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.
There are analogues to our approach in the field of machine learning. OptNet @cite_20 incorporates a QP solver as a layer into the neural network and is able to provide analytical gradients of the solution to the QP w.r.t. input parameters for back propagation. Gould @cite_2 presents results on differentiating argmin optimization problems w.r.t. optimization variables in the context of bilevel optimization. Their results give exact gradients for the . However, works described in @cite_20 , @cite_2 do not consider finding the gradient of the objective function. Moreover, methods proposed by Gould @cite_2 involves inverting a Hessian matrix which could be computationally expensive and often not even invertible, and they only describe the case of equality constraints and use a log-barrier function to tackle inequality constraints. The gradient of the objective function is even cheaper to compute, making our approach very suitable for speeding up trajectory optimization.
@cite_20 @cite_2
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We propose a partially learned approach for the solution of ill-posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularisation theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularising functional. The method results in a gradient-like iterative scheme, where the 'gradient' component is learned using a convolutional network that includes the gradients of the data discrepancy and regulariser as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of pixel images in about 0.4 s using a single graphics processing unit (GPU). This paper discusses a new method to perform propagation over a (two-layer, feed-forward) Neural Network embedded in a Constraint Programming model. The method is meant to be employed in Empirical Model Learning, a technique designed to enable optimal decision making over systems that cannot be modeled via conventional declarative means. The key step in Empirical Model Learning is to embed a Machine Learning model into a combinatorial model. It has been showed that Neural Networks can be embedded in a Constraint Programming model by simply encoding each neuron as a global constraint, which is then propagated individually. Unfortunately, this decomposition approach may lead to weak bounds. To overcome such limitation, we propose a new network-level propagator based on a non-linear Lagrangian relaxation that is solved with a subgradient algorithm. The method proved capable of dramatically reducing the search tree size on a thermal-aware dispatching problem on multicore CPUs. The overhead for optimizing the Lagrangian multipliers is kept within a reasonable level via a few simple techniques. This paper is an extended version of [27], featuring an improved structure, a new filtering technique for the network inputs, a set of overhead reduction techniques, and a thorough experimentation. Maximization of mutual information of voxel intensities has been demonstrated to be a very powerful criterion for three-dimensional medical image registration, allowing robust and accurate fully automated affine registration of multimodal images in a variety of applications, without the need for segmentation or other preprocessing of the images. In this paper, we investigate the performance of various optimization methods and multiresolution strategies for maximization of mutual information, aiming at increasing registration speed when matching large high-resolution images. We show that mutual information is a continuous function of the affine registration parameters when appropriate interpolation is used and we derive analytic expressions of its derivatives that allow numerically exact evaluation of its gradient. Various multiresolution gradient- and non-gradient-based optimization strategies, such as Powell, simplex, steepest-descent, conjugate-gradient, quasi-Newton and Levenberg—Marquardt methods, are evaluated for registration of computed tomography (CT) and magnetic resonance images of the brain. Speed-ups of a factor of 3 on average compared to Powell's method at full resolution are achieved with similar precision and without a loss of robustness with the simplex, conjugate-gradient and Levenberg—Marquardt method using a two-level multiresolution scheme. Large data sets such as 2562 × 128 MR and 5122 × 48 CT images can be registered with subvoxel precision in < 5 min CPU time on current workstations. We propose a very simple modification of gradient descent and stochastic gradient descent. We show that when applied to a variety of machine learning models including softmax regression, convolutional neural nets, generative adversarial nets, and deep reinforcement learning, this very simple surrogate can dramatically reduce the variance and improve the accuracy of the generalization. The new algorithm, (which depends on one nonnegative parameter) when applied to non-convex minimization, tends to avoid sharp local minima. Instead it seeks somewhat flatter local (and often global) minima. The method only involves preconditioning the gradient by the inverse of a tri-diagonal matrix that is positive definite. The motivation comes from the theory of Hamilton-Jacobi partial differential equations. This theory demonstrates that the new algorithm is almost the same as doing gradient descent on a new function which (a) has the same global minima as the original function and (b) is "more convex". Again, the programming effort in doing this is minimal, in cost, complexity and effort. We implement our algorithm into both PyTorch and Tensorflow platforms, which will be made publicly available.