We consider the problem of approximating the singular value decomposition of a bidiagonal matrix by a one-parameter family of differentiable matrix flows. It is shown that this approach can be fully expressed as an autonomous, homogeneous, and cubic dynamical system. Asymptotic behavior is justified by the established theory of the Toda lattice.

The minimum sum-of-squares clustering problem is considered. The mathematical modeling of this problem leads to a min− sum−min formulation which, in addition to its intrinsic bi-level nature, has the significant characteristic of being strongly nondifferentiable. To overcome these difficulties, the resolution method proposed adopts a smoothing strategy using a special C differentiable class fun...

Model interpretability has become an important problem in machine learning (ML) due to the increased effect algorithmic decisions have on humans. Counterfactual explanations can help users understand not only why ML models make certain decisions, but also how these be changed. We frame of finding counterfactual as optimization task and extend previous work that could applied differentiable mode...

Let the cardinal invariant sn denote the least number of continuously smooth n-dimensional surfaces into which (n+1)-dimensional Euclidean space can be decomposed. It will be shown to be consistent that sn is greater than sn+1. These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.

Abstract: We consider a semi-infinite optimization problem in Banach spaces, where both the objective functional and the constraint operator are compositions of convex nonsmooth mappings and differentiable mappings. We derive necessary optimality conditions for these problems. Finally, we apply these results to nonconvex stochastic optimization problems with stochastic dominance constraints, ge...

In this paper, we establish first-order optimality conditions for the problem of minimizing a function f on the solution set of an inclusion 0 ∈ F (x) where f and the support function of a set-valued mapping F are epi-differentiable at x.

This paper studies the problem of obtaining complexifications of a differentiable manifold which have desirable analytic or algebraic properties and which are minimal in the sense described below. It is seen that there is a significant difference between analytic and algebraic complexifications.

In this paper, we establish the existence of at least two distinct weak solutions for some singular elliptic problems involving a p-biharmonic operator, subject to Navier boundary conditions in a smooth bounded domain in RN . A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least two distinct nontrivial weak solutions.

In this paper the differentiable structure of the Shannon wavelets is defined and the projection of a linear differential operators is given for any order. As application, the wavelet solution of a heat propagation problem is computed and the contribution of the different scale components is explicitly shown.

This paper develops a genetic algorithm based multivariable system identification technique from plant step response data directly. Using this technique, globally optimised models for linear and nonlinear systems can be identified without the need of a differentiable cost function or linearly separable parameters. Results are validated against a benchmark identification problem and a laboratory...