A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory

The first motivation for this note is to obtain a general version of the following result: let E be a Banach space and f : E → R be a differentiable function, bounded below and satisfying the Palais-Smale condition; then, f is coercive, i.e., f(x) goes to infinity as ||x|| goes to infinity. In recent years, many variants and extensions of this result appeared, see [3], [5], [6], [9], [14], [18], [19] and the references therein. A general result of this type was given in [3, Theorem 5.1] for a lower semicontinuous function defined on a Banach space, through an approach based on an abstract notion of subdifferential operator, and taking into account the “smoothness” of the Banach space. Here, we give (Theorem 1) an extension in a metric setting, based on the notion of slope from [11] and coercivity is considered in a generalized sense, inspired by [9]; our result allows to recover, for example, the coercivity result of [19], where a weakened version of the Palais-Smale condition is used. Our main tool (Proposition 1) is a consequence of Ekeland’s variational principle extending [12, Corollary 3.4], and deals with a function f which is, in some sense, the “uniform” Γ-limit of a sequence of functions.

Abstract Subdifferential Calculus and Semi-Convex Functions

∗ The work is partially supported by NSFR Grant No MM 409/94.

Subdifferentials of Performance Functions and Calculus of Coderivatives of Set-Valued Mappings

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gˆateaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key element of the unified approach to obtaining various calculus rules for various types of derivatives presented in the paper are simple formulas for subdifferentials of marginal, or performance functions.

A Mean Value Theorem for non Differentiable Mappings in Banach Spaces

We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii.

Well-Behavior, Well-Posedness and Nonsmooth Analysis

AMS subject classification: 90C30, 90C33.

Dense Subdifferentiability and Trustworthiness for Arbitrary Subdifferentials

2000 Mathematics Subject Classification: 49J52, 49J50, 58C20, 26B09.

Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

On the stability of the optimal value and the optimal set in optimization problems

The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in [17], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property).

Not all inflammatory markers are linked to kidney function: results from a population-based study.

BACKGROUND: Several studies have reported increased levels of inflammatory biomarkers in chronic kidney disease (CKD), but data from the general population are sparse. In this study, we assessed levels of the inflammatory markers C-reactive protein (hsCRP), tumor necrosis factor α (TNF-α), interleukin (IL)-1β and IL-6 across all ranges of renal function. METHODS: We conducted a cross-sectional study in a random sample of 6,184 Caucasian subjects aged 35-75 years in Lausanne, Switzerland. Serum levels of hsCRP, TNF-α, IL-6, and IL-1β were measured in 6,067 participants (98.1%); serum creatinine-based estimated glomerular filtration rate (eGFR(creat), CKD-EPI formula) was used to assess renal function, and albumin/creatinine ratio on spot morning urine to assess microalbuminuria (MAU). RESULTS: Higher serum levels of IL-6, TNF-α and hsCRP and lower levels of IL-1β were associated with a lower renal function, CKD (eGFR(creat) <60 ml/min/1.73 m(2); n = 283), and MAU (n = 583). In multivariate linear regression analysis adjusted for age, sex, hypertension, smoking, diabetes, body mass index, lipids, antihypertensive and hypolipemic therapy, only log-transformed TNF-α remained independently associated with lower renal function (β -0.54 ±0.19). In multivariate logistic regression analysis, higher TNF-α levels were associated with CKD (OR 1.17; 95% CI 1.01-1.35), whereas higher levels of IL-6 (OR 1.09; 95% CI 1.02-1.16) and hsCRP (OR 1.21; 95% CI 1.10-1.32) were associated with MAU. CONCLUSION: We did not confirm a significant association between renal function and IL-6, IL-1β and hsCRP in the general population. However, our results demonstrate a significant association between TNF-α and renal function, suggesting a potential link between inflammation and the development of CKD. These data also confirm the association between MAU and inflammation.

Inclusões Dinâmicas em Escalas Temporais: Existência de Soluções sob a Hipótese de Semicontinuidade Inferior

In this paper we consider the problem of differential inclusion in time scales whose vector field is a multifunction, that is, a function that maps points to sets. It is provided conditions of existence without requiring compactness of the vector field; it is required that the vector field is closed, convex, and lower semicontinuous. In previous work in literature, it is required that the field is either scalar or compact, convex, and has closed graph.

