This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability ...

A B S T RAe T This paper considers the problem of minimizing a convex-differentiable function subject to convex differentiable constraints. Necessary and sufficient conditions (not requiring any constraints qualification) for a point to be an optimal solution are given in terms of a parametric I inear program. Dual characterization theorems are then derived, which generalizes the classical resu...

The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally differentiable Lagrangians. Here, we formulate a Legendre condition and an extension of the classical theorem of Emmy Noether, thus obtaining first integrals for ...

An algorithm is described for finding the minimum of any convex, not necessarily differentiable, function f of several variables. The algorithm yields a sequence of points tending to the solution of the problem, if any, requiring only the calculation o f f and one subgradient o f f at designated points. Its rate of convergence is estimated for convex and for differentiable convex functions. For...

We present a differentiable formulation of rigid-body contact dynamics for objects and robots represented as compositions convex primitives. Classical physics engines rely on non-differentiable collision detection modules. More recent optimization-based approaches simulating between primitives bilevel that separates simulation. These latest are unreliable in realistic simulation scenarios becau...

We firstly establish an identity for $n$ time differentiable mappings Then, a new inequality for $n$ times differentiable functions is deduced. Finally, some perturbed Ostrowski type inequalities for functions whose $n$th derivatives are of bounded variation are obtained.

The aim of present paper is to study a constrained programming with generalized $alpha-$univex fuzzy mappings. In this paper we introduce the concepts of $alpha-$univex, $alpha-$preunivex, pseudo $alpha-$univex and $alpha-$unicave fuzzy mappings, and we discover that $alpha-$univex fuzzy mappings are more general than univex fuzzy mappings. Then, we discuss the relationships of generalized $alp...

This paper addresses the minimum sum-of-squares clustering problem. 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 solution method proposed adopts a smoothing strategy using a special C ∞ differentiable cla...

This article considers the minimum sum of distances clustering (MSDC) problem, where the distances are measured through the L1 or Manhattan metric. 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 proposed r...

Imbed a n w 1 sphere in an n sphere, and the complement is divided into two components. I t seems that the closure of each of the resulting components should be a topological w-cell. This statement isn't true. The classical counterexample (in dimension 3) is the Alexander Horned Sphere. I t was conjectured, however, that if one restricts one's attention to some class of well-behaved imbeddings,...