Practical methods for approximating shortest paths on a convex polytope in R3

AbstractWe propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p, q on its surface, let dP(p, q) denote the shortest path distance between p and q on the surface of P. Our algorithm produces a path of length at most 2dP(p, q) in time O(n). Extending this result, we can also compute an approximation of the shortest path tree rooted at an arbitrary point x ∈ P in time O(n log n). In the approximate tree, the distance between a vertex v ∈ P and x is at most cdP(x, v), where c = 2.38(1 + ε) for any fixed ε > 0. The best algorithms for computing an exact shortest path on a convex polytope take Ω(n2) time in the worst case; in addition, they are too complicated to be suitable in practice. We can also get a weak approximation result in the general case of k disjoint convex polyhedra: in O(n) time our algorithm gives a path of length at most 2k times the optimal

Nonlinear Set Membership Filter with State Estimation Constraints via Consensus-ADMM

This paper considers the state estimation problem for nonlinear dynamic systems with unknown but bounded noises. Set membership filter (SMF) is a popular algorithm to solve this problem. In the set membership setting, we investigate the filter problem where the state estimation requires to be constrained by a linear or nonlinear equality. We propose a consensus alternating direction method of multipliers (ADMM) based SMF algorithm for nonlinear dynamic systems. To deal with the difficulty of nonlinearity, instead of linearizing the nonlinear system, a semi-infinite programming (SIP) approach is used to transform the nonlinear system into a linear one, which allows us to obtain a more accurate estimation ellipsoid. For the solution of the SIP, an ADMM algorithm is proposed to handle the state estimation constraints, and each iteration of the algorithm can be solved efficiently. Finally, the proposed filter is applied to typical numerical examples to demonstrate its effectiveness

Compact union of disjoint boxes: An efficient decomposition model for binary volumes

This paper presents in detail the CompactUnion of Disjoint Boxes (CUDB), a decomposition modelfor binary volumes that has been recently but brieflyintroduced. This model is an improved version of aprevious model called Ordered Union of Disjoint Boxes(OUDB). We show here, several desirable features thatthis model has versus OUDB, such as less unitary basicelements (boxes) and thus, a better efficiency in someneighborhood operations. We present algorithms forconversion to and from other models, and for basiccomputations as area (2D) or volume (3D). We alsopresent an efficient algorithm for connected-componentlabeling (CCL) that does not follow the classical two-passstrategy. Finally we present an algorithm for collision (oradjacency) detection in static environments. We test theefficiency of CUDB versus existing models with severaldatasets.Peer ReviewedPostprint (published version

Visualization And Collision Detection Of Direct Metal Deposition

Direct metal deposition (DMD) is a manufacturing technique that manufactures solid metal parts from bottom to top using powdered metal and a focused laser. In this research, the swept volume technique was used as framework to develop a computer program to perform volumetric visualization of the deposition process as a pre-processor, before the actual metal deposition commences

On contact numbers of totally separable unit sphere packings

Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n>1 and d>1.Comment: 11 page

Learning Solutions of Similar Linear Programming Problems using Boosting Trees

In many optimization problems, similar linear programming (LP) problems occur in the nodes of the branch and bound trees that are used to solve integer (mixed or pure, deterministic or stochastic) programming problems. Similar LP problems are also found in problem domains where the objective function and constraint coefficients vary due to uncertainties in the operating conditions. In this report, we present a regression technique for learning a set of functions that map the objective function and the constraints to the decision variables of such an LP system by modifying boosting trees, an algorithm we term the Boost-LP algorithm. Matrix transformations and geometric properties of boosting trees are utilized to provide theoretical performance guarantees on the predicted values. The standard form of the loss function is altered to reduce the possibility of generating infeasible LP solutions. Experimental results on three different problems, one each on scheduling, routing, and planning respectively, demonstrate the effectiveness of the Boost-LP algorithm in providing significant computational benefits over regular optimization solvers without generating solutions that deviate appreciably from the optimum values