We investigate the convergence of subgradient-oriented descent methods in non-smooth non-convex optimization. We prove convergence in the sense of subsequences for functions with a strict standard model, and we show that convergence to a single critical point may be guaranteed if the Kurdyka–Łojasiewicz inequality is satisfied. We show, by way of an example, that the Kurdyka–Łojasiewicz inequality alone is not sufficient to prove the convergence to critical points.

/pdf/convergence-of-non-smooth-descent-methods-using-the-kurdyka-4oxwcsm2cv.pdf

Convergence of Non-smooth Descent Methods Using the Kurdyka–Łojasiewicz Inequality

We discuss a bundle method to minimize locally Lipschitz functions which are both nonconvex and nonsmooth. We analyze situations where only inexact subgradients or function values are available. For suitable classes of such non-smooth functions we prove convergence of our algorithm to approximate critical points.

/pdf/bundle-method-for-non-convex-minimization-with-inexact-3dscrk7vol.pdf

Bundle Method for Non-Convex Minimization with Inexact Subgradients and Function Values

The H∞ control problem was posed by G. Zames in 1981 [1], and various attempts to address it had been made over the years. Ultimately, in 2006, the authors presented their solution, which is based on a tailored non-smooth optimization technique [2]. In this treatise we present the rationale of H∞ control, give a brief history, and recall the milestones reached before our 2006 solution. We clarify why our novel approach is welcomed in the high-tech and aerospace industry. Recent MATLAB functions, hinfstruct and systune, based on work by Apkarian, Noll and Gahinet (The MathWorks) are available in the Robust Control Toolbox, since R2010b and R2012b respectively.

https://hal.archives-ouvertes.fr/hal-01653161/document

The H∞ Control Problem is Solved

Delamination is a typical failure mode of composite materials caused by weak bonding. It arises when a crack initiates and propagates under a destructive loading. Given the physical law characterizing the properties of the interlayer adhesive between the bonded bodies, we consider the problem of computing the propagation of the crack front and the stress field along the contact boundary. This leads to a hemivariational inequality, which after discretization by finite elements we solve by a nonconvex bundle method, where upper-$C^1$ criteria have to be minimized. As this is in contrast with other classes of mechanical problems with non-monotone friction laws and in other applied fields, where criteria are typically lower-$C^1$, we propose a bundle method suited for both types of nonsmoothness. We prove its global convergence in the sense of subsequences and test it on a typical delamination problem of material sciences.

Nonconvex bundle method with application to a delamination problem

Proximal bundle method has usually been presented for unconstrained convex optimization problems. In this paper, we develop an infeasible proximal bundle method for nonsmooth nonconvex constrained optimization problems. Using the improvement function we transform the problem into an unconstrained one and then we build a cutting plane model. The resulting algorithm allows effective control of the size of quadratic programming subproblems via the aggregation techniques. The novelty in our approach is that the objective and constraint functions can be any arbitrary (regular) locally Lipschitz functions. In addition the global convergence, starting from any point, is proved in the sense that every accumulation point of the iterative sequence is stationary for the improvement function. At the end, some encouraging numerical results with a MATLAB implementation are also reported.

A new infeasible proximal bundle algorithm for nonsmooth nonconvex constrained optimization

We design a feedback control architecture for longitudinal flight of an aircraft. The multi-level architecture includes the flight control loop to govern the short-term dynamics of the aircraft, and the autopilot to control the long-term modes. Using $$H_\infty $$
 performance and robustness criteria, the problem is cast as a non-convex and non-smooth optimization program. We present a non-convex bundle method, prove its convergence, and show that it is apt to solve the longitudinal flight control problem.

/pdf/design-of-a-flight-control-architecture-using-a-non-convex-47lk77o9s5.pdf

Design of a flight control architecture using a non-convex bundle method

Structured flight control law design using non-smooth optimization

Nous etendons les concepts sous-jacents et fondateurs de la methode des faisceaux afin d'aborder des problemes d’optimisation non convexes et non lisses Ce type de problemes survient par exemple lors de la conception d'une loi de controle par retour de sortie Cette nouvelle approche est ensuite appliquee a la synthese de lois de commande de vol longitudinal d'un avion de transport Cet
exemple met en evidence le besoin et l'interet de disposer
d'un outil d'optimisation de lois de commande structurees
tenant compte de diverses contraintes frequentielles sur des bandes de frequences limitees

Marion Gabarrou

Papers

Design of a flight control architecture using a non-convex bundle method

Structured flight control law design using non-smooth optimization

Optimisation non lisse appliquée à la synthèse de lois de commande de vol structurées