Semismooth and Semiconvex Functions in Constrained Optimization

We introduce semismooth and semiconvex functions and discuss their properties with respect to nonsmooth nonconvex constrained optimization problems. These functions are locally Lipschitz, and hence have generalized gradients. The author has given an optimization algorithm that uses generalized gradients of the problem functions and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then we show that a stationary point is an optimal point. We show that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function and that the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. Furthermore, we show that a semismooth composition of semismooth functions is semismooth and gives a type of chain rule for generalized gradients

An Algorithm for Constrained Optimization with Semismooth Functions

We present an implementable algorithm for solving constrained optimization problems defined by functions that are not everywhere differentiable. The method is based on combining, modifying and extending the nonsmooth optimization work of Wolfe, Lemarechal, Feuer, Poljak, and Merrill. It can be thought of as a generalized reset conjugate gradient algorithm. We also introduce the class of weakly upper semismooth functions. These functions are locally Lipschitz and have a semicontinuous relationship between their generalized gradient sets and their directional derivatives. The algorithm is shown to converge to stationary points of the optimization problem if the objective and constraint functions are weakly upper semismooth. Such points are optimal points if the problem functions are also semiconvex and a constraint qualification is satisfied. Under stronger convexity assumptions, bounds on the deviation from optimality of the algorithm iterates are given

Nonsmooth Optimization; Proceedings of an IIASA Workshop, March 28 - April 8, 1977

Optimization, a central methodological tool of systems analysis, is used in many of IIASA's research areas, including the Energy Systems and Food and Agriculture Programs. IIASA's activity in the field of optimization is strongly connected with nonsmooth or nondifferentiable extreme problems, which consist of searching for conditional or unconditional minima of functions that, due to their complicated internal structure, have no continuous derivatives. Particularly significant for these kinds of extreme problems in systems analysis is the strong link between nonsmooth or nondifferentiable optimization and the decomposition approach to large-scale programming. This volume contains the report of the IIASA workshop held from March 28 to April 8, 1977, entitled Nondifferentiable Optimization. However, the title was changed to Nonsmooth Optimization for publication of this volume as we are concerned not only with optimization without derivatives, but also with problems having functions for which gradients exist almost everywhere but are not continous, so that the usual gradient-based methods fail. Because of the small number of participants and the unusual length of the workshop, a substantial exchange of information was possible. As a result, details of the main developments in nonsmooth optimization are summarized in this volume, which might also be considered a guide for inexperienced users. Eight papers are presented: three on subgradient optimization, four on descent methods, and one on applicability. The report also includes a set of nonsmooth optimization test problems and a comprehensive bibliography

Convergence rates for a method of centers algorithm

Convergence of centers algorithm for solving nonlinear programming problem

Computing Economic Equilibria through Nonsmooth Optimization

In the first section the problem of nonsmooth optimization is described in general terms, setting the precise hypothesis in mathematical language. Section 2 describes the principles of an example which arises in the context of the linkage of national models of food and agriculture. The general methodology is presented in Section 3, where the algorithm of solution is outlined. Section 4 reports on an extensive set of numerical experiments, both on problems known in the literature, and on the example of Section 2. Finally the paper concludes with some remarks about improvements of the algorithm, which motivate further research on the subject

Yours ever (well, maybe): Studies and signposts in letter writing

Electronic mail and other digital communications technologies seemingly threaten to end the era of handwritten and typed letters, now affectionately seen as part of snail mail. In this essay, I analyze a group of popular and scholarly studies about letter writing-including examples of pundits critiquing the use of e-mail, etiquette manuals advising why the handwritten letter still possesses value, historians and literary scholars studying the role of letters in the past and what it tells us about our present attitudes about digital communications technologies, and futurists predicting how we will function as personal archivists maintaining every document including e-mail. These are useful guideposts for archivists, providing both a sense of the present and the past in the role, value and nature of letters and their successors. They also provide insights into how such documents should be studied, expanding our gaze beyond the particular letters, to the tools used to create them and the traditions dictating their form and function. We also can discern a role for archivists, both for contributing to the literature about documents and in using these studies and commentaries, suggesting not a new disciplinary realm but opportunities for new interdisciplinary work. Examining a documentary form makes us more sensitive to both the innovations and traditions as it shifts from the analog to the digital; we can learn not to be caught up in hysteria or nostalgia about one form over another and archivists can learn about what they might expect in their labors to document society and its institutions. At one time, paper was part of an innovative technology, with roles very similar to the Internet and e-mail today. It may be that the shifts are far less revolutionary than is often assumed. Reading such works also suggests, finally, that archivists ought to rethink how they view their own knowledge and how it is constructed and used. © 2010 Springer Science+Business Media B.V

