Related papers: Order and disorder in energy minimization
We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity.…
Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing…
In this paper, we derive some new invariant solutions of dark energy models in cylindrically symmetric space-time. To quantify the deviation of pressure from isotropy, we introduce three different time dependent skewness parameters along…
Many systems exhibit a mixture of continuous and discrete dynamics. We consider a family of mixed-integer non-convex non-linear optimisation problems obtained in discretisations of optimal control of such systems. For this family, a…
Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a…
This paper presents a geometric approach to the classical isoperimetric problem by analysing the efficiency of regular polygons in enclosing maximum area for a fixed perimeter. Using efficiency metrics, it proves that regular polygons…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
Effectively rearranging heterogeneous objects constitutes a high-utility skill that an intelligent robot should master. Whereas significant work has been devoted to the grasp synthesis of heterogeneous objects, little attention has been…
We consider an ultra-weak first order system discretization of the Helmholtz equation. When employing the optimal test norm, the `ideal' method yields the best approximation to the pair of the Helmholtz solution and its scaled gradient…
We develop an approach for solving rooted orienteering problems with category constraints as found in tourist trip planning and logistics. It is based on expanding partial solutions in a systematic way, prioritizing promising ones, which…
Optimal transport has been used extensively in resource matching to promote the efficiency of resources usages by matching sources to targets. However, it requires a significant amount of computations and storage spaces for large-scale…
We apply the Dijkstra algorithm to generate optimal paths between two given sites on a lattice representing a disordered energy landscape. We study the geometrical and energetic scaling properties of the optimal path where the energies are…
This paper addresses an optimal control problem for a robot that has to find and collect a finite number of objects and move them to a depot in minimum time. The robot has fourth-order dynamics that change instantaneously at any pick-up or…
We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at…
We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the…
This chapter describes techniques for the numerical resolution of optimal transport problems. We will consider several discretizations of these problems, and we will put a strong focus on the mathematical analysis of the algorithms to solve…
In this short note we review results on equilibrium shapes of minimizers to the sessile drop problem. More precisely, we study the Winterbottom problem and prove that the Winterbottom shape is indeed optimal. The arguments presented here…
We study the ordered equilibrium structures of patchy particles where the patches are located on the surface of the colloid such that they form a regular tetrahedron. Using optimization techniques based on ideas of evolutionary algorithms…
We investigate approximation algorithms for several fundamental optimization problems on geometric packing. The geometric objects considered are very generic, namely $d$-dimensional convex fat objects. Our main contribution is a versatile…
Inspired by, and using methods of optimization derived from classical three dimensional electrostatics, we note a novel beautiful symmetric four dimensional polytope we have found with 80 vertices. We also describe how the method used to…