Related papers: The local minimality of differentiable functions
This work establishes a rigorous connection between stability properties of discrete-time algorithms (DTAs) and corresponding continuous-time dynamical systems derived through $ O(s^r) $-resolution ordinary differential equations (ODEs). We…
In this paper we study the right differentiability of a parametric infimum function over a parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to…
We show that on almost complex surfaces plurisubharmonic functions can be locally approximated by smooth plurisubharmonic functions. The main tool is the Poletsky type theorem due to U. Kuzman.
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local…
We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the…
A sufficient condition for the stability of a system resulting from the interconnection of dynamical systems is given by the small gain theorem. Roughly speaking, to apply this theorem, it is required that the gains composition is…
The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the…
We consider the gradient method with variable step size for minimizing functions that are definable in o-minimal structures on the real field and differentiable with locally Lipschitz gradients. We prove that global convergence holds if…
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
We provide a generalization of first-order necessary conditions of optimality for infinite-dimensional optimization problems with a finite number of inequality constraints and with a finite number of inequality and equality constraints. Our…
We establish the higher differentiability for the minimizers of the following non-autonomous integral functionals \begin{equation*} \mathcal{F}(u,\Omega):= \, \int_\Omega \sum_{i=1}^{n} \, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx,…
A necessary and sufficient condition for local solvability is presented for the linear partial differential operators $-X^2-Y^2+ia(x)[X,Y]$ in $\bold R^3=\{(x,y,t)\}$, where $X=\partial_x,\; Y=\partial_y+x^k\partial_t$, and $a\in…
In this note we generate two local median oscillation decompositions of an arbitrary measurable function and discuss some applications to Calder\'{o}n-Zygmund singular integral operators $T$. These applications rely on the inequality…
We propose a derivative-free trust-region method based on finite-difference gradient approximations for smooth optimization problems with convex constraints. The proposed method does not require computing an approximate stationarity…
When a parameter of interest is defined to be a nondifferentiable transform of a regular parameter, the parameter does not have an influence function, rendering the existing theory of semiparametric efficient estimation inapplicable.…
We study the regularity properties of H\"older continuous minimizers to non-autonomous functionals satisfying $(p,q)$-growth conditions, under Besov assumptions on the coefficients. In particular, we are able to prove higher integrability…
We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…
In this paper it is shown that any regular critical point of the Mumford-Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the L^1-topology.
We propose a unifying framework for characterizing pure and mixed state phases of matter across equilibrium, non equilibrium, and metastable regimes. We introduce the concept of locally stable states, defined by the operational property…
Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can…