformulation for this class of optimization problems, along with sufficient conditions for the convexity of the proposed reformulation. The con…

This https://arxiv.org/abs/2309.15922 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Capping the positivity cone: dimension-8 Higgs operators in the SMEFT
SMEFT Wilson coefficients are subject to various positivity bounds in order to be consistent with the fundamental principles of S-matrix. Previous bounds on dimension-8 SMEFT operators have been obtained using the positivity part of UV partial wave unitarity and form a (projective) convex cone. We derive a set of linear UV unitarity conditions that go beyond positivity and are easy to implement in an optimization scheme with dispersion relations in a multi-field EFT. Using Higgs scattering as a…

Continuity of HYM connections with respect to metric variations
R\'emi Delloque
https://arxiv.org/abs/2403.16814 https://arxiv.or…

Continuity of HYM connections with respect to metric variations
We investigate the set of (real Dolbeault classes of) balanced metrics $Θ$ on a balanced manifold $X$ with respect to which a torsion-free coherent sheaf $\mathcal{E}$ on $X$ is slope stable. We prove that the set of all such $[Θ] \in H^{n - 1,n - 1}(X,\mathbb{R})$ is a convex cone defined by a finite number of linear inequalities. For this, we prove that the set of all $c_1(\mathcal{S})$ for $\mathcal{S} \subset \mathcal{E}$ saturated and coherent is finite. When $\mathcal{E}$ is a Hermiti…

Monitoring the Convergence Speed of PDHG to Find Better Primal and Dual Step Sizes
Olivier Fercoq (S2A, LTCI)
https://arxiv.org/abs/2403.19202 https://

Monitoring the Convergence Speed of PDHG to Find Better Primal and Dual Step Sizes
Primal-dual algorithms for the resolution of convex-concave saddle point problems usually come with one or several step size parameters. Within the range where convergence is guaranteed, choosing well the step size can make the difference between a slow or a fast algorithm. A usual way to adaptively set step sizes is to ensure that there is a fair balance between primal and dual variable's amount of change. In this work, we show how to find even better step sizes for the primal-dual hybrid grad…

Extreme points of general transportation polytopes
Patrice Koehl
https://arxiv.org/abs/2404.16791 https://arxiv.org/pdf/2404.16791

Extreme points of general transportation polytopes
Transportation matrices are $m\times n$ non-negative matrices whose row sums and row columns are equal to, or dominated above with given integral vectors $R$ and $C$. Those matrices belong to a convex polytope whose extreme points have been previously characterized. In this article, a more general set of non-negative transportation matrices is considered, whose row sums are bounded by two integral non-negative vectors $R_{min}$ and $R_{max}$ and column sums are bounded by two integral non-negat…

The $L_p$ dual Minkowski problem for unbounded closed convex sets
Wen Ai, Yunlong Yang, Deping Ye
https://arxiv.org/abs/2404.09804 https://

The $L_p$ dual Minkowski problem for unbounded closed convex sets
The central focus of this paper is the $L_p$ dual Minkowski problem for $C$-compatible sets, where $C$ is a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior. Such a problem deals with the characterization of the $(p, q)$-th dual curvature measure of a $C$-compatible set. It produces new Monge-Ampère equations for unbounded convex hypersurface, often defined over open domains and with non-positive unknown convex functions. Within the family of $C$-determined sets, the $L_p$ d…

Real plane separating (M-2)-curves of degree d and totally real pencils of degree (d-3)
Matilde Manzaroli
https://arxiv.org/abs/2404.09671 https://<…

Real plane separating (M-2)-curves of degree d and totally real pencils of degree (d-3)
It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context and generalised to all real plane separating (M-2)-curves.

On linearisation and existence of preduals
Karsten Kruse
https://arxiv.org/abs/2402.12615 https://arxiv.org/pdf/2402.12615

On linearisation and existence of preduals
We study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space $X$. Then we turn to the case that $X=\mathcal{F}(Ω)$ is a space of scalar-valued functions on a non-empty set $Ω$ and characterise those among them which admit a special predual, namely a strong linearisation, i.e. there are a locally convex Hausdorff space $Y$, a…

This https://arxiv.org/abs/2311.18438 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Solution-Set Geometry and Regularization Path of a Nonconvexly Regularized Convex Sparse Model
The generalized minimax concave (GMC) penalty is a nonconvex sparse regularizer which can preserve the overall-convexity of the regularized least-squares problem. In this paper, we focus on a significant instance of the GMC model termed scaled GMC (sGMC), and present various notable findings on its solution-set geometry and regularization path. Our investigation indicates that while the sGMC penalty is a nonconvex extension of the LASSO penalty (i.e., the $\ell_1$-norm), the sGMC model preserve…

