Nonlinear programming

About: Nonlinear programming is a research topic. Over the lifetime, 19486 publications have been published within this topic receiving 656602 citations. The topic is also known as: non-linear programming & NLP.

Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms

Abstract: This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinate-wise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparse-simplex methods. The first algorithm is essentially a gradient projection method while the remaining two algorithms are of coordinate descent type. The theoretical convergence of these methods and their relations to the derived optimality conditions are studied. The algorithms and results are illustrated by several numerical examples.

Nonlinear Constraint Network Optimization for Efficient Map Learning

Abstract: Learning models of the environment is one of the fundamental tasks of mobile robots since maps are needed for a wide range of robotic applications, such as navigation and transportation tasks, service robotic applications, and several others. In the past, numerous efficient approaches to map learning have been proposed. Most of them, however, assume that the robot lives on a plane. In this paper, we present a highly efficient maximum-likelihood approach that is able to solve 3-D and 2-D problems. Our approach addresses the so-called graph-based formulation of simultaneous localization and mapping (SLAM) and can be seen as an extension of Olson's algorithm toward non-flat environments. It applies a novel parameterization of the nodes of the graph that significantly improves the performance of the algorithm and can cope with arbitrary network topologies. The latter allows us to bound the complexity of the algorithm to the size of the mapped area and not to the length of the trajectory. Furthermore, our approach is able to appropriately distribute the roll, pitch, and yaw error over a sequence of poses in 3-D mapping problems. We implemented our technique and compared it with multiple other graph-based SLAM solutions. As we demonstrate in simulated and real-world experiments, our method converges faster than the other approaches and yields accurate maps of the environment.

Parallel MR Image Reconstruction Using Augmented Lagrangian Methods

Abstract: Magnetic resonance image (MRI) reconstruction using SENSitivity Encoding (SENSE) requires regularization to suppress noise and aliasing effects. Edge-preserving and sparsity-based regularization criteria can improve image quality, but they demand computation-intensive nonlinear optimization. In this paper, we present novel methods for regularized MRI reconstruction from undersampled sensitivity encoded data-SENSE-reconstruction-using the augmented Lagrangian (AL) framework for solving large-scale constrained optimization problems. We first formulate regularized SENSE-reconstruction as an unconstrained optimization task and then convert it to a set of (equivalent) constrained problems using variable splitting. We then attack these constrained versions in an AL framework using an alternating minimization method, leading to algorithms that can be implemented easily. The proposed methods are applicable to a general class of regularizers that includes popular edge-preserving (e.g., total-variation) and sparsity-promoting (e.g., -norm of wavelet coefficients) criteria and combinations thereof. Numerical experiments with synthetic and in vivo human data illustrate that the proposed AL algorithms converge faster than both general-purpose optimization algorithms such as nonlinear conjugate gradient (NCG) and state-of-the-art MFISTA.

Simultaneous robot-world and hand-eye calibration

Abstract: Zhuang et al. (1994) proposed a method that allows simultaneous computation of the rigid transformations from world frame to robot base frame and from hand frame to camera frame. Their method attempts to solve a homogeneous matrix equation of the for, AX=ZB. They use quaternions to derive explicit linear solution for X and Z. In this paper, we present two new solutions that attempt to solve the homogeneous matrix equation mentioned above: 1) a closed-form method which uses quaternion algebra and a positive quadratic error function associated with this representation; 2) a method based on nonlinear constrained minimization and which simultaneously solves for rotations and translations. These results may be useful to other problems that can be formulated in the same mathematical form. We perform a sensitivity analysis for both our two methods and the linear method developed by Zhuang et al. This analysis allows the comparison of the three methods. In the light of this comparison, the nonlinear optimization method, which solves for rotations and translations simultaneously, seems to be the most stable one with respect to noise and to measurement errors.

Mixed integer models for the stationary case of gas network optimization

Abstract: A gas network basically consists of a set of compressors and valves that are connected by pipes. The problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost-efficiently such that all demands of the gas network are satisfied. This problem leads to a complex mixed integer nonlinear optimization problem. We describe techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program. We study sub-polyhedra linking these piece-wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms. Suitable branching strategies complementing the separation algorithms are also presented. Our computational results demonstrate the success of this approach.