semialgebraic – Optimization Online

Splitting Methods for Nonconvex Optimisation: Convergence and Saddle Point Avoidance Through Shadow Sequences

Published: 2024/01/18, Updated: 2026/09/29

Felipe Atenas

Generalized Convexity/Monoticity, Nonsmooth Optimization Davis-Yin splitting, douglas-rachford splitting, moreau envelope, nonconvex optimisation, semialgebraic, strict saddle, weak convexity

In nonconvex optimisation, commonly used methods are usually shown to converge to stationary points, while guarantees of convergence to local minimisers have comparatively received less attention. To close this gap, we analyse the convergence behaviour of the Davis–Yin three-operator splitting method. More precisely, for structured weakly convex semialgebraic optimisation problems, we establish convergence to local … Read more

A nearly linearly convergent first-order method for nonsmooth functions with quadratic growth

Published: 2022/04/29, Updated: 2022/05/04

Convex and Nonsmooth Optimization active manifold, decomposable, Goldstein, Kurdyka-Lojasiewicz, max-of-smooth, nonsmooth, parameter-free, quadratic growth, semialgebraic, subgradient method

Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro’s decomposable … Read more

Stochastic subgradient method converges on tame functions

Published: 2018/05/25

Nonsmooth Optimization differential inclusion, lyapunov function, semialgebraic, stochastic subgradient method, subdifferential

This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces limit points that are all first-order stationary. More generally, our result applies to any function with a Whitney stratifiable graph. … Read more

Continuity of set-valued maps revisited in the light of tame geometry

Published: 2009/05/04

Nonsmooth Optimization aubin property, inner) continuity, outer, piecewise polyhedral, semialgebraic, set-valued map, tame optimization

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued … Read more