This https://arxiv.org/abs/1709.05504 has been replaced.
link: https://scholar.google.com/scholar?q=a

A priori bounds for geodesic diameter. Part III. A Sobolev-Poincaré inequality and applications to a variety of geometric variational problems
Based on a novel type of Sobolev-Poincaré inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of the mean curvatures of the surface and its boundary. Our varifold setting includes smooth immersions, surfaces with finite Willmore energy, two-convex hypersurfaces in level-set mean curvature flow, integral curr…

This https://arxiv.org/abs/2308.11393 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Strong limit theorems for empirical halfspace depth trimmed regions
We study empirical variants of the halfspace (Tukey) depth of a probability measure $μ$, which are obtained by replacing $μ$ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of $μ$ being the uniform distribution on a convex b…

Greedy Monochromatic Island Partitions
Steven van den Broek, Wouter Meulemans, Bettina Speckmann
https://arxiv.org/abs/2402.13340 https://

Greedy Monochromatic Island Partitions
Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in the plane where each point has one of $k \geq 2$ colors. A set of points is monochromatic if it contains points of only one color. An island $I$ is a subset of $S$ such that $\mathcal{CH}(I) \cap S = I$, where $\mathcal{CH}(I)$ denotes the convex hull of $I$…

Differential inclusions, polycrystals and stability under lamination
Nathan Albin, Vincenzo Nesi, Mariapia Palombaro
https://arxiv.org/abs/2402.06401 https…

Differential inclusions, polycrystals and stability under lamination
We study approximate solutions to a differential inclusion associated to a certain system of pdes in dimension three. The only datum is a set of three positive numbers identified with a positive definite diagonal matrix $S$. The average fields naturally belong to the convex hull of the set of points obtained by the triple $S$ and its permutations. We find a set of attainable average fields strictly contained in the convex hull and stable under lamination. The corresponding microgeometries are l…

Fully chaotic conservative models for some torus homeomorphisms
Alejo Garc\'ia-Sassi, F\'abio Armando Tal
https://arxiv.org/abs/2404.02341 https://…

Fully chaotic conservative models for some torus homeomorphisms
We study homotopic-to-the-identity torus homeomorphisms, whose rotation set has nonempty interior. We prove that any such map is monotonically semiconjugate to a homeomorphism that preserves the Lebesgue measure, and that has the same rotation set. Furthermore, the dynamics of the quotient map has several interesting chaotic traits: for instance, it is topologically mixing, has a dense set of periodic points and is continuum-wise expansive. In particular, this shows that a convex compact set of…

This https://arxiv.org/abs/2305.10620 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Soft-Minimum and Soft-Maximum Barrier Functions for Safety with Actuation Constraints
This paper presents two new control approaches for guaranteed safety (remaining in a safe set) subject to actuator constraints (the control is in a convex polytope). The control signals are computed using real-time optimization, including linear and quadratic programs subject to affine constraints, which are shown to be feasible. The first control method relies on a soft-minimum barrier function that is constructed using a finite-time-horizon prediction of the system trajectories under a known …

This https://arxiv.org/abs/2302.13616 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

ResQPASS: an algorithm for bounded variable linear least squares with asymptotic Krylov convergence
We present the Residual Quadratic Programming Active-Set Subspace (ResQPASS) method that solves large-scale linear least-squares problems with bound constraints on the variables. The problem is solved by creating a series of small problems of increasing size by projecting on the basis of residuals. Each projected problem is solved by the active-set method for convex quadratic programming, warm-started with a working set and solution of the previous problem. The method coincides with conjugate g…

This https://arxiv.org/abs/2212.03014 has been replaced.
link: https://scholar.google.com/scholar?q=a

Lower Bounding Ground-State Energies of Local Hamiltonians Through the Renormalization Group
Given a renormalization scheme, we show how to formulate a tractable convex relaxation of the set of feasible local density matrices of a many-body quantum system. The relaxation is obtained by introducing a hierarchy of constraints between the reduced states of ever-growing sets of lattice sites. The coarse-graining maps of the underlying renormalization procedure serve to eliminate a vast number of those constraints, such that the remaining ones can be enforced with reasonable computational m…

