The optimal value function is one of the basic objects in the field of mathematical optimization, as it allows the evaluation of the variations in the cost/revenue generated while minimizing/maximizing a given function under some constraints. In the context of stability/sensitivity analysis, a large number of publications have been dedicated to the study of continuity and differentiability prop...

Motivated by fatigue damage models, this paper addresses optimal control problems governed a non-smooth system featuring two non-differentiable mappings. This consists of coupling between doubly history-dependent evolution and an elliptic PDE. After proving the directional differentiability associated solution mapping, optimality which is stronger than one obtained classical smoothening procedu...

We examine the formulae for the first and second derivatives of spectral functions of real symmetric matrices to understand why they have the form that they do. The development, which involves only elementary linear algebra and calculus, is based on computing directional derivatives of the spectral function and its gradient along a canonical set of orthonormal matrices. A particular aim of our ...

We investigate the problem of estimating derivatives of expected steady-state performance measures in parametric systems. Unlike most of the existing work in the area, we allow those functions to be nonsmooth and study the estimation of directional derivatives. For the class of regenerative Markovian systems we provide conditions under which we can obtain consistent estimators of those directio...

We consider convex optimization problems with kth order stochastic dominance constraints for k ≥ 2. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets of the optimization problems in terms of a suitably selected probability metric. Moreover, we provide conditions ensuring Hadamar...

In the traditional approach to differentiability, found in almost all university textbooks, this notion is considered only for interior points of domain function or functions with an open domain. This leads fact that differentiability has usually been Rn, which severely limits possibility applying potential techniques and tools differential calculus a broader class functions. Although there gre...

This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models stationary incompressible shear-thickening fluid. We start studying directional differentiability non-smooth term within state as prior step to demonstrate solution operator. Thereafter, we establish primal first order necessary optimality condition (Bou...

We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with denoising lower-level constraints. For problem, we able to derive M-stationarity conditions, after characterizing corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions....