Applications of Variational Analysis to a Generalized Heron Problem

This paper is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to $n$ given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point C such that the sum of the distances from C to the given points A and B is minimal. We show that the advanced variational techniques allow us to completely solve optimal location problems of this type in some important settings

Well-posedness of minimal time problems with constant dynamics in Banach spaces

This paper concerns the study of a general minimal time problem with a convex constant dynamics and a closed target set in Banach spaces. We pay the main attention to deriving sufficient conditions for the major well-posedness properties that include the existence and uniqueness of optimal solutions as well as certain regularity of the optimal value function with respect to state variables. Most of the results obtained are new even in finite-dimensional spaces. Our approach is based on advanced tools of variational analysis and generalized differentiation

Well-Posedness of Minimal Time Problem with Constant Dynamics in Banach Spaces

This paper concerns the study of a general minimal time problem with a convex constant dynamic and a closed target set in Banach spaces. We pay the main attention to deriving efficient conditions for the major well-posedness properties that include the existence and uniqueness of optimal solutions as well as certain regularity of the optimal value function with respect to state variables. Most of the results obtained are new even in finite-dimensional spaces. Our approach is based on advanced tools of variational analysis and generalized differentiation

Well-posedness of minimal time problems with constant dynamics in Banach spaces

This paper concerns the study of a general minimal time problem with a convex constant dynamics and a closed target set in Banach spaces. We pay the main attention to deriving sufficient conditions for the major well-posedness properties that include the existence and uniqueness of optimal solutions as well as certain regularity of the optimal value function with respect to state variables. Most of the results obtained are new even in finite-dimensional spaces. Our approach is based on advanced tools of variational analysis and generalized differentiation

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

Minimal time functions and the smallest intersecting ball problem generated by unbounded dynamics

The smallest enclosing circle problem introduced in the 19th century by J. J. Sylvester [20] aks for the circle of smallest radius enclosing a given set of finite points in the plane. An extension of the smallest enclosing circle problem called the smallest intersecting ball problem was considered in [17,18]: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that intersects all of the sets. In this paper we initiate the study of minimal time functions generated by unbounded dynamics and discuss their applications to extensions of the smallest intersecting ball problem. This approach continues our effort in applying convex and nonsmooth analysis to the well-established field of facility location