Sparse systems with high local multiplicity
Fr\'ed\'eric Bihan, Alicia Dickenstein, Jens Forsg{\aa}rd
https://arxiv.org/abs/2402.08410 https://

Sparse systems with high local multiplicity
Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the …

The Price of Adaptivity in Stochastic Convex Optimization
Yair Carmon, Oliver Hinder
https://arxiv.org/abs/2402.10898 https://arxiv.o…

The Price of Adaptivity in Stochastic Convex Optimization
We prove impossibility results for adaptivity in non-smooth stochastic convex optimization. Given a set of problem parameters we wish to adapt to, we define a "price of adaptivity" (PoA) that, roughly speaking, measures the multiplicative increase in suboptimality due to uncertainty in these parameters. When the initial distance to the optimum is unknown but a gradient norm bound is known, we show that the PoA is at least logarithmic for expected suboptimality, and double-logarithmic for median…

Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4
Maarten L\"offler, Tamara Mchedlidze, David Orden, Josef Tkadlec, Jules Wulms
https://arxiv.org/abs/2402.12357

Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4
A pseudo-triangle is a simple polygon with exactly three convex vertices, and all other vertices (if any) are distributed on three concave chains. A pseudo-triangulation~$\mathcal{T}$ of a point set~$P$ in~$\mathbb{R}^2$ is a partitioning of the convex hull of~$P$ into pseudo-triangles, such that the union of the vertices of the pseudo-triangles is exactly~$P$. We call a size-4 pseudo-triangle a dart. For a fixed $k\geq 1$, we study $k$-dart pseudo-triangulations ($k$-DPTs), that is, pseudo-tri…

Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones
Jes\'us A. De Loera, Brittney Marsters, Luze Xu, Shixuan Zhang
https://arxiv.org/abs/2403.09927

Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones
We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a f…

An example of an "unlinked" set of $2k 3$ points in $2k$-space
M. Starkov
https://arxiv.org/abs/2402.09002 https://arxiv.…

An example of an "unlinked" set of $2k + 3$ points in $2k$-space
Take any $d + 3$ points in $\mathbb{R}^d$. It is known that (a) if $d = 2k + 1$, then there are two linked $(k + 1)$-simplices with the vertices at these points; (b) if $d = 2k$, then there are two disjoint $(k + 1)$-tuples of these points such that their convex hulls intersect. The analogue of (b) for $d = 2k + 1$, which is also the analogue of (a) for intersections (instead of linkings), states that there are two disjoint $(k + 1)$- and $(k + 2)$-tuples of these points such that their c…

This https://arxiv.org/abs/2006.11568 has been replaced.
link: https://scholar.google.com/scholar?q=a

Quasiconvex functions on regular trees
We introduce a definition of a quasiconvex function on an infinite directed regular tree that depends on what we understood by a segment on the tree. Our definition is based on thinking on segments as sub-trees with the root as the midpoint of the segment. A convex set in the tree is then a subset such that it contains every midpoint of every segment with terminal nodes in the set. Then a quasiconvex function is a real map on the tree such that every level set is a convex set. For this concept …

Dynamic Convex Hulls for Simple Paths
Bruce Brewer, Gerth St{\o}lting Brodal, Haitao Wang
https://arxiv.org/abs/2403.05697 https://ar…

Dynamic Convex Hulls for Simple Paths
We consider the planar dynamic convex hull problem. In the literature, solutions exist supporting the insertion and deletion of points in poly-logarithmic time and various queries on the convex hull of the current set of points in logarithmic time. If arbitrary insertion and deletion of points are allowed, constant time updates and fast queries are known to be impossible. This paper considers two restricted cases where worst-case constant time updates and logarithmic time queries are possible. …

Projection-free computation of robust controllable sets with constrained zonotopes
Abraham P. Vinod, Avishai Weiss, Stefano Di Cairano
https://arxiv.org/abs/2403.13730

Projection-free computation of robust controllable sets with constrained zonotopes
We study the problem of computing robust controllable sets for discrete-time linear systems with additive uncertainty. We propose a tractable and scalable approach to inner- and outer-approximate robust controllable sets using constrained zonotopes, when the additive uncertainty set is a symmetric, convex, and compact set. Our least-squares-based approach uses novel closed-form approximations of the Pontryagin difference between a constrained zonotopic minuend and a symmetric, convex, and compa…

This https://arxiv.org/abs/2302.13616 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

