380 research outputs found
Complexity Certification of the Fast Alternating Minimization Algorithm for Linear MPC
In this technical note, the fast alternating minimization algorithm (FAMA) is proposed to solve model predictive control (MPC) problems with polytopic and second-order cone constraints. Two splitting strategies with efficient implementations for MPC problems are presented. We derive computational complexity certificates for both splitting strategies, by providing complexity upper-bounds on the number of iterations required to provide a certain accuracy of the dual function value and, most importantly, of the primal solution. This is of particular relevance in the context of real-time MPC in order to bound the required on-line computation time. We further address the computation of the complexity bounds, requiring the solution of a non-convex minimization problem. Finally, we demonstrate the performance of FAMA compared to other splitting methods using a quadrotor example
Newton-type Alternating Minimization Algorithm for Convex Optimization
We propose NAMA (Newton-type Alternating Minimization Algorithm) for solving
structured nonsmooth convex optimization problems where the sum of two
functions is to be minimized, one being strongly convex and the other composed
with a linear mapping. The proposed algorithm is a line-search method over a
continuous, real-valued, exact penalty function for the corresponding dual
problem, which is computed by evaluating the augmented Lagrangian at the primal
points obtained by alternating minimizations. As a consequence, NAMA relies on
exactly the same computations as the classical alternating minimization
algorithm (AMA), also known as the dual proximal gradient method. Under
standard assumptions the proposed algorithm possesses strong convergence
properties, while under mild additional assumptions the asymptotic convergence
is superlinear, provided that the search directions are chosen according to
quasi-Newton formulas. Due to its simplicity, the proposed method is well
suited for embedded applications and large-scale problems. Experiments show
that using limited-memory directions in NAMA greatly improves the convergence
speed over AMA and its accelerated variant
Complexity Certification of the Fast Alternating Minimization Algorithm for Linear Model Predictive Control
In this paper, the fast alternating minimization algorithm (FAMA) is proposed to solve model predictive control (MPC) problems with polytopic and second-order cone constraints. We extend previous theoretical results of FAMA to a more general case, where convex constraints are allowed to be imposed on the strongly convex objective and all convergence properties of FAMA are still preserved. Two splitting strategies for MPC problems are presented. Both of them satisfy the assumptions of FAMA and result in efficient implementations by reducing each iteration of FAMA to simple operations. We derive computational complexity certificates for both splitting strategies, by providing bounds on the number of iterations for both primal and dual variables, which are of particular relevance in the context of real-time MPC to bound the required online computation time. For MPC problems with polyhedral and ellipsoidal constraints, an off-line preconditioning method is presented to further improve the convergence speed of FAMA by reducing the complexity bound and enlarging the step-size of the algorithm. Finally, we demonstrate the performance of FAMA compared to other splitting methods using a quadrotor example
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
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