Generic identifiability and second-order sufficiency in tame convex optimization

We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, "tame"). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is "partly smooth", ensuring finite identification of the manifold by many optimization algorithms. Furthermore, second-order optimality conditions hold, guaranteeing smooth behavior of the optimal solution under small perturbations to the objective

Clarke subgradients of stratifiable functions

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $x\in{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz inequality for functions definable in an arbitrary o-minimal structure

Motion Estimation Using a Spherical Camera

Robotic navigation algorithms increasingly make use of the panoramic field of view provided by omnidirectional images to assist with localization tasks. Since the images taken by a particular class of omnidirectional sensors can be mapped to the sphere, the problem of attitude estimation arising from 3D motions of the camera can be treated as a problem of estimating the camera motion between spherical images. This problem has traditionally been solved by tracking points or features between images. However, there are many natural scenes where the features cannot be tracked with confidence. We present an algorithm that uses image features to estimate ego-motion without explicitly searching for correspondences. We formulate the problem as a correlation of functions defined on the product of spheres S2 × S2 which are acted upon by elements of the direct product group SO(3) × SO(3). We efficiently compute this correlation and obtain our solution using the spectral information of functions in S2 × S2

Stability in Linear Optimization Under Perturbations of the Left-Hand Side Coefficients

This paper studies stability properties of linear optimization problems with finitely many variables and an arbitrary number of constraints, when only left hand side coefficients can be perturbed. The coefficients of the constraints are assumed to be continuous functions with respect to an index which ranges on certain compact Hausdorff topological space, and these properties are preserved by the admissible perturbations. More in detail, the paper analyzes the continuity properties of the feasible set, the optimal set and the optimal value, as well as the preservation of desirable properties (boundedness, uniqueness) of the feasible and of the optimal sets, under sufficiently small perturbations.Research supported by the grants: BASAL PFB-03 (Chile), FONDECYT 1130176 (Chile) and MTM2011-29064-C03-01 (Spain). Research supported by the grant MTM2014-59179-C2-1-P (Spain) and the Discovery Projects DP120100467 and DP110102011 (Australian Research Council). Research supported by the MIUR project ’Variational and Topological Methods in the Study of Nonlinear Phenomena” (2009)

Characterizations of Super-regularity and its Variants

Convergence of projection-based methods for nonconvex set feasibility problems has been established for sets with ever weaker regularity assumptions. What has not kept pace with these developments is analogous results for convergence of optimization problems with correspondingly weak assumptions on the value functions. Indeed, one of the earliest classes of nonconvex sets for which convergence results were obtainable, the class of so-called super-regular sets introduced by Lewis, Luke and Malick (2009), has no functional counterpart. In this work, we amend this gap in the theory by establishing the equivalence between a property slightly stronger than super-regularity, which we call Clarke super-regularity, and subsmootheness of sets as introduced by Aussel, Daniilidis and Thibault (2004). The bridge to functions shows that approximately convex functions studied by Ngai, Luc and Th\'era (2000) are those which have Clarke super-regular epigraphs. Further classes of regularity of functions based on the corresponding regularity of their epigraph are also discussed.Comment: 15 pages, 2 figure