Related papers: Double Well Potential Function and Its Optimizatio…
A functional data depth provides a center-outward ordering criterion which allows the definition of measures such as median, trimmed means, central regions or ranks in a functional framework. A functional data depth can be global or local.…
We have an $\m\x\n$ real-valued arbitrary matrix $A$ (e.g. a dictionary) with $\m<\n$ and data $d$ describing the sought-after object with the help of $A$. This work provides an in-depth analysis of the (local and global) minimizers of an…
Cavity flow problems in two dimensions, as well as in the axially symmetric three-dimensional case, have been extensively studied in the literature from a qualitative perspective. While numerous results exist concerning minimizers or stable…
We study the double bubble problem where the perimeter is taken with respect to the hexagonal norm, i.e. the norm whose unit circle in $\mathbb{R}^2$ is the regular hexagon. We provide an elementary proof for the existence of minimizing…
In this paper we present necessary and sufficient conditions (in terms of {\L}ojasiewicz inequalities) for the stability of local minimum points in smooth unconstrained optimization. In particular, we derive a sufficient condition for which…
Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler, Burgers, and 2D surface quasi-geostrophic equations. Many questions about regularity and properties of solutions of…
This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We…
This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex…
The optimized effective potential method is formulated as a convex minimization problem. This formulation does not require assumptions about $v$-representability nor functional differentiability. The formulation provides a natural framework…
Local solutions for variational and quasi-variational inequalities are usually the best type of solutions that could practically be obtained when in case of lack of convexity or else when available numerical techniques are too limited for…
In this article, we prove the existence of at least two positive weak solutions for a mixed local-nonlocal singular problem in the presence of critical exponential nonlinearity in dimension two. The novelty of this work is the inclusion of…
Hidden convex optimization is such a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we focus on checking local optimality in hidden…
We prove sharp local and global variation bounds for the centred Hardy--Littlewood maximal functions of indicator functions in one dimension. We characterise maximisers, treat both the continuous and discrete settings and extend our results…
We investigate the existence of positive solutions to fractional equations presenting a double criticality: a multi-polar Hardy-type potential and a Sobolev critical nonlinearity. The nonlocal nature of the operator and the absence of…
In this paper we summarize our results in infinite horizon optimal control. We present optimality conditions for weak local minimizer in the framework of weighted functions. Moreover we formulate the Pontryagin Maximum Principle for strong…
This paper is devoted to the study of tilt stability of local minimizers for classical nonlinear programs with equality and inequality constraints in finite dimensions described by twice continuously differentiable functions. The importance…
In the recent paper [J. Funct. Anal. {\bf 259} (2010)], Naor and Tao introduce a new class of measures with a so-called micro-doubling property and present, via martingale theory and probability methods, a localization theorem for the…
This paper considers the distributed smooth optimization problem in which the objective is to minimize a global cost function formed by a sum of local smooth cost functions, by using local information exchange. The standard assumption for…
We study global minimizers of biharmonic analogues of the Alt-Caffarelli functional. It turns out that half-space solutions are global minimizers for the two-sided Alt-Caffarelli functional, but not in the one-sided case. In addition, we…
Here we consider stationary states for nonlinear Schrodinger equations with symmetric double well potentials. These stationary states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures…