The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F’(uJj* < l , i.e., its de...

The optimal control problem for the nonlinear elliptic equation is considered. The corresponding control-state mapping is not Gataux differentiable. Necessary conditions of optimality are obtained by means of the extended derivatives theory.

This work deals with shape optimization of an elastic body in sliding contact (Signorini) a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using classical iterative approach. For practical reasons we are interested applying the process respect to intermediate solution produced by method. Because projection operator involved at each it...

In natural language processing tasks performance of the models is often measured with some non-differentiable metric, such as BLEU score. To use efficient gradientbased methods for optimization, it is a common workaround to optimize some surrogate loss function. This approach is effective if optimization of such loss also results in improving target metric. The corresponding problem is referred...

We consider the minimization of a function G defined on R , which is the sum of a (non necessarily convex) differentiable function and a (non necessarily differentiable) convex function. Moreover, we assume that G satisfies the KurdykaLojasiewicz property. Such a problem can be solved with the Forward-Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose...

In this paper, we consider an optimization problem arising in multidimensional scaling with city-block distances. The objective function of this problem has many local minimum points and may be even non-differentiable at a minimum point. We reformulate the problem into a problem with convex quadratic objective function, linear and complementarity constraints. In addition, we propose an algorith...

The alternating direction method of multipliers (ADMM) is a versatile tool for solving a wide range of constrained optimization problems, with differentiable or non-differentiable objective functions. Unfortunately, its performance is highly sensitive to a penalty parameter, which makes ADMM often unreliable and hard to automate for a non-expert user. We tackle this weakness of ADMM by proposin...

Regular and singular Sturm-Liouville problems (SLP) are studied including the continuous and differentiable dependence of eigenvalues on the problem. Also initial value problems (IVP) are considered for the SL equation and for general first order systems.

A multidimesional function y(~r) defined by a sample of points {~ri, yi} is approximated by a differentiable function ỹ(~r). The problem is solved by using the GaussHermite folding method developed in the nuclear shell correction method by Strutinsky.

Considering that the movements of complex system entities take place on continuous, but non-differentiable, curves, concepts, like non-differentiable entropy, informational non-differentiable entropy and informational non-differentiable energy, are introduced. First of all, the dynamics equations of the complex system entities (Schrödinger-type or fractal hydrodynamic-type) are obtained. The la...