High sensitivity measurement of 224Ra and 226Ra in water with an improved hydrous titanium oxide technique at the Sudbury Neutrino Observatory

The existing hydrous titanium oxide (HTiO) technique for the measurement of 224Ra and 226Ra in the water at the Sudbury Neutrino Observatory (SNO) has been changed to make it faster and less sensitive to trace impurities in the HTiO eluate. Using HTiO-loaded filters followed by cation exchange adsorption and HTiO co-precipitation, Ra isotopes from 200-450 tonnes of heavy water can be extracted and concentrated into a single sample of a few millilitres with a total chemical efficiency of 50%. Combined with beta-alpha coincidence counting, this method is capable of measuring 2.0x10^3 uBq/kg of 224Ra and 3.7x10^3 uBq/kg of 226Ra from the 232Th and 238U decay chains, respectively, for a 275 tonne D2O assay, which are equivalent to 5x10^16 g Th/g and 3x10^16 g U/g in heavy water.Comment: 8 Pages, 2 figures and 2 table

Embodying compassion: A systematic review of the views of nurses and patients

© 2018 John Wiley & Sons Ltd Aims and objectives: To provide a review of empirical research investigating how compassion is expressed by nurses and received by patients in hospital settings. Background: Compassion is viewed as an important and fundamental part of a health professional practice. Universally, reports from both media and government agencies have addressed perceived deficits of compassion in healthcare with nurses accused of a lack of compassion. Research into compassion to date has largely focused on the problematic nature of compassion such as burnout, fatigue and other negative personal and work-related outcomes. Design: A systematic literature review of empirical research guided by a meta-ethnographic approach supported the systematic comparison and translation of the included studies. Six online databases were searched from January 2006–December 2016. Methods: This review was carried out according to the PRISMA-P reporting guidelines. How compassion in healthcare was defined was extracted alongside findings on how compassion was expressed by nurses and received by patients. Synthesis of the research was completed resulting in new interpretations. Results: Eleven papers met the inclusion criteria and were included in the review. Multiple differing definitions of compassion in healthcare were applied. Nurses embody and enact compassion through behaviours such as spending time with patients and communicating effectively with patients. Patients experience compassion through a sense of togetherness with nurses. Conclusion: Existing research demonstrated dissonance between the expression of compassion by nurses and how compassion is experienced by patients. The themes identified in this review should be considered by health professionals providing patient care. Relevance to clinical practice: Health providers should acknowledge and account for the time that nurses need with patients to demonstrate compassion in practice. Nursing education relating to the expression of compassion should articulate both the subjectivity and ambiguity of the term and examine the relationship between compassion and suffering

The Radius of Metric Subregularity

There is a basic paradigm, called here the radius of well-posedness, which quantifies the "distance" from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings.Comment: 20 page

Appetite and gut hormone responses to moderate-intensity continuous exercise versus high-intensity interval exercise, in normoxic and hypoxic conditions.

This study investigated the effects of continuous moderate-intensity exercise (MIE) and high-intensity interval exercise (HIIE) in combination with short exposure to hypoxia on appetite and plasma concentrations of acylated ghrelin, peptide YY (PYY), and glucagon-like peptide-1 (GLP-1). Twelve healthy males completed four, 2.6 h trials in a random order: 1) MIE-normoxia, 2) MIE-hypoxia, 3) HIIE-normoxia, and 4) HIIE-hypoxia. Exercise took place in an environmental chamber. During MIE, participants ran for 50 min at 70% of altitude-specific maximal oxygen uptake ( 2max) and during HIIE performed 6 x 3 min running at 90% 2max interspersed with 6 x 3 min active recovery at 50% 2max with a 7 min warm-up and cool-down at 70% 2max (50 min total). In hypoxic trials, exercise was performed at a simulated altitude of 2,980 m (14.5% O2). Exercise was completed after a standardised breakfast. A second meal standardised to 30% of participants’ daily energy requirements was provided 45 min after exercise. Appetite was suppressed more in hypoxia than normoxia during exercise, post-exercise, and for the full 2.6 h trial period (linear mixed modelling, p 0.05). These findings demonstrate that short exposure to hypoxia causes suppressions in appetite and plasma acylated ghrelin concentrations. Furthermore, appetite responses to exercise do not appear to be influenced by exercise modality