Fast Second-order Cone Programming for Safe Mission Planning

This paper considers the problem of safe mission planning of dynamic systems operating under uncertain environments. Much of the prior work on achieving robust and safe control requires solving second-order cone programs (SOCP). Unfortunately, existing general purpose SOCP methods are often infeasible for real-time robotic tasks due to high memory and computational requirements imposed by existing general optimization methods. The key contribution of this paper is a fast and memory-efficient algorithm for SOCP that would enable robust and safe mission planning on-board robots in real-time. Our algorithm does not have any external dependency, can efficiently utilize warm start provided in safe planning settings, and in fact leads to significant speed up over standard optimization packages (like SDPT3) for even standard SOCP problems. For example, for a standard quadrotor problem, our method leads to speedup of 1000x over SDPT3 without any deterioration in the solution quality. Our method is based on two insights: a) SOCPs can be interpreted as optimizing a function over a polytope with infinite sides, b) a linear function can be efficiently optimized over this polytope. We combine the above observations with a novel utilization of Wolfe's algorithm to obtain an efficient optimization method that can be easily implemented on small embedded devices. In addition to the above mentioned algorithm, we also design a two-level sensing method based on Gaussian Process for complex obstacles with non-linear boundaries such as a cylinder

SECOND ORDER CONE (SOC) DAN SIFAT-SIFAT KENDALA SECOND ORDER CONE PROGRAMMING DENGAN NORMA 1

Pada makalah ini dikembangkan pengertian Second Order Cone (SOC) dan sifat-sifat kendala Second Order Cone Programming dengan Norma 1. Kata kunci: Second Order Cone (SOC), Second Order Cone Programming

Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming

Elfving's Theorem is a major result in the theory of optimal experimental design, which gives a geometrical characterization of $c-$optimality. In this paper, we extend this theorem to the case of multiresponse experiments, and we show that when the number of experiments is finite, $c-,A-,T-$ and $D-$optimal design of multiresponse experiments can be computed by Second-Order Cone Programming (SOCP). Moreover, our SOCP approach can deal with design problems in which the variable is subject to several linear constraints. We give two proofs of this generalization of Elfving's theorem. One is based on Lagrangian dualization techniques and relies on the fact that the semidefinite programming (SDP) formulation of the multiresponse $c-$optimal design always has a solution which is a matrix of rank $1$. Therefore, the complexity of this problem fades. We also investigate a \emph{model robust} generalization of $c-$optimality, for which an Elfving-type theorem was established by Dette (1993). We show with the same Lagrangian approach that these model robust designs can be computed efficiently by minimizing a geometric mean under some norm constraints. Moreover, we show that the optimality conditions of this geometric programming problem yield an extension of Dette's theorem to the case of multiresponse experiments. When the number of unknown parameters is small, or when the number of linear functions of the parameters to be estimated is small, we show by numerical examples that our approach can be between 10 and 1000 times faster than the classic, state-of-the-art algorithms