Related papers: The subdifferentials of the first and second order…

Consider the problem of minimizing functions that are Lipschitz and strongly convex, but not necessarily differentiable. We prove that after $T$ steps of stochastic gradient descent, the error of the final iterate is $O(\log(T)/T)$ with…

In this paper, an upper semismooth function is defined to be a lower semicontinuous function whose radial subderivative satisfies a mild directional upper semicontinuity property. Examples of upper semismooth functions are the proper lower…

Block-coordinate algorithms are recognized to furnish efficient iterative schemes for addressing large-scale problems, especially when the computation of full derivatives entails substantial memory requirements and computational efforts. In…

We present a simple transformation of any linear program or semidefinite program into an equivalent convex optimization problem whose only constraints are linear equations. The objective function is defined on the whole space, making…

In this paper, we study the construction of Lyapunov functions based on first order approximations. In a first part, the study of local exponential stability property of a transverse invariant manifold is considered. This part is mainly a…

We present an accelerated gradient method for non-convex optimization problems with Lipschitz continuous first and second derivatives. The method requires time $O(\epsilon^{-7/4} \log(1/ \epsilon) )$ to find an $\epsilon$-stationary point,…

We will prove a global estimate for the gradient of the solution to the {\it Poisson differential inequality} $|\Delta u(x)|\le a|\nabla u(x)|^2+b$, $x\in B^{n}$, where $a,b<\infty$ and $u|_{S^{n-1}}\in C^{1,\alpha}(S^{n-1}, \Bbb R^m)$. If…

Second order approximate ancillaries have evolved as the primary ingredient for recent likelihood development in statistical inference. This uses quantile functions rather than the equivalent distribution functions, and the intrinsic…

The derivations of a left coideal subalgebra B of a Hopf algebra A which are compatible with the comultiplication of A (that is, the covariant first order differential calculi, as defined by Woronowicz, on a quantum homogeneous space) are…

We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second…

In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type…

In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of…

This work presents a universal accelerated first-order primal-dual method for affinely constrained convex optimization problems. It can handle both Lipschitz and H\"{o}lder gradients but does not need to know the smoothness level of the…

In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both…

We propose a level proximal subdifferential for a proper lower semicontinuous function. Level proximal subdifferential is a uniform refinement of the well-known proximal subdifferential, and has the pleasant feature that its resolvent…

We show that the optimal complexity of Nesterov's smooth first-order optimization algorithm is preserved when the gradient is only computed up to a small, uniformly bounded error. In applications of this method to semidefinite programs,…

The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this method are mainly derived for Lipschitz continuous objective functions. In this…

Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in…

We give a new proof of the boundedness of bilinear Schur multipliers of second order divided difference functions, as obtained earlier by Potapov, Skripka and Sukochev in their proof of Koplienko's conjecture on the existence of higher…

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…