arXiv subjects
Publications and source records attributed to Dominikus Noll.
We investigate convergence of the expectation maximization algorithm by representing it as a generalized proximal method. Convergence of iterates and not just in value is investigated under natural hypotheses such as definability of the incomplete data log-likelihood in the sense of o-minimal structure theory.
We investigate convergence of alternating Bregman projections between non-convex sets and prove convergence to a point in the intersection, or to points realizing a gap between the two sets. The speed of convergence is generally sub-linear, but may be linear under transversality. We apply our analysis to prove convergence of versions of the expectation maximization algorithm for non-convex parameter sets.
We propose a vector linear programming formulation for a non-stationary, finite-horizon Markov decision process with vector-valued rewards. Pareto efficient policies are shown to correspond to efficient solutions of the linear program, and vector linear programming theory allows us to fully characterize deterministic efficient policies. An algorithm for enumerating all efficient deterministic policies is presented then tested numerically in an engineering application.
We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed graph theorem, and we compare closed graph and open mapping spaces.
We introduce system norms which assess transient behavior of stable Linear Time-Invariant (LTI) systems. This allows us to address undesired responses to initial conditions, finite resource consumption signals, or persistent perturbations. We then consider the challenging problem of minimizing these norms in closed loop using structured linear feedback. The computed controllers mitigate transients in a linearized closed loop, with the potential side effect of enlarging the region of stability of the underlying non-linear controlled system. In applications this helps to prevent transition to undesired nonlinear regimes, limit cycles or chaotic behavior. The success of our approach is certified a posteriori using Lyapunov-like techniques and simulations, as we demonstrate through a variety of applications.
We discuss strategies to bring $H_\infty$-control techniques into play when the system dynamics are modeled by hyperbolic partial differential equations, or more generally, by systems with non-sectorial pole pattern.
We discuss a variant of the Banach-Mazur game which has applications to topological open mapping and closed graph theorems.
We consider stabilization and performance optimization of non-linear controlled systems, where the non-linearity satisfies a sector constraint asymptotically. This leads to optimization of the closed loop peak-to-peak system norm subject to $H_\infty$-performance constraints. Non-linear controlled systems tuned successfully by this novel approach are locally exponentially stable and globally BIBO-stable.
We consider convergence of alternating projections between non-convex sets and obtain applications to convergence of the Gerchberg-Saxton error reduction method, of the Gaussian expectation-maximization algorithm, and of Cadzow's algorithm.
We discuss boundary control of a wave equation with a non-linear anti-damping boundary condition. We design structured finite-dimensional $H_\infty$-output feedback controllers which stabilize the infinite dimensional system exponentially in closed loop. The method is applied to control torsional vibrations in drilling systems with the goal to avoid slip-stick.
We present a frequency domain based $H_\infty$-control strategy to solve boundary control problems for systems governed by parabolic or hyperbolic partial differential equation, where controllers are constrained to be physically implementable and of simple structure suited for practical applications. The efficiency of our technique is demonstrated by controlling a reaction-diffusion equation with input delay, and a wave equation with boundary anti-damping.
We prove global convergence of a bundle trust region algorithm for non-smooth non-convex optimization, where cutting planes are generated by oracles respecting four basic rules. The benefit is that convergence theory applies to a large variety of methods encountered in practice. This includes in particular the method of downshifted tangents, for which previously no convergence result in the trust region framework was known. We also show that certain splitting techniques can be seen as special cases of bundle trust region techniques.
The Douglas-Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater's condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the Douglas-Rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection-projection.
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.
Many iterative methods for solving optimization or feasibility problems have been invented, and often convergence of the iterates to some solution is proven. Under favourable conditions, one might have additional bounds on the distance of the iterate to the solution leading thus to worst case estimates, i.e., how fast the algorithm must converge. Exact convergence estimates are typically hard to come by. In this paper, we consider the complementary problem of finding best case estimates, i.e., how slow the algorithm has to converge, and we also study exact asymptotic rates of convergence. Our investigation focuses on convex feasibility in the Euclidean plane, where one set is the real axis while the other is the epigraph of a convex function. This case study allows us to obtain various convergence rate results. We focus on the popular method of alternating projections and the Douglas-Rachford algorithm. These methods are connected to the proximal point algorithm which is also discussed. Our findings suggest that the Douglas-Rachford algorithm outperforms the method of alternating projections in the absence of constraint qualifications. Various examples illustrate the theory.
The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets $A,B$ under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is O$(k^{-ρ})$ for some $ρ\in (0,\infty)$.
We introduce regularity notions for averaged nonexpansive operators. Combined with regularity notions of their fixed point sets, we obtain linear and strong convergence results for quasicyclic, cyclic, and random iterations. New convergence results on the Borwein-Tam method (BTM) and on the cylically anchored Douglas-Rachford algorithm (CADRA) are also presented. Finally, we provide a numerical comparison of BTM, CADRA and the classical method of cyclic projections for solving convex feasibility problems.
We present a mathematical model for solvated crystallization of alpha -lactose monohydrate in semi-batch mode. The process dynamics are governed by conservation laws including population, molar and energy balance equations. We present and discuss the model and then control the process with the goal to privilege the production of small particles in specific the range. We compare several specific and unspecific cost functions leading to optimal strategies with significantly different effects on product quality. Control inputs are temperature, feed rate, and the choice of an appropriate crystal seed.