Multisegment Scheme Applications to Modified Chebyshev Picard Iteration Method for Highly Elliptical Orbits

A modified Chebyshev Picard iteration method is proposed for solving orbit propagation initial/boundary value problems. Cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes, are used to reduce Runge’s phenomenon that plagues many series approximations. The key benefit of using the CGL data sampling is that the nodal points are distributed nonuniformly, with dense sampling at the beginning and ending times. This problem can be addressed by a nonlinear time transformation and/or by utilizing multiple time segments over an orbit. This paper suggests a method, called a multisegment method, to obtain accurate solutions overall regardless of initial states and albeit eccentricity by dividing the given orbit into two or more segments based on the true anomaly

Sampling-Based Methods for Factored Task and Motion Planning

This paper presents a general-purpose formulation of a large class of discrete-time planning problems, with hybrid state and control-spaces, as factored transition systems. Factoring allows state transitions to be described as the intersection of several constraints each affecting a subset of the state and control variables. Robotic manipulation problems with many movable objects involve constraints that only affect several variables at a time and therefore exhibit large amounts of factoring. We develop a theoretical framework for solving factored transition systems with sampling-based algorithms. The framework characterizes conditions on the submanifold in which solutions lie, leading to a characterization of robust feasibility that incorporates dimensionality-reducing constraints. It then connects those conditions to corresponding conditional samplers that can be composed to produce values on this submanifold. We present two domain-independent, probabilistically complete planning algorithms that take, as input, a set of conditional samplers. We demonstrate the empirical efficiency of these algorithms on a set of challenging task and motion planning problems involving picking, placing, and pushing

An adaptive sampling sequential quadratic programming method for nonsmooth stochastic optimization with upper-C2\mathcal{C}^2 objective

We propose an optimization algorithm that incorporates adaptive sampling for stochastic nonsmooth nonconvex optimization problems with upper-$\mathcal{C}^2$ objective functions. Upper-$\mathcal{C}^2$ is a weakly concave property that exists naturally in many applications, particularly certain classes of solutions to parametric optimization problems, e.g., recourse of stochastic programming and projection into closed sets. Our algorithm is a stochastic sequential quadratic programming (SQP) method extended to nonsmooth problems with upper$\mathcal{C}^2$ objectives and is globally convergent in expectation with bounded algorithmic parameters. The capabilities of our algorithm are demonstrated by solving a joint production, pricing and shipment problem, as well as a realistic optimal power flow problem as used in current power grid industry practice.Comment: arXiv admin note: text overlap with arXiv:2204.0963

Control Theoretic Approach To Sampling And Approximation Problems

Thesis (Ph.D.) University of Alaska Fairbanks, 2009We present applications of some methods of control theory to problems of signal processing and optimal quadrature problems. The following problems are considered: construction of sampling and interpolating sequences for multi-band signals; spectral estimation of signals modeled by a finite sum of exponentials modulated by polynomials; construction of optimal quadrature formulae for integrands determined by solutions of initial boundary value problems. A multi-band signal is a function whose Fourier transform is supported on a finite union of intervals. The approach used in Chapter I is based on connections between the sampling and interpolation problem and the problem of the controllability of a dynamical system. We prove that there exist infinitely many sampling and interpolating sequences for signals whose spectra are supported on a union of two disjoint intervals, and provide an algorithm for construction of such sequences. There exist numerous methods for solving the spectral estimation problem. In Chapter II we introduce a new approach to this problem based on the Boundary Control method, which uses the connection between inverse problems of mathematical physics and control theory for partial differential equations. Using samples of the signal at integer moments of time we construct a convolution operator regarded as an input-output map of a linear discrete dynamical system. This system can be identified, and the exponents and amplitudes of the signal can be found from the parameters of the system. We show that the coefficients of the signal can be recovered by solving a generalized eigenvalue problem as in the Matrix Pencil method. Our method allows to consider signals with polynomial amplitudes, and we obtain an exact formula for these amplitudes. In the third chapter we consider an optimal quadrature problem for solutions of initial boundary value problems. The problem of optimization of an error functional over the set of solutions and quadrature weights is a problem of optimal control of partial differential equations. We obtain estimates for the error in quadrature formulae and an optimality condition for quadrature weights

Informed anytime fast marching tree for asymptotically-optimal motion planning

In many applications, it is necessary for motion planning planners to get high-quality solutions in high-dimensional complex problems. In this paper, we propose an anytime asymptotically-optimal sampling-based algorithm, namely Informed Anytime Fast Marching Tree (IAFMT*), designed for solving motion planning problems. Employing a hybrid incremental search and a dynamic optimal search, the IAFMT* fast finds a feasible solution, if time permits, it can efficiently improve the solution toward the optimal solution. This paper also presents the theoretical analysis of probabilistic completeness, asymptotic optimality, and computational complexity on the proposed algorithm. Its ability to converge to a high-quality solution with the efficiency, stability, and self-adaptability has been tested by challenging simulations and a humanoid mobile robot

Optimal staffing under an annualized hours regime using Cross-Entropy optimization

This paper discusses staffing under annualized hours. Staffing is the selection of the most cost-efficient workforce to cover workforce demand. Annualized hours measure working time per year instead of per week, relaxing the restriction for employees to work the same number of hours every week. To solve the underlying combinatorial optimization problem this paper develops a Cross-Entropy optimization implementation that includes a penalty function and a repair function to guarantee feasible solutions. Our experimental results show Cross-Entropy optimization is efficient across a broad range of instances, where real-life sized instances are solved in seconds, which significantly outperforms an MILP formulation solved with CPLEX. In addition, the solution quality of Cross-Entropy closely approaches the optimal solutions obtained by CPLEX. Our Cross-Entropy implementation offers an outstanding method for real-time decision making, for example in response to unexpected staff illnesses, and scenario analysis

A Collection of Challenging Optimization Problems in Science, Engineering and Economics

Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE Congress on Evolutionary Computation (at Sendai International Center, Sendai, Japan, from 25th to 28th May, 2015.