Optimization under uncertainty and risk: Quadratic and copositive approaches

Robust optimization and stochastic optimization are the two main paradigms for dealing with the uncertainty inherent in almost all real-world optimization problems. The core principle of robust optimization is the introduction of parameterized families of constraints. Sometimes, these complicated semi-infinite constraints can be reduced to finitely many convex constraints, so that the resulting optimization problem can be solved using standard procedures. Hence flexibility of robust optimization is limited by certain convexity requirements on various objects. However, a recent strain of literature has sought to expand applicability of robust optimization by lifting variables to a properly chosen matrix space. Doing so allows to handle situations where convexity requirements are not met immediately, but rather intermediately. In the domain of (possibly nonconvex) quadratic optimization, the principles of copositive optimization act as a bridge leading to recovery of the desired convex structures. Copositive optimization has established itself as a powerful paradigm for tackling a wide range of quadratically constrained quadratic optimization problems, reformulating them into linear convex-conic optimization problems involving only linear constraints and objective, plus constraints forcing membership to some matrix cones, which can be thought of as generalizations of the positive-semidefinite matrix cone. These reformulations enable application of powerful optimization techniques, most notably convex duality, to problems which, in their original form, are highly nonconvex. In this text we want to offer readers an introduction and tutorial on these principles of copositive optimization, and to provide a review and outlook of the literature that applies these to optimization problems involving uncertainty

Functional and Psychosocial Outcomes of Hand Transplantation Compared with Prosthetic Fitting in Below-Elbow Amputees:A Multicenter Cohort Study

BACKGROUND:Hand-transplantation and improvements in the field of prostheses opened new frontiers in restoring hand function in below-elbow amputees. Both concepts aim at restoring reliable hand function, however, the indications, advantages and limitations for each treatment must be carefully considered depending on level and extent of amputation. Here we report our findings of a multi-center cohort study comparing hand function and quality-of-life of people with transplanted versus prosthetic hands. METHODS:Hand function in amputees with either transplant or prostheses was tested with Action Research Arm Test (ARAT), Southampton Hand Assessment Procedure (SHAP) and the Disabilities of the Arm, Shoulder and Hand measure (DASH). Quality-of-life was compared with the Short-Form 36 (SF-36). RESULTS:Transplanted patients (n = 5) achieved a mean ARAT score of 40.86 ± 8.07 and an average SHAP score of 75.00 ± 11.06. Prosthetic patients (n = 7) achieved a mean ARAT score of 39.00 ± 3.61 and an average SHAP score of 75.43 ± 10.81. There was no significant difference between transplanted and prosthetic hands in ARAT, SHAP or DASH. While quality-of-life metrics were equivocal for four scales of the SF-36, transplanted patients reported significantly higher scores in "role-physical" (p = 0.006), "vitality" (p = 0.008), "role-emotional" (p = 0.035) and "mental-health" (p = 0.003). CONCLUSIONS:The indications for hand transplantation or prosthetic fitting in below-elbow amputees require careful consideration. As functional outcomes were not significantly different between groups, patient's best interests and the route of least harm should guide treatment. Due to the immunosuppressive side-effects, the indication for allotransplantation must still be restrictive, the best being bilateral amputees

Sparse Conic Reformulation of Structured QCQPs based on Copositive Optimization with Applications in Stochastic Optimization

In an effort to develop a new alternative approach, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is similar to Burer's reformulation, but lifts the variables into a comparatively lower dimensional space. The reformulation rests on a generalization of the set-completely positive matrix cone. This cone can then be approximated via inner and out approximations in order to obtain upper and lower bounds, which potentially close the optimality gap, and hence can give a certificate of exactness for these sparse reformulations outside of traditional, known sufficient conditions. Finally, we provide some numerical experiments, where we asses the quality of the inner and outer approximations, thereby showing that the approximations may indeed close the optimality gap in interesting cases.Comment: 31 Page

Experimental Hydraulic Investigation of Angled Fish Protection Systems—Comparison of Circular Bars and Cables

The requirements for fish protection at hydro power plants have led to a significant decrease of the bar spacing at trash racks as well as the need of an inclined or angled design to improve the guidance effect (fish-friendly trash racks). The flexible fish fence (FFF) is a new developed fish protection and guidance system, created by horizontally arranged steel cables instead of bars. The presented study investigated experimentally the head loss coefficient of an angled horizontal trash rack with circular bars (CBTR) and the FFF with identical cross sections in a flume (scale 1:2). Nine configurations of different bar and cable spacing (blockage ratio) and rack angles were studied for CBTR and FFF considering six different stationary flow conditions. The results demonstrate that head loss coefficient is independent from the studied Bar−Reynolds number range and increases with increasing blockage ratio and angle. At an angle of 30 degrees, a direct comparison between the two different rack options was conducted to investigate the effect of cable vibrations. At the lowest blockage ratio, head loss for both options are in similar very low ranges, while the head loss coefficient of the FFF increases significantly compared to the CBTR with an increase of blockage. Further, the results indicate a moderate overestimation with the predicted head loss by common head loss equations developed for inclined vertical trash racks. Thus, an adaption of the design equation is proposed to improve the estimation of head loss on both rack options

