A parallel Bernstein algorithm for global optimization based on the implicit Bernstein form

Dhabe, Priyadarshan; Nataraj, P. S. V.

doi:10.1007/s13198-017-0639-z

Cited by 5 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then In other words (by taking the contrapositive), the distance between x * and any subbox contained in B R (x * ) cannot exceed R , meaning min y∈y y − x * ≤ R ∀ y ∈ L and y ⊂ B R (x * ). Using (38), we can choose a large enough n such that R < R. Therefore, all subboxes contained in B R (x * ) must also be contained in a hypercube centered at x * with widths 2R +2•2 −n . Since the maximum width of all subboxes is 2 −n (Remark 8), the number of subboxes contained in B R (x * ) is bounded from above by…”

Section: B Theorem 16 (Unconstrained Memory Usage)mentioning

confidence: 99%

“…Note, one can use the Bernstein basis to transform a POP into a linear program (LP) on each subdivided portion of the problem domain, which allows one to find tighter solution bounds that given by the Bernstein coefficients alone [35]. Since subdivision can be parallelized [38], the time required to solve a POP can be greatly reduced by implementing BA on a Graphics Processing Unit (GPU). However, a parallelized implementation or bounds on the rate of convergence of BA with constraints has not yet been shown in the literature.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Safe, Optimal, Real-time Trajectory Planning with a Parallel Constrained Bernstein Algorithm

Kousik¹,

Zhao²,

Vasudevan³

2020

Preprint

View full text Add to dashboard Cite

To move through the world, mobile robots typically use a receding-horizon strategy, wherein they execute an old plan while computing a new plan to incorporate new sensor information. A plan should be dynamically feasible, meaning it obeys constraints like the robot's dynamics and obstacle avoidance; it should have liveness, meaning the robot does not stop to plan so frequently that it cannot accomplish tasks; and it should be optimal, meaning that the robot tries to satisfy a user-specified cost function such as reaching a goal location as quickly as possible. Reachability-based Trajectory Design (RTD) is a planning method that can generate provably dynamically-feasible plans. However, RTD solves a nonlinear polynmial optimization program at each planning iteration, preventing optimality guarantees; furthermore, RTD can struggle with liveness because the robot must brake to a stop when the solver finds local minima or cannot find a feasible solution. This paper proposes RTD*, which certifiably finds the globally optimal plan (if such a plan exists) at each planning iteration. This method is enabled by a novel Parallelized Constrained Bernstein Algorithm (PCBA), which is a branch-andbound method for polynomial optimization. The contributions of this paper are: the implementation of PCBA; proofs of bounds on the time and memory usage of PCBA; a comparison of PCBA to state of the art solvers; and the demonstration of PCBA/RTD* on a mobile robot. RTD* outperforms RTD in terms of optimality and liveness for real-time planning in a variety of environments with randomly-placed obstacles.

show abstract

Section: B Theorem 16 (Unconstrained Memory Usage)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Safe, Optimal, Real-time Trajectory Planning with a Parallel Constrained Bernstein Algorithm

Kousik¹,

Zhao²,

Vasudevan³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…. , κ, see (3). In Table 1 the number of the operations of this method and Ray and Nataraj s method (as reported in [23]) is presented.…”

Section: A) Unit Boxmentioning

confidence: 99%

“…Before we detail the topics of our paper, we mention three points: The fast determination of the Bernstein enclosure for polynomials over a simplex and for multivariate rational functions based in part on the three tests in [25] is presented in [28]. The parallel computation of the Bernstein coefficients is first considered in [6] and in [3] combined with the implicit Bernstein form [25]. Extensions of the Bernstein enclosure for multivariate complex polynomials and rational functions may be found in [30].…”

Section: Introductionmentioning

confidence: 99%

Matrix methods for the tensorial Bernstein form

Titi

Garloff

2019

Applied Mathematics and Computation

View full text Add to dashboard Cite

In this paper, multivariate polynomials in the Bernstein basis over a box (tensorial Bernstein representation) are considered. A new matrix method for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented and compared with existing methods. Also matrix methods for the calculation of the Bernstein coefficients over subboxes generated by subdivision of the original box are proposed. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. In the case that the coefficients of the polynomial are due to uncertainties and can be represented in the form of intervals it is shown that the developed methods can be extended to compute the set of the Bernstein coefficients of all members of the polynomial family.

show abstract

The Bernstein algorithm using the modified implicit Bernstein form and its GPU parallelization using CUDA

Dhabe

Nataraj

2017

Int J Syst Assur Eng Manag

View full text Add to dashboard Cite

A parallel Bernstein algorithm for global optimization based on the implicit Bernstein form

Cited by 5 publications

References 13 publications

Safe, Optimal, Real-time Trajectory Planning with a Parallel Constrained Bernstein Algorithm

Safe, Optimal, Real-time Trajectory Planning with a Parallel Constrained Bernstein Algorithm

Matrix methods for the tensorial Bernstein form

The Bernstein algorithm using the modified implicit Bernstein form and its GPU parallelization using CUDA

Contact Info

Product

Resources

About