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

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

Sexual Violence against Girls in Schools: Addressing the Gaps between Policy and Practice in Awaso, Ghana

Despite the established relationship between girls\u27 education and several social development outcomes, gender disparities in education remain particularly concerning. Among the many obstacles that still hinder girls’ access to quality education, sexual violence against girls in schools (SVAGS) is one of the most worrying but also one that has received the least attention in light of recent efforts to increase girls’ attendance in school. This article explores the interface between the seemingly solid Ghanaian legal and policy framework to protect children in educational institutions and the high incidence of SVAGS in such institutions. Its purpose is twofold: to identify the major barriers to fighting SVAGS in Awaso, a rural Ghanaian town, and to highlight strategies for lifting those barriers. Using classroom observation, focus group discussions and interviews with students, teachers, parents, NGO staff and government representatives, it explains how lack of knowledge, lack of financial resources, deep-set values and popular perceptions of masculinity, femininity and violence against women and girls contribute to SVAGS

Winter Habitat Use by Moose, Alces alces, in Central Interior British Columbia

In central British Columbia, recent epidemics of Mountain Pine Beetle (Dendroctonus ponderosae) have resulted in the use of expansive clearcut areas to remove infested mature and old Lodgepole Pine (Pinus contorta) stands. This study aimed to determine if Moose (Alces alces) use late-successional Lodgepole Pine stands in mid- to late-winter. Moose activity and habitat use was determined during February-March track surveys in 2000 (60 km) and 2001 (55.7 km). In 2000 (69 tracks) and 2001 (313 tracks), Moose track distribution differed significantly (P < 0.05) from random. They were significantly more abundant than predicted in young stands (dominated by Picea spp.), or early seral stages; they were less abundant than predicted in mature and old Lodgepole Pine stands. It is unlikely that harvesting late-successional Lodgepole Pine stands would affect Moose winter habitat supply

Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization

This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev family of node points, be it Lobatto, Radau or Gauss, then, the resulting collection of trajectory optimization methods satisfy the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an $\mathcal{O}(1)$ computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin's- and Covector-Mapping Principles, where the latter was developed in Part~I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the full differential-algebraic boundary value problem that results from an application of Pontryagin's Principle. Numerical problems are solved to illustrate all these ideas. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: (1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector trajectories can be well approximated, and (3) extremal solutions over dense grids can be computed in a stable and efficient manner

The expectancy bias: Expectancy-violating faces evoke earlier pupillary dilation than neutral or negative faces

Humans maintain a negativity bias, whereby they perceive threatening stimuli to be more salient than rewarding or neutral stimuli. Across 6 within-subject experimental comparisons, we tested the hypothesis that humans maintain an even stronger expectancy bias, preferentially processing stimuli that violate mental representations of expected associations. To assess this bias, we measured variations in pupillary dilation as a means of determin- ing attentional arousal in response to neutral, negative and expectancy-violating versions of the same social stimuli: human faces. We conducted three baseline manipulation checks that directly compared neutral faces with threatening (angry) and expectancy-violating (upside-down and Thatcherized) faces, and three bias comparisons that directly compared threatening and expectancy-violating faces with one another. Across these experiments, we found evidence for a dominant expectancy bias in pupillary arousal for social stimuli, whereby expectancy-violating faces produced pupillary dilation earlier than neutral and threatening faces, with Thatcherized faces producing the greatest magnitude of dilation

Experimentally induced pain does not influence updating of peripersonal space and body representations following tool-use

This is the final version. Available from the publisher via the DOI in this record.The data is available on the Open Science Framework. URL: https://osf.io/4v938/. DOI: 10.17605/OSF.IO/4V938.Representations of the body and peripersonal space can be distorted for people with some chronic pain conditions. Experimental pain induction can give rise to similar, but transient distortions in healthy individuals. However, spatial and bodily representations are dynamic, and constantly update as we interact with objects in our environment. It is unclear whether induced pain disrupts the mechanisms involved in updating these representations. In the present study, we sought to investigate the effect of induced pain on the updating of peripersonal space and body representations during and following tool-use. We compared performance under three conditions (pain, active placebo, neutral) on a visuotactile crossmodal congruency task and a tactile distance judgement task to measure updating of peripersonal space and body representations, respectively. Consistent with previous findings, the difference in crossmodal interference from visual distractors in the same compared to opposite visual field to the tactile target was less when tools were crossed than uncrossed. This suggests an extension of peripersonal space to incorporate the tips of the tools. Also consistent with previous findings, estimates of the felt tactile distance judgements decreased after active tool-use. In contrast to our predictions, however, we found no evidence that pain interfered with performance on either task when compared to the control conditions. Our findings suggest that the updating of peripersonal space and body representations is not disrupted by induced pain. That is, experiencing acute pain does not give rise to distorted representations of the body and peripersonal space that can be present in people with chronic pain conditions.Medical Research Council (MRC