Uncontrolled inexact information within bundle methods

International audienceWe consider convex nonsmooth optimization problems where additional information with uncontrolled accuracy is readily available. It is often the case when the objective function is itself the output of an optimization solver, as for large-scale energy optimization problems tackled by decomposition. In this paper, we study how to incorporate the uncontrolled linearizations into (proximal and level) bundle algorithms in view of generating better iterates and possibly accelerating the methods. We provide the convergence analysis of the algorithms using uncontrolled linearizations, and we present numerical illustrations showing they indeed speed up resolution of two stochastic optimization problems coming from energy optimization (two-stage linear problems and chance-constrained problems in reservoir management)

Inexact Model: A Framework for Optimization and Variational Inequalities

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities.Comment: 41 page

Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

Valuing intellectual property and calculating infringement damages : a nonauthoritative guide; Consulting services practice aid, 99-2

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International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

On Breakdown Criteria for Nonvacuum Einstein Equations

The recent "breakdown criterion" result of S. Klainerman and I. Rodnianski stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the author's Ph.D. thesis. In this paper, we state the main results of the thesis, and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature and the coupling between the Einstein and the field equations.Comment: 62 pages This version: corrected minor typos, expanded Section 6 (geometry of null cones

Active chatter control in high-speed milling processes

In present day manufacturing industry, an increasing demand for highprecision products at a high productivity level is seen. High-speed milling is a manufacturing technique which is commonly exploited to produce highprecision parts at a high productivity level for the aeroplane, automotive and mould and dies industry. The performance of a manufacturing process such as high-speed milling, indicated by the material removal rate, is limited by the occurrence of a dynamic instability phenomenon called chatter. The occurrence of chatter results in an inferior workpiece quality due to heavy vibrations of the cutter. Moreover, a high level of noise is produced and the tool wears out rapidly. Although different types of chatter exist, regenerative chatter is recognised as the most prevalent type of chatter. The occurrence of (regenerative) chatter has such a devastating effect on workpiece quality and tool wear that it should be avoidedat all times. The occurrence of chatter can be visualised in so-called stability lobes diagrams (sld). In an sld the chatter stability boundary between a stable cut (i.e. without chatter) and an unstable cut (i.e. with chatter) is visualised in terms of spindle speed and depth of cut. Using the information gathered in a sld, the machinist can select a chatter free operating point. In this thesis two problems are tackled. Firstly, due to e.g. heating of the spindle, tool wear, etc., the sld may vary in time. Consequently, a stable working point that was originally chosen by the machinist may become unstable. This requires a (controlled) adaptation of process parameters such that stability of the milling process is ensured (i.e. chatter is avoided) even under such changing process conditions. Secondly, the ever increasing demand for high-precision products at a high productivity level requires dedicated shaping of the chatter stability boundary. Such shaping of the sld should render working points (in terms of spindle speed and depth of cut) of high productivity feasible, while avoiding chatter. These problems require the design of dedicated control strategies that ensure stable high-speed milling operations with increased performance. In this work, two chatter control strategies are developed that guarantee high-speed chatter-free machining operations. The goal of the two chatter control strategies is, however, different. The first chatter control strategy guarantees chatter-free high-speed milling operations by automatic adaptation of spindle speed and feed (i.e. the feed is not stopped during the spindle speed transition). In this way, the high-speed milling process will remain stable despite changes in the process, e.g. due to heating of the spindle, tool wear, etc. To do so, an accurate and fast chatter detection algorithm is presented which predicts the occurrence of chatter before chatter marks are visible on the workpiece. Once the onset of chatter is detected, the developed controller adapts the spindle speed and feed such that a new chatter-free working point is attained. Experimental results confirm that by using this control strategy chatter-free machining is ensured. It is also shown experimentally that the detection algorithm is able to detect chatter before it is fully developed. Furthermore, the control strategy ensures that chatter is avoided, thereby ensuring a robust machining operation and a high surface quality. The second chatter control strategy is developed to design controllers that guarantee chatter-free cutting operations in an a priori defined range of process parameters (spindle speed and depth of cut) such that a higher productivity can be attained. Current (active) chatter control strategies for the milling process cannot provide such a strong guarantee of a priori stability for a predefined range of working points. The methodology is based on a robust control approach using µ-synthesis, where the most important process parameters (spindle speed and depth of cut) are treated as uncertainties. The proposed methodology will allow the machinist to define a desired working range (in spindle speed and depth of cut) and lift the sld locally in a dedicated fashion. Finally, experiments have been performed to validate the working principle of the active chatter control strategy in practice. Hereto, a milling spindle with an integrated active magnetic bearing is considered. Based on the obtained experimental results, it can be stated that the active chatter control methodology, as presented in this thesis, can indeed be applied to design controllers, which alter the sld such that a pre-defined domain of working points is stabilised. Results from milling tests underline this conclusion. By using the active chatter controller working points with a higher material removal rate become feasible while avoiding chatter. To summarise, the control strategies developed in this thesis, ensure robust chatter-free high-speed milling operations where, by dedicated shaping of the chatter stability boundary, working points with a higher productivity are attained

Discrimination of Dynamical System Models for Biological and Chemical Processes

In technical chemistry, systems biology and biotechnology, the construction of predictive models has become an essential step in process design and product optimization. Accurate modelling of the reactions requires detailed knowledge about the processes involved. However, when concerned with the development of new products and production techniques for example, this knowledge often is not available due to the lack of experimental data. Thus, when one has to work with a selection of proposed models, the main tasks of early development is to discriminate these models. In this article, a new statistical approach to model discrimination is described that ranks models wrt. the probability with which they reproduce the given data. The article introduces the new approach, discusses its statistical background, presents numerical techniques for its implementation and illustrates the application to examples from biokinetics