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

Polycyclic aromatic hydrocarbons and postmenopausal breast cancer: An evaluation of effect measure modification by body mass index and weight change

Polycyclic aromatic hydrocarbons (PAHs) have been linked to breast cancer in many, but not all, previous studies. PAHs are lipophilic and stored in fat tissue, which we hypothesized may result in constant low-dose exposure to these carcinogens. No previous studies have evaluated whether obesity modifies associations between multiple measures of PAHs and breast cancer incidence

Regularity of a kind of marginal functions in Hilbert spaces

We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Vehicular Traffic–Related Polycyclic Aromatic Hydrocarbon Exposure and Breast Cancer Incidence: The Long Island Breast Cancer Study Project (LIBCSP)

BackgroundPolycyclic aromatic hydrocarbons (PAHs) are widespread environmental pollutants, known human lung carcinogens, and potent mammary carcinogens in laboratory animals. However, the association between PAHs and breast cancer in women is unclear. Vehicular traffic is a major ambient source of PAH exposure.ObjectivesOur study aim was to evaluate the association between residential exposure to vehicular traffic and breast cancer incidence.MethodsResidential histories of 1,508 participants with breast cancer (case participants) and 1,556 particpants with no breast cancer (control participants) were assessed in a population-based investigation conducted in 1996–1997. Traffic exposure estimates of benzo[a]pyrene (B[a]P), as a proxy for traffic-related PAHs, for the years 1960–1995 were reconstructed using a model previously shown to generate estimates consistent with measured soil PAHs, PAH–DNA adducts, and CO readings. Associations between vehicular traffic exposure estimates and breast cancer incidence were evaluated using unconditional logistic regression.ResultsThe odds ratio (95% CI) was modestly elevated by 1.44 (0.78, 2.68) for the association between breast cancer and long-term 1960–1990 vehicular traffic estimates in the top 5%, compared with below the median. The association with recent 1995 traffic exposure was elevated by 1.14 (0.80, 1.64) for the top 5%, compared with below the median, which was stronger among women with low fruit/vegetable intake [1.46 (0.89, 2.40)], but not among those with high fruit/vegetable intake [0.92 (0.53, 1.60)]. Among the subset of women with information regarding traffic exposure and tumor hormone receptor subtype, the traffic–breast cancer association was higher for those with estrogen/progesterone-negative tumors [1.67 (0.91, 3.05) relative to control participants], but lower among all other tumor subtypes [0.80 (0.50, 1.27) compared with control participants].ConclusionsIn our population-based study, we observed positive associations between vehicular traffic-related B[a]P exposure and breast cancer incidence among women with comparatively high long-term traffic B[a]P exposures, although effect estimates were imprecise.CitationMordukhovich I, Beyea J, Herring AH, Hatch M, Stellman SD, Teitelbaum SL, Richardson DB, Millikan RC, Engel LS, Shantakumar S, Steck SE, Neugut AI, Rossner P Jr., Santella RM, Gammon MD. 2016. Vehicular traffic–related polycyclic aromatic hydrocarbon exposure and breast cancer incidence: the Long Island Breast Cancer Study Project (LIBCSP). Environ Health Perspect 124:30–38; http://dx.doi.org/10.1289/ehp.130773

Genome-wide Association Study of Susceptibility to Particulate Matter–Associated QT Prolongation

BACKGROUND: Ambient particulate matter (PM) air pollution exposure has been associated with increases in QT interval duration (QT). However, innate susceptibility to PM-associated QT prolongation has not been characterized. OBJECTIVE: To characterize genetic susceptibility to PM-associated QT prolongation in a multi-racial/ethnic, genome-wide association study (GWAS). METHODS: Using repeated electrocardiograms (1986–2004), longitudinal data on PM<10 μm in diameter (PM10), and generalized estimating equations methods adapted for low-prevalence exposure, we estimated approximately 2.5×106 SNP×PM10 interactions among nine Women’s Health Initiative clinical trials and Atherosclerosis Risk in Communities Study subpopulations (n=22,158), then combined subpopulation-specific results in a fixed-effects, inverse variance-weighted meta-analysis. RESULTS: A common variant (rs1619661; coded allele: T) significantly modified the QT-PM10 association (p=2.11×10−8). At PM10 concentrations >90th percentile, QT increased 7 ms across the CC and TT genotypes: 397 (95% confidence interval: 396, 399) to 404 (403, 404) ms. However, QT changed minimally across rs1619661 genotypes at lower PM10 concentrations. The rs1619661 variant is on chromosome 10, 132 kilobase (kb) downstream from CXCL12, which encodes a chemokine, stromal cell-derived factor 1, that is expressed in cardiomyocytes and decreases calcium influx across the L-type Ca2+ channel. CONCLUSIONS: The findings suggest that biologically plausible genetic factors may alter susceptibility to PM10-associated QT prolongation in populations protected by the U.S. Environmental Protection Agency’s National Ambient Air Quality Standards. Independent replication and functional characterization are necessary to validate our findings. https://doi.org/10.1289/EHP34