Sistemas de controle em domínios estratificados

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

The Lower Limb Functional Index: development and clinimetric properties

Background Existing lower-limb, region-specific, patient-reported outcome measures have clinimetric limitations, including limitations in psychometric characteristics (eg, lack of internal consistency, lack of responsiveness, measurement error) and the lack of reported practical and general characteristics. A new patient-reported outcome measure, the Lower Limb Functional Index (LLFI), was developed to address these limitations. Objective The purpose of this study was to overcome recognized deficiencies in existing lower-limb, region-specific, patient-reported outcome measures through: (1) development of a new lower-extremity outcome scale (ie, the LLFI) and (2) evaluation of the clinimetric properties of the LLFI using the Lower Extremity Functional Scale (LEFS) as a criterion measure. Design This was a prospective observational study. Methods The LLFI was developed in a 3-stage process of: (1) item generation, (2) item reduction with an expert panel, and (3) pilot field testing (n=18) for reliability, responsiveness, and sample size requirements for a larger study. The main study used a convenience sample (n=127) from 10 physical therapy clinics. Participants completed the LLFI and LEFS every 2 weeks for 6 weeks and then every 4 weeks until discharge. Data were used to assess the psychometric, practical, and general characteristics of the LLFI and the LEFS. The characteristics also were evaluated for overall performance using the Measurement of Outcome Measures and Bot clinimetric assessment scales. Results The LLFI and LEFS demonstrated a single-factor structure, comparable reliability (intraclass correlation coefficient [2,1]=.97), scale width, and high criterion validity (Pearson r=.88, with 95% confidence interval [CI]). Clinimetric performance was higher for the LLFI compared with the LEFS on the Measurement of Outcome Measures scale (96% and 95%, respectively) and the Bot scale (100% and 83%, respectively). The LLFI, compared with the LEFS, had improved responsiveness (standardized response mean=1.75 and 1.64, respectively), minimal detectable change with 90% CI (6.6% and 8.1%, respectively), and internal consistency (α=.91 and .95, respectively), as well as readability with reduced user error and completion and scoring times. Limitations Limitations of the study were that only participants recruited from outpatient physical therapy clinics were included and that no specific conditions or diagnostic subgroups were investigated. Conclusion The LLFI demonstrated sound clinimetric properties. There was lower response error, efficient completion and scoring, and improved responsiveness and overall performance compared with the LEFS. The LLFI is suitable for assessment of lower-limb function.

From the Farkas Lemma to the Hahn–Banach Theorem

This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ◦ g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn–Banach–Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn–Banach theorem and the Mazur–Orlicz theorem for extended sublinear functions.

Socioeconomic position, cognitive function, and clustering of cardiovascular risk factors in adolescence: Findings from the Mater University Study of Pregnancy and its outcomes

Objectives: The objectives of this study were to examine the extent of clustering of smoking, high levels of television watching, overweight, and high blood pressure among adolescents and whether this clustering varies by socioeconomic position and Cognitive function. Methods: This study was a cross-sectional analysis of 3613 (1742 females) participants of an Australian birth cohort who were examined at age 14. Results: Three hundred fifty-three (9.8%) of the participants had co-occurrence of three or four risk factors. Risk factors clustered in these adolescents with a greater number of participants than would be predicted by assumptions of independence having no risk factors and three or four risk factors. The extent of clustering tended to be greater in those from lower-income families and among those with lower cognitive function. The age-adjusted ratio of observed to expected cooccurrence of three or four risk factors was 2.70 (95% confidence interval [Cl], 1.80-4.06) among those from low-income families and 1.70 (95% Cl, 1.34-2.16) among those from more affluent families. The ratio among those with low Raven's scores (nonverbal reasoning) was 2.36 (95% Cl, 1.69-3.30) and among those with higher scores was 1.51 (95% Cl, 1.19-1.92); similar results for the WRAT 3 score (reading ability) were 2.69 (95% Cl, 1.85-3.94) and 1.68 (95% Cl, 1.34-2.11). Clustering did not differ by sex. Conclusion: Among adolescents, coronary heart disease risk factors cluster, and there is some evidence that this clustering is greater among those from families with low income and those who have lower cognitive function.

Porosity and Variational Principles

We prove that in some classes of optimization problems, like lower semicontinuous functions which are bounded from below, lower semi-continuous or continuous functions which are bounded below by a coercive function and quasi-convex continuous functions with the topology of the uniform convergence, the complement of the set of well-posed problems is σ-porous. These results are obtained as realization of a theorem extending a variational principle of Ioffe-Zaslavski.