An Optimal Control Theory for the Traveling Salesman Problem and Its Variants

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.Comment: 24 pages, 8 figure

Electricity from photovoltaic solar cells: Flat-Plate Solar Array Project final report. Volume VI: Engineering sciences and reliability

The Flat-Plate Solar Array (FSA) Project, funded by the U.S. Government and managed by the Jet Propulsion Laboratory, was formed in 1975 to develop the module/array technology needed to attain widespread terrestrial use of photovoltaics by 1985. To accomplish this, the FSA Project established and managed an Industry, University, and Federal Government Team to perform the needed research and development. This volume of the series of final reports documenting the FSA Project deals with the Project's activities directed at developing the engineering technology base required to achieve modules that meet the functional, safety and reliability requirements of large-scale terrestrial photovoltaic systems applications. These activities included: (1) development of functional, safety, and reliability requirements for such applications; (2) development of the engineering analytical approaches, test techniques, and design solutions required to meet the requirements; (3) synthesis and procurement of candidate designs for test and evaluation; and (4) performance of extensive testing, evaluation, and failure analysis to define design shortfalls and, thus, areas requiring additional research and development. During the life of the FSA Project, these activities were known by and included a variety of evolving organizational titles: Design and Test, Large-Scale Procurements, Engineering, Engineering Sciences, Operations, Module Performance and Failure Analysis, and at the end of the Project, Reliability and Engineering Sciences. This volume provides both a summary of the approach and technical outcome of these activities and provides a complete Bibliography (Appendix A) of the published documentation covering the detailed accomplishments and technologies developed

Flat-plate solar array project. Volume 6: Engineering sciences and reliability

The Flat-Plate Solar Array (FSA) Project activities directed at developing the engineering technology base required to achieve modules that meet the functional, safety, and reliability requirements of large scale terrestrial photovoltaic systems applications are reported. These activities included: (1) development of functional, safety, and reliability requirements for such applications; (2) development of the engineering analytical approaches, test techniques, and design solutions required to meet the requirements; (3) synthesis and procurement of candidate designs for test and evaluation; and (4) performance of extensive testing, evaluation, and failure analysis of define design shortfalls and, thus, areas requiring additional research and development. A summary of the approach and technical outcome of these activities are provided along with a complete bibliography of the published documentation covering the detailed accomplishments and technologies developed

Instability of fixed, low-thrust drag compensation

FORCED drag compensation using continuous low-thrustpropulsion has been considered for satellites in low Earth orbit. This simple, but nonoptimal, scheme merely requires that the thrust vector is directed opposite to the drag vector and that the magnitude of the two are equal. In principle, the drag force acting on the spacecraft could be determined onboard using accurate accelerometers. However, for small, low-cost spacecraft such sensors may beunavailable. An alternative strategy would be to Ž x the thrust magnitude equal to the expected air drag that would be experienced by the spacecraft. The thrust levelwould be periodically updated based on ground-based orbit determination. In this Engineering Note, it is shown that such a forced circular orbit with a Ž fixed thrust levelis exponentially unstable for all physically reasonable atmosphere models

A Universal Birkhoff Theory for Fast Trajectory Optimization

Over the last two decades, pseudospectral methods based on Lagrange interpolants have flourished in solving trajectory optimization problems and their flight implementations. In a seemingly unjustified departure from these highly successful methods, a new starting point for trajectory optimization is proposed. This starting point is based on the recently-developed concept of universal Birkhoff interpolants. The new approach offers a substantial computational upgrade to the Lagrange theory in completely flattening the rapid growth of the condition numbers from O(N2) to O(1), where N is the number of grid points. In addition, the Birkhoff-specific primal-dual computations are isolated to a well-conditioned linear system even for nonlinear, nonconvex problems. This is part I of a two-part paper. In part I, a new theory is developed on the basis of two hypotheses. Other than these hypotheses, the theoretical development makes no assumptions on the choices of basis functions or the selection of grid points. Several covector mapping theorems are proved to establish the mathematical equivalence between direct and indirect Birkhoff methods. In part II of this paper (with Proulx), it is shown that a select family of Gegenbauer grids satisfy the two hypotheses required for the theory to hold. Numerical examples in part II illustrate the power and utility of the new theory

Derivation of Coordinate Descent Algorithms from Optimal Control Theory

Recently, it was posited that disparate optimization algorithms may be coalesced in terms of a central source emanating from optimal control theory. Here we further this proposition by showing how coordinate descent algorithms may be derived from this emerging new principle. In particular, we show that basic coordinate descent algorithms can be derived using a maximum principle and a collection of max functions as "control" Lyapunov functions. The convergence of the resulting coordinate descent algorithms is thus connected to the controlled dissipation of their corresponding Lyapunov functions. The operational metric for the search vector in all cases is given by the Hessian of the convex objective function

Fast Mesh Refinement in Pseudospectral Optimal Control

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy --- simply increase the order $N$ of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as $N$ increases, the condition number of the resulting linear algebra increases as $N^2$; hence, spectral efficiency and accuracy are lost in practice. In this paper, we advance Birkhoff interpolation concepts over an arbitrary grid to generate well-conditioned PS optimal control discretizations. We show that the condition number increases only as $\sqrt{N}$ in general, but is independent of $N$ for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as $N$ increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using \underline{polynomials of over a thousandth order} to solve a low-thrust, long-duration orbit transfer problem.Comment: 27 pages, 12 figures, JGCD April 201

Charge-Focusing Readout of Time Projection Chambers

Time projection chambers (TPCs) have found a wide range of applications in particle physics, nuclear physics, and homeland security. For TPCs with high-resolution readout, the readout electronics often dominate the price of the final detector. We have developed a novel method which could be used to build large-scale detectors while limiting the necessary readout area. By focusing the drift charge with static electric fields, we would allow a small area of electronics to be sensitive to particle detection for a much larger detector volume. The resulting cost reduction could be important in areas of research which demand large-scale detectors, including dark matter searches and detection of special nuclear material. We present simulations made using the software package Garfield of a focusing structure to be used with a prototype TPC with pixel readout. This design should enable significant focusing while retaining directional sensitivity to incoming particles. We also present first experimental results and compare them with simulation.Comment: 5 pages, 17 figures, Presented at IEEE Nuclear Science Symposium 201