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Optimality conditions in convex multiobjective SIP

The purpose of this paper is to characterize the weak efficient solutions, the efficient solutions, and the isolated efficient solutions of a given vector optimization problem with finitely many convex objective functions and infinitely many convex constraints. To do this, we introduce new and already known data qualifications (conditions involving the constraints and/or the objectives) in order to get optimality conditions which are expressed in terms of either Karusk–Kuhn–Tucker multipliers or a new gap function associated with the given problem.This research was partially cosponsored by the Ministry of Economy and Competitiveness (MINECO) of Spain, and by the European Regional Development Fund (ERDF) of the European Commission, Project MTM2014-59179-C2-1-P

Exposure to multiple sources of polycyclic aromatic hydrocarbons and breast cancer incidence

Background: Despite studies having consistently linked exposure to single-source polycyclic aromatic hydrocarbons (PAHs) to breast cancer, it is unclear whether single sources or specific groups of PAH sources should be targeted for breast cancer risk reduction. Objectives: This study considers the impact on breast cancer incidence from multiple PAH exposure sources in a single model, which better reflects exposure to these complex mixtures. Methods: In a population-based case-control study conducted on Long Island, New York (N = 1508 breast cancer cases/1556 controls), a Bayesian hierarchical regression approach was used to estimate adjusted posterior means and credible intervals (CrI) for the adjusted odds ratios (ORs) for PAH exposure sources, considered singly and as groups: active smoking; residential environmental tobacco smoke (ETS); indoor and outdoor air pollution; and grilled/smoked meat intake. Results: Most women were exposed to PAHs from multiple sources, and the most common included active/passive smoking and grilled/smoked food intake. In multiple-PAH source models, breast cancer incidence was associated with residential ETS from a spouse (OR = 1.20, 95%CrI = 1.03, 1.40) and synthetic firelog burning (OR = 1.29, 95%CrI = 1.06, 1.57); these estimates are similar, but slightly attenuated, to those from single-source models. Additionally when we considered PAH exposure groups, the most pronounced significant associations included total indoor sources (active smoking, ETS from spouse, grilled/smoked meat intake, stove/fireplace use, OR = 1.45, 95%CrI = 1.02, 2.04). Conclusions: Groups of PAH sources, particularly indoor sources, were associated with a 30–50% increase in breast cancer incidence. PAH exposure is ubiquitous and a potentially modifiable breast cancer risk factor

Associations between Polycyclic Aromatic Hydrocarbon–Related Exposures and p53 Mutations in Breast Tumors

Background: Previous studies have suggested that polycyclic aromatic hydrocarbons (PAHs) may be associated with breast cancer. However, the carcinogenicity of PAHs on the human breast remains unclear. Certain carcinogens may be associated with specific mutation patterns in the p53 tumor suppressor gene, thereby contributing information about disease etiology. Objectives: We hypothesized that associations of PAH-related exposures with breast cancer would differ according to tumor p53 mutation status, effect, type, and number. Methods: We examined this possibility in a population-based case–control study using polytomous logistic regression. As previously reported, 151 p53 mutations among 859 tumors were identified using Surveyor nuclease and confirmed by sequencing. Results: We found that participants with p53 mutations were less likely to be exposed to PAHs (assessed by smoking status in 859 cases and 1,556 controls, grilled/smoked meat intake in 822 cases and 1,475 controls, and PAH–DNA adducts in peripheral mononuclear cells in 487 cases and 941 controls) than participants without p53 mutations. For example, active and passive smoking was associated with p53 mutation–negative [odds ratio (OR) = 1.55; 95% confidence interval (CI), 1.11–2.15] but not p53 mutation–positive (OR = 0.77; 95% CI, 0.43–1.38) cancer (ratio of the ORs = 0.50, p < 0.05). However, frameshift mutations, mutation number, G:C→A:T transitions at CpG sites, and insertions/deletions were consistently elevated among exposed subjects. Conclusions: These findings suggest that PAHs may be associated with specific breast tumor p53 mutation subgroups rather than with overall p53 mutations and may also be related to breast cancer through mechanisms other than p53 mutation