Nonsmooth Bundle Trust-region Algorithm with Applications to Robust Stability

Abstract

We propose a bundle trust-region algorithm to minimize locally Lipschitz functions which are potentially nonsmooth and nonconvex. We prove global convergence of our method and show by way of an example that the classical convergence argument in trust-region methods based on the Cauchy point fails in the nonsmooth setting. Our method is tested experimentally on three problems in automatic control.The question is then whether there are more restrictive classes of nonsmooth functions, where the Cauchy point can be salvaged. In response we show that the classical trust-region strategy with Cauchy point is still valid for upper C 1 -functions, and at least partially, for functions having a strict standard model. It turns out that several problems in control and in contact mechanics are in this class, which justifies the disquisition. Nonetheless, the class of functions where the Cauchy point works remains exceptional in the nonsmooth framework, which is corroborated by the fact that it does not include nonsmooth convex functions.A strong incentive for the present work comes indeed from applications in automatic control. In the experimental part we will apply our novel bundle trust-region method to compute locally optimal solutions to three NP-hard problems in the theory of systems with uncertain parameters. This includes (i) computing the worst-case H ∞ -norm of a system over a given uncertain parameter range, (ii) checking robust stability of an uncertain system over a given parameter range, and (iii) computing the distance to instability of a nominally stable system with uncertain parameters. In these applications the versatility of the bundle trust-region approach with regard to the choice of the norm is exploited.Nonsmooth trust-region methods which do not include the possibility of bundling are more common, see for instance Dennis et al. [17], where the authors present an axiomatic approach, and [13, Chap. 11], where that idea is further expanded. A recent trust-region method for DC-functions is [26].The structure of the paper is as follows. The algorithm is developed in section 2, and its global convergence is proved in section 3. Applications of the model approach are discussed in section 5, where we also discuss failure of the Cauchy point. Numerical experiments with three problems in automatic control are presented in section 6. NotationFor nonsmooth optimization we follow [12]. The Clarke directional derivative of f is

Computing the structured distance to instability

“…The proof is outside the scope of the present contribution and can be found in [3]. By an observation first made in [9], it is possible to say more when f is upper-C…”

“…Following classical lines in bundle methods, g k is called the aggregate subgradient. The convergence proof in [3] shows that if g j is the aggregate subgradient at acceptance of z k = x j+1 for the corresponding y k = x j+1 , then g j → 0. That leads to the following stopping test.…”

Branch and bound algorithm with applications to robust stability

“…We use a local optimization technique Trust based on a non-smooth bundling trust-region algorithm [2] to compute a locally optimal solution α of (5), which gives a lower bound α α * for (5). More details on Trust are given in Sect.…”

“…The fact that α computed by Trust is much more accurate gives our method an advantage for pruning. For a detailed analysis of the trust-region method we refer to [1][2][3]19,20]. For the current analysis it suffices to know that if Trust is started at an initial guess δ 0 , then it finds a locally optimal solution δ * such that α(A(δ * )) α(A(δ 0 )).…”

“…5) Let us now assume that the algorithm does not halt and creates a sequence α (n) of lower bounds with α (1) α (2) . .…”