Submerged Wall Instead of a Penstock Shutoff Valve—Alternative Protection as Part of a Refurbishment

Hydropower is an important source of renewable energy. Due to ageing infrastructure, more and more existing hydropower plants have to be refurbished and modernised. This includes a complete review of the design parameters as well as the change of specific parts. Investments should be targeted to improve the overall performance of hydropower plants and ensure a long lasting life extension. This paper presents the concept of the submerged wall as a local high point in the headrace tunnel, which can—in combination with the intake gates—replace existing penstock shutoff valves. Such a replacement was conducted for the hydropower plant Schneiderau in Austria, which also allowed us to prove the concept based on measurements including a simulated break of the penstock. The presented solution can help to reduce investment costs and also minimise maintenance efforts and therefore is an attractive option for classic penstock shutoff valves for comparable projects

Adaptation of an Existing Intake Structure Caused by Increased Sediment Level

An unexpected and massive redistribution of fine sediment in a large Alpine reservoir was triggered by a further lowering of the water level to conduct maintenance work. This caused the need of a total redesign of the existing head race intake for a high head power plant in the Austrian Alps. Two main geometry options for the trash rack support structure are compared with numerical simulations (ANSYS-CFX) as well as with a scale model test (scale 1:20). The laboratory experiment substantially benefited from the preliminary numerical investigation in respect of the location and implementation of the model boundaries. In return was the confidence in the numerics strengthened by the successful validation of the local head loss and the velocity distribution for the main operation cases. This allowed that the main inputs for the structural design and the further optimisation is conducted only with the 3D-numerical tool. The paper presents the interlaced concept as well as the main finding of the investigation, which lead to a successful adaptation of the intake structure

3D-numerische Modellansätze für die Berechnung von Lawineneinstößen in Speicher

Speicherbecken im Pumpbetrieb oder Beschneiungsspeicher können in Zeiträumen großer Lawinengefahr einen hohen Wasserstand aufweisen. In Hinblick auf eine Gefahrenbetrachtung und Risikoabschätzung ist deshalb auch ein möglicher Einstoß einer Lawine in einen gefüllten Speicher zu untersuchen. Der Einstoß einer großen Lawine kann dabei eine Impulswelle erzeugen, welche eine Gefährdung für den Uferbereich, das Absperrbauwerk und in weiterer Folge auch für die Unterlieger darstellt. Um den Effekt abschätzen und mögliche Gegenmaßnahmen (Speicherabsenkung oder bauliche Maßnahmen) überprüfen zu können, haben sich Modellversuche bewährt. Vermehrt kommen aber auch numerische Methoden zum Einsatz, die von allgemein gültigen Berechnungstabellen bis hin zu projektspezifischen 3D-numerischen Berechnungen reichen können. Der Beitrag fasst die Entwicklungen bei der Modellbildung für die Randbedingung und Erzeugung dieser Impulswelle für die 3D-numerische Software FLOW-3D zusammen. Diese umfassen die Definition von (i) Partikelgemischen, (ii) dynamischen Festkörpern und (iii) ausschließlich aus Wasser bestehenden Fluidkörpern in der Sturzbahn der Lawine. Mögliche Konzepte sowie deren Vor- und Nachteile werden aufgezeigt. Zudem wird die Möglichkeit der Modifikation der „Wasserlawine“ durch Modifikation der Startdichte vorgestellt.Avalanche impacts into reservoirs can cause an impulse wave which endanger the area around the reservoir, the dam itself and as well the area located downstream of the reservoir. Especially reservoirs for artificial snow production or pump storage can be filled up to high water levels in periods with significant avalanche hazard potential. The investigation of possible impulse wave scenarios is part of the global risk analysis and can lead to a limitation of the maximum permitted water level in the reservoir under specific weather condition or structural measures. To quantify such an impulse wave, general equations, scale model tests or numerical simulations can be applied. Within the present paper focus is put on different approaches to model an avalanche impact into a reservoir and the corresponding impulse wave generation by use of the 3D-numerical simulation tool FLOW-3D. Different modelling concepts are presented and discussed, consisting of (i) particle-water-mixtures, (ii) moving objects and (iii) pure water avalanches. In addition to this review on the development, preliminary results of the advanced water avalanche are compared to the equations provided by the ETH Zürich. Therefore, the possibility to define a different density for each initial fluid region is used and two parameters (initial velocity of the avalanche at the start and the adapted density of the model avalanche) are varied.(VLID)456579