Cutting Plane Oracles to Minimize Non-smooth Non-convex Functions

Abstract

We discuss a bundle method for non-smooth non-convex optimization programs. In the absence of convexity, a substitute for the cutting plane mechanism has to be found. We propose such a mechanism and prove convergence of our method in the sense that every accumulation point of the sequence of serious iterates is critical.

Robust control via sequential semidefinite programming

“…An alternative way to address (17) uses a succession of structured H ∞ problems. Indeed, locally program (7) is equivalent to the following unconstrained structured H ∞ -program minimize max{ T w→z (K(θ)) ∞ , β T e w→e z (K(θ)) ∞ } subject to K(θ) closed-loop stabilizing (26) which for fixed β > 0 is a specific form of (5) if a suitable weighting is introduced in (4) and the performance channel is (w, w) → (z, z). We have the following Proposition 3.…”

“…We have the following Proposition 3. Programs (17), (19) and (26) are locally equivalent in the following sense. Let θ r be a KKT point of (17) where the constraint R(θ) ≤ r −1 is active.…”

“…Let θ r be a KKT point of (17) where the constraint R(θ) ≤ r −1 is active. Then θ r is a critical point for (26) for the value β(r) = rP(θ r ), and it is a KKT point for (19) for the value α(r) = P(θ r )/p 1 −1. If θ β is a critical point of (26) for parameter β, then it is also a KKT point for (17) with r(β) = (24)).…”

“…Every point corresponds to a run of (19) with different α and R. r HINFSTRUCT made available through the MATLAB R2010b Prerelease, Robust Control Toolbox Version 3.5 developed by The MathWorks, Inc., and based on nonsmooth H ∞ -synthesis according to [3]. Evolutions of this method are presented and discussed in [6,26,27].…”

“…As programs (17), (26) are nonconvex, we use local optimization methods, which in some rare cases may cause discontinuity or rupture of the correspondences θ r ↔ θ β ↔ θ α despite a seemingly continuous behavior β → r(β) or r → β(r), etc., because we may jump from one branch of local minima to another. This cannot be avoided unless global optimization techniques are applied, but those are usually prohibitively expensive.…”