Mathematics of Operations Research
A New Sequential Optimality Condition for Constrained Optimization and Algorithmic Consequences
SIAM Journal on Optimization
Łojasiewicz inequality and exponential convergence of the full-range model of CNNs
International Journal of Circuit Theory and Applications
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Given a real-analytic function $f:\mathbb{R}^{n} \rightarrow\mathbb{R}$ and a critical point $a \in \mathbb{R}^{n}$, theŁojasiewicz inequality asserts that there exists$\theta\in\lbrack\frac{1}{2},1)$ such that the function$|f-f(a)|^{\theta}\,\Vert\nabla f\Vert^{-1}$ remains bounded around$a$. In this paper, we extend the above result to a wide class ofnonsmooth functions (that possibly admit the value $+\infty$), byestablishing an analogous inequality in which the derivative$\nabla f(x)$ can be replaced by any element $x^{\ast}$ of thesubdifferential $\partial f(x)$ of $f$. Like its smooth version,this result provides new insights into the convergence aspects ofsubgradient-type dynamical systems. Provided that the function $f$is sufficiently regular (for instance, convex or lower-$C^{2}$),the bounded trajectories of the corresponding subgradient dynamicalsystem can be shown to be of finite length. Explicit estimates ofthe rate of convergence are also derived.