ResQPASS: an algorithm for bounded variable linear least squares with asymptotic Krylov convergence
We present the Residual Quadratic Programming Active-Set Subspace (ResQPASS) method that solves large-scale linear least-squares problems with bound constraints on the variables. The problem is solved by creating a series of small problems of increasing size by projecting on the basis of residuals. Each projected problem is solved by the active-set method for convex quadratic programming, warm-started with a working set and solution of the previous problem. The method coincides with conjugate g…

Polynomial Fourier Decay For Patterson-Sullivan Measures
Osama Khalil
https://arxiv.org/abs/2404.09424 https://arxiv.org/pdf/2404.094…

Polynomial Fourier Decay For Patterson-Sullivan Measures
We show that the Fourier transform of Patterson-Sullivan measures associated to convex cocompact groups of isometries of real hyperbolic space decays polynomially quickly at infinity. The proof is based on the $L^2$-flattening theorem obtained in prior work of the author, combined with a method based on dynamical self-similarity for ruling out the sparse set of potential frequencies where the Fourier transform can be large.

On the Set of Possible Minimizers of a Sum of Convex Functions
Moslem Zamani, Fran\c{c}ois Glineur, Julien M. Hendrickx
https://arxiv.org/abs/2403.05467 ht…

On the Set of Possible Minimizers of a Sum of Convex Functions
Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results cover several types of assumptions on the summands, such as smoothness or strong convexity. Our main tool is the use of necessary and sufficient conditions for interpolating the considered function classes, which leads to shorter and more direct proofs in compar…

This https://arxiv.org/abs/2106.11085 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the Maximal Monotone Operators in Hadamard Spaces
In this paper, some topics of monotone operator theory in the setting of Hadamard spaces are investigated. For a fixed element $p$ in a Hadamard space $X$, the notion of $p$-Fenchel conjugate is introduced and a type of the Fenchel-Young inequality is proved. Moreover, we examine the $p$-Fitzpatrick transform and its main properties for monotone set-valued operators in Hadamard spaces. Furthermore, some relations between maximal monotone operators and certain classes of proper, convex, l.s.c. e…

This https://arxiv.org/abs/2302.13970 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications
We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our …

This https://arxiv.org/abs/2310.00397 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Powered Descent Guidance via First-Order Optimization with Expansive Projection
This paper introduces a first-order method for solving optimal powered descent guidance (PDG) problems, that directly handles the nonconvex constraints associated with the maximum and minimum thrust bounds with varying mass and the pointing angle constraints on thrust vectors. This issue has been conventionally circumvented via lossless convexification (LCvx), which lifts a nonconvex feasible set to a higher-dimensional convex set, and via linear approximation of another nonconvex feasible set …

Rainbow ortho-convex 4-sets in k-colored point sets
David Flores-Pe\~naloza (Departamento de Matem\'aticas, Facultad de Ciencias, Universidad Nacional Aut\'onoma de M\'exico, Mexico), Mario A. Lopez (Department of Computer Science, University of Denver, US), Nestaly Mar\'in (Departamento de Matem\'aticas, Facultad de Ciencias, Universidad Nacional Aut\'onoma de M\'exico, Mexico), David Orden (Departamento de F\'isica y Matem\'aticas, Universidad de A…

Rainbow ortho-convex 4-sets in k-colored point sets
Let $P$ be a $k$-colored set of $n$ points in the plane, $4 \leq k \leq n$. We study the problem of deciding if $P$ contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We provide an $O(n \log n)$-time algorithm for this problem, where the hidden constant does not depend on $k$; then, we prove that this problem has time complexity $Ω(n \log n)$ in the algebraic computation tree model. No general position assumptions for $P$ are required.

This https://arxiv.org/abs/2310.04176 has been replaced.
link: https://scholar.google.com/scholar?q=a

Approximating the set of Nash equilibria for convex games
In Feinstein and Rudloff (2023), it was shown that the set of Nash equilibria for any non-cooperative $N$ player game coincides with the set of Pareto optimal points of a certain vector optimization problem with non-convex ordering cone. To avoid dealing with a non-convex ordering cone, an equivalent characterization of the set of Nash equilibria as the intersection of the Pareto optimal points of $N$ multi-objective problems (i.e.\ with the natural ordering cone) is proven. So far, algorithms …

This https://arxiv.org/abs/2209.14802 has been replaced.
link: https://scholar.google.com/scholar?q=a

