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We establish the following result: if the graph of a (nonsmooth) real-extended-valued function f : R n → R ∪ {+∞} is closed and admits a Whitney stratification, then the norm of the gradient of f at x ∈ dom f relative to the stratum containing x bounds from below all norms of Clarke subgradients of f at x. As a consequence, we obtain some Morse-Sard type(More)
The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry.(More)
We consider the separation problem for sets X that are inverse images of a given set S by a linear mapping. Classical examples occur in integer programming, complementarity problems and other optimization problems. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. Formulas for such(More)
In this work we prove that every locally symmetric smooth submanifold M of R n gives rise to a naturally defined smooth submanifold of the space of n × n symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to M. We also present an explicit formula for the dimension of the spectral manifold in(More)
In this note, we prove the equivalence, under appropriate conditions, between several dynamical formalisms: projected dynamical systems, two types of differential inclusions, and a class of complementarity dynamical systems. Each of these dynamical systems can also be considered as a hybrid dynamical system. This work both generalizes previous results and(More)
We consider linear optimization over a nonempty convex semi-algebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique " active " manifold, around which F is " partly smooth " , and the second-order sufficient conditions hold. Perturbing(More)
In optimization problems such as integer programs or their relaxations, one encounters feasible regions of the form {x ∈ R n + : Rx ∈ S} where R is a general real matrix and S ⊂ R q is a specific closed set with 0 / ∈ S. For example, in a relaxation of integer programs introduced in [ALWW2007], S is of the form Z q − b where b ∈ Z q. One would like to(More)