Doubly resonant semilinear elliptic problems via nonsmooth critical point theory

AbstractWe consider the existence of weak solutions for classical doubly resonant semilinear elliptic problems. We show how the main technical assumptions can be used to define appropriate metrics on the underlying function space, so that extensions of the results already known in the literature can be obtained using only basic facts from critical point theory for continuous functionals on complete metric spaces

Diffeomorphism-invariant properties for quasi-linear elliptic operators

For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and non-degenerate coerciveness.Comment: 16 page

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization

Organizing risk: organization and management theory for the risk society

Risk has become a crucial part of organizing, affecting a wide range of organizations in all sectors. We identify, review and integrate diverse literatures relevant to organizing risk, building on an existing framework that describes how risk is organized in three ‘modes’ – prospectively, in real-time, and retrospectively. We then identify three critical issues in the existing literature: its fragmented nature; its neglect of the tensions associated with each of the modes; and its tendency to assume that the meaning of an object in relation to risk is singular and stable. We provide a series of new insights with regard to each of these issues. First, we develop the concept of a risk cycle that shows how organizations engage with all three modes and transition between them over time. Second, we explain why the tensions have been largely ignored and show how studies using a risk work perspective can provide further insights into them. Third, we develop the concept of risk translation to highlight the ways in the meanings of risks can be transformed and to identify the political consequences of such translations. We conclude the paper with a research agenda to elaborate these insights and ideas further

Nonlinear local error bounds via a change of metric

International audienceIn this work, we improve the approach of the second author and V. Motreanu [Math. Program. 114 (2008), 291–319] to nonlinear error bounds for lower semicontinuous functions on complete metric spaces, an approach consisting in reducing the nonlinear case to the linear one through a change of metric. This improvement is basically a technical one, and it allows dealing with local error bounds in an appropriate way. We present some consequences of the general results in the framework of classical nonsmooth analysis, involving Banach spaces and subdifferential operators. In particular, we describe connections between local quadratic growth of a function and metric regularity of its subdifferential

Nonlinear Error Bounds via a Change of Function

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