Steiner Cut Dominants
For a subset T of nodes of an undirected graph G, a T-Steiner cut is a cut δ(S) where S intersects both T and the complement of T. The T-Steiner cut dominant} of G is the dominant CUT_+(G,T) of the convex hull of the incidence vectors of the T-Steiner cuts of G. For T={s,t}, this is the well-understood s-t-cut dominant. Choosing T as the set of all nodes of G, we obtain the \emph{cut dominant}, for which an outer description in the space of the original variables is still not known. We prove t…

Happy Ending: An Empty Hexagon in Every Set of 30 Points
Marijn J. H. Heule, Manfred Scheucher
https://arxiv.org/abs/2403.00737 https://

Happy Ending: An Empty Hexagon in Every Set of 30 Points
Satisfiability solving has been used to tackle a range of long-standing open math problems in recent years. We add another success by solving a geometry problem that originated a century ago. In the 1930s, Esther Klein's exploration of unavoidable shapes in planar point sets in general position showed that every set of five points includes four points in convex position. For a long time, it was open if an empty hexagon, i.e., six points in convex position without a point inside, can be avoided.…

Two models forsandpile growth in weighted graphs
J. M. Mazon, J. Toledo
https://arxiv.org/abs/2403.02900 https://arxiv.org/pdf/2403.0…

Two models forsandpile growth in weighted graphs
In this paper we study $\infty$-Laplacian type diffusion equations in weighted graphs obtained as limit as $p\to \infty$ to two types of $p$-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set $$K^G_{\infty}:= \left\{ u \in L^2(V, ν_G) \ : \ \vert u(y) - u(x) \vert \leq 1 \ \ \hbox{if} \ \ x \sim y \right\}$$ and the set $$K^w_{\infty}:= \left\…

Bicolored point sets admitting non-crossing alternating Hamiltonian paths
Jan Soukup
https://arxiv.org/abs/2404.06105 https://arxiv.o…

Bicolored point sets admitting non-crossing alternating Hamiltonian paths
Consider a bicolored point set $P$ in general position in the plane consisting of $n$ blue and $n$ red points. We show that if a subset of the red points forms the vertices of a convex polygon separating the blue points, lying inside the polygon, from the remaining red points, lying outside the polygon, then the points of $P$ can be connected by non-crossing straight-line segments so that the resulting graph is a properly colored closed Hamiltonian path.

This https://arxiv.org/abs/2301.05932 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On continuation and convex Lyapunov functions
Suppose that the origin is globally asymptotically stable under a set of continuous vector fields on Euclidean space and suppose that all those vector fields come equipped with -- possibly different -- convex Lyapunov functions. We show that this implies there is a homotopy between any two of those vector fields such that the origin remains globally asymptotically stable along the homotopy. Relaxing the assumption on the origin to any compact convex set or relaxing convexity to geodesic convexi…

This https://arxiv.org/abs/2307.01361 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Quadruple Inequalities: Between Cauchy-Schwarz and Triangle
We prove a set of inequalities that interpolate the Cauchy-Schwarz inequality and the triangle inequality. Every nondecreasing, convex function with a concave derivative induces such an inequality. They hold in any metric space that satisfies a metric version of the Cauchy-Schwarz inequality, including all CAT(0) spaces and, in particular, all Euclidean spaces. Because these inequalities establish relations between the six distances of four points, we call them quadruple inequalities. In this c…

This https://arxiv.org/abs/2402.10898 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The Price of Adaptivity in Stochastic Convex Optimization
We prove impossibility results for adaptivity in non-smooth stochastic convex optimization. Given a set of problem parameters we wish to adapt to, we define a "price of adaptivity" (PoA) that, roughly speaking, measures the multiplicative increase in suboptimality due to uncertainty in these parameters. When the initial distance to the optimum is unknown but a gradient norm bound is known, we show that the PoA is at least logarithmic for expected suboptimality, and double-logarithmic for median…

This https://arxiv.org/abs/2306.13174 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Finite element approximation of time-dependent mean field games with nondifferentiable Hamiltonians
The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonian. However in many cases, the structure of the underlying optimal problem leads to a convex but nondifferentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusion (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniquen…

A note on lenses in arrangements of pairwise intersecting circles in the plane
Rom Pinchasi
https://arxiv.org/abs/2403.05270 https://…

A note on lenses in arrangements of pairwise intersecting circles in the plane
Let $\F$ be a family of $n$ pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by $\F$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in $\F$ touch, then the geometric graph $G$ on the set of centers of the circles in $\F$ whose edges correspond to the lenses generated by $\F$ does not contain pairs of avoiding edges. That is, $G$ does not contain pairs of edges that are opposite edges in a con…

