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We consider linear optimization over a nonempty convex semialgebraic 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 the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F. [ABSTRACT FROM AUTHOR]

Given a real-analytic function f : ℝn → ℝ and a critical point a ∈ ℝn, the Łojasiewicz inequality asserts that there exists θ ∈ [½ , 1) such that the function |f - f(a)|θ||∇f||-1 remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value +∞), by establishing an analogous inequality in which the derivative ∇f(x) can be replaced by any element x* of the subdifferential ∂f(x) of f. Like its smooth version, this result provides new insights into the convergence aspects of subgradient-type dynamical systems. Provided that the function f is sufficiently regular (for instance, convex or lower-C²), the bounded trajectories of the corresponding subgradient dynamical system can be shown to be of finite length. Explicit estimates of the rate of convergence are also derived. [ABSTRACT FROM AUTHOR]