8 research outputs found
Quantitative Stability of Variational Systems: II. A Framework for Nonlinear Conditioning
It is shown that for well-conditioned problems (local) optima are holderian with respect to the epi-distance
Stability and Error Analysis for Optimization and Generalized Equations
Stability and error analysis remain challenging for problems that lack
regularity properties near solutions, are subject to large perturbations, and
might be infinite dimensional. We consider nonconvex optimization and
generalized equations defined on metric spaces and develop bounds on solution
errors using the truncated Hausdorff distance applied to graphs and epigraphs
of the underlying set-valued mappings and functions. In the process, we extend
the calculus of such distances to cover compositions and other constructions
that arise in nonconvex problems. The results are applied to constrained
problems with feasible sets that might have empty interiors, solution of KKT
systems, and optimality conditions for difference-of-convex functions and
composite functions
Convergence of the Forward-Backward Algorithm: Beyond the Worst Case with the Help of Geometry
We provide a comprehensive study of the convergence of forward-backward
algorithm under suitable geometric conditions leading to fast rates. We present
several new results and collect in a unified view a variety of results
scattered in the literature, often providing simplified proofs. Novel
contributions include the analysis of infinite dimensional convex minimization
problems, allowing the case where minimizers might not exist. Further, we
analyze the relation between different geometric conditions, and discuss novel
connections with a priori conditions in linear inverse problems, including
source conditions, restricted isometry properties and partial smoothness
Set-Convergence and Its Application: A Tutorial
Optimization problems, generalized equations, and the multitude of other
variational problems invariably lead to the analysis of sets and set-valued
mappings as well as their approximations. We review the central concept of
set-convergence and explain its role in defining a notion of proximity between
sets, especially for epigraphs of functions and graphs of set-valued mappings.
The development leads to an approximation theory for optimization problems and
generalized equations with profound consequences for the construction of
algorithms. We also introduce the role of set-convergence in variational
geometry and subdifferentiability with applications to optimality conditions.
Examples illustrate the importance of set-convergence in stability analysis,
error analysis, construction of algorithms, statistical estimation, and
probability theory
Recent trends and views on elliptic quasi-variational inequalities
We consider state-of-the-art methods, theoretical limitations, and open problems in elliptic Quasi-Variational Inequalities (QVIs). This involves the development of solution algorithms in function space, existence theory, and the study of optimization problems with QVI constraints. We address the range of applicability and theoretical limitations of fixed point and other popular solution algorithms, also based on the nature of the constraint, e.g., obstacle and gradient-type. For optimization problems with QVI constraints, we study novel formulations that capture the multivalued nature of the solution mapping to the QVI, and generalized differentiability concepts appropriate for such problems
Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry
We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective
functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those
geometric notions over arbitrary sets. In turn, this allows us to present several new results as well
as collect in a unified view a variety of results scattered in the literature. Our contributions include
the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz
inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach
allows to establish unexpected connections between geometry and a priori conditions in inverse
problems, such as source conditions, or restricted isometry properties