This paper presents an efficient optimization framework that solves trajectory optimization problems efficiently by decoupling state variables from timing variables, thereby decomposing a challenging nonlinear programming (NLP) problem into two easier subproblems. With timing fixed, the state variables can be optimized efficiently using convex optimization, and so the time variables can be optimized using a separate outer optimization. This is a bilevel optimization in which the outer objective function itself requires an optimization to compute. The challenge is that gradient optimization methods require the gradient of the objective function with respect to the time variables, which is not available. Whereas the finite difference method must solve many optimization problems to compute a gradient, this paper proposes a more efficient method: the dual solution (Lagrange multipliers) of the convex optimization problem is exploited to calculate the analytical gradient. Since the dual solution is a by-product of the convex optimization problem, the gradient can be obtained for free' with high accuracy. The framework is demonstrated on solving minimum-jerk trajectory optimization problems in safety corridors for unmanned aerial vehicles (UAVs). Experiments demonstrate that bilevel optimization improves performance over a standard NLP solver, and analytical gradients outperforms finite differences. With a 40 ,ms cutoff time, our approach achieves over 8 times better suboptimality than the current state-of-the-art.
One way to improve a non-optimal time allocation scheme is to refine it iteratively using gradient descent @cite_11 @cite_13 . The refinement process computes the gradient of the objective w.r.t. the time allocation scheme, takes a step after choosing a suitable descent direction and step length. To the best of our knowledge, finite difference is the only way being used for gradient calculation in this scenario. However, estimating gradient with finite difference is time-consuming since the objective function is actually the objective of an optimization problem. For each gradient evaluation, this approach has to solve the same number of optimization problems as the number of segments in the spline, thus making it intractable for real-time performance. Moreover, choosing step sizes for finite difference is hard and the result may not necessarily be a good approximation to the actual gradient.
@cite_13 @cite_11
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We study a general online convex optimization problem. We have a convex set S and an unknown sequence of cost functions c 1 , c 2 ,..., and in each period, we choose a feasible point x t in S, and learn the cost c t (x t ). If the function c t is also revealed after each period then, as Zinkevich shows in [25], gradient descent can be used on these functions to get regret bounds of O(√n). That is, after n rounds, the total cost incurred will be O(√n) more than the cost of the best single feasible decision chosen with the benefit of hindsight, min x Σ ct(x).We extend this to the "bandit" setting, where, in each period, only the cost c t (x t ) is revealed, and bound the expected regret as O(n3 4).Our approach uses a simple approximation of the gradient that is computed from evaluating c t at a single (random) point. We show that this biased estimate is sufficient to approximate gradient descent on the sequence of functions. In other words, it is possible to use gradient descent without seeing anything more than the value of the functions at a single point. The guarantees hold even in the most general case: online against an adaptive adversary.For the online linear optimization problem [15], algorithms with low regrets in the bandit setting have recently been given against oblivious [1] and adaptive adversaries [19]. In contrast to these algorithms, which distinguish between explicit explore and exploit periods, our algorithm can be interpreted as doing a small amount of exploration in each period. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. We propose a family of nonconvex optimization algorithms that are able to save gradient and negative curvature computations to a large extent, and are guaranteed to find an approximate local minimum with improved runtime complexity. At the core of our algorithms is the division of the entire domain of the objective function into small and large gradient regions: our algorithms only perform gradient descent based procedure in the large gradient region, and only perform negative curvature descent in the small gradient region. Our novel analysis shows that the proposed algorithms can escape the small gradient region in only one negative curvature descent step whenever they enter it, and thus they only need to perform at most @math negative curvature direction computations, where @math is the number of times the algorithms enter small gradient regions. For both deterministic and stochastic settings, we show that the proposed algorithms can potentially beat the state-of-the-art local minima finding algorithms. For the finite-sum setting, our algorithm can also outperform the best algorithm in a certain regime. In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.