A Parallel Riccati Factorization Algorithm with Applications to Model Predictive Control

Model