This https://arxiv.org/abs/2311.11489 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Universal Trust-Region Method for Convex and Nonconvex Optimization
This paper presents a universal trust-region method simultaneously incorporating quadratic regularization and the ball constraint. We introduce a novel mechanism to set the parameters in the proposed method that unifies the analysis for convex and nonconvex optimization. Our method exhibits an iteration complexity of $\tilde O(ε^{-3/2})$ to find an approximate second-order stationary point for nonconvex optimization. Meanwhile, the analysis reveals that the universal method attains an $O(ε^{-…

This https://arxiv.org/abs/2401.14189 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Form Convex Hull to Concavity: Surface Contraction Around a Point Set
This paper investigates the transformation of a convex hull, derived from a d-dimensional point cloud, into a concave surface. Our primary focus is on the development of a methodology that ensures all points in the point cloud are encapsulated within a closed, non-intersecting concave surface. The study begins with the initial convex hull and employs an iterative process of facet replacement and expansion to evolve the surface into Scc, which accurately conforms to the complex geometry of the p…

The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes
Yingying Cai, Jinping Zhuge
https://arxiv.org/abs/2403.00279 https://

The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes
Let $P$ be a bounded $n$-dimensional Lipschitz polytope, and let $φ_λ$ be a Dirichlet Laplace eigenfunction in $P$ corresponding to the eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the nodal set of $φ_λ$ does not exceed $C(P)\sqrtλ$. Our result extends the previous ones in quaisconvex domains (including $C^1$ and convex domains) to general polytopes that are not necessarily quasiconvex.

Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators
Peijing Liu, Alper Atamt\"urk, Andr\'es G\'omez, Simge K\"u\c{c}\"ukyavuz
https://arxiv.org/abs/2404.04236

Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators
In this paper, we consider convex quadratic optimization problems with indicators on the continuous variables. In particular, we assume that the Hessian of the quadratic term is a Stieltjes matrix, which naturally appears in sparse graphical inference problems and others. We describe an explicit convex formulation for the problem by studying the Stieltjes polyhedron arising as part of an extended formulation and exploiting the supermodularity of a set function defined on its extreme points. Our…

This https://arxiv.org/abs/2311.14857 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Optimality conditions for bilevel programs via Moreau envelope reformulation
For bilevel programs with a convex lower level program, the classical approach replaces the lower level program with its Karush-Kuhn-Tucker condition and solve the resulting mathematical program with complementarity constraint (MPCC). It is known that when the set of lower level multipliers is not unique, MPCC may not be equivalent to the original bilevel problem, and many MPCC-tailored constraint qualifications do not hold. In this paper, we study bilevel programs where the lower level is gene…

This https://arxiv.org/abs/2311.02557 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging
Consider the problem of minimizing an expected logarithmic loss over either the probability simplex or the set of quantum density matrices. This problem includes tasks such as solving the Poisson inverse problem, computing the maximum-likelihood estimate for quantum state tomography, and approximating positive semi-definite matrix permanents with the currently tightest approximation ratio. Although the optimization problem is convex, standard iteration complexity guarantees for first-order meth…

On the controllability of nonlinear systems with a periodic drift
Jean-Baptiste CaillauUniversit\'e C\^ote d'Azur, Inria, CNRS, Lamberto Dell'ElceUniversit\'e C\^ote d'Azur, Inria, CNRS, Alesia HerasimenkaUniversit\'e C\^ote d'Azur, Inria, CNRS, Jean-Baptiste PometUniversit\'e C\^ote d'Azur, Inria, CNRS
https://

On the controllability of nonlinear systems with a periodic drift
Sufficient and necessary conditions are established for controllability of affine control systems, motivated by these with a drift all of whose solutions are periodic. In contrast with previously known results, these conditions encompass the case of a control set whose convex hull contains the origin but is not a neighborhood of the origin. The condition is expressed by means of pushforwards along the flow of the drift, rather than in terms of Lie brackets. It turns out that this also amounts t…

The Nearest Graph Laplacian in Frobenius Norm
Kazuhiro Sato, Masato Suzuki
https://arxiv.org/abs/2404.03371 https://arxiv.org/pdf/240…

The Nearest Graph Laplacian in Frobenius Norm
We address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush-Kuhn-Tucker (KKT) points. The proposed algorithms can be applied to system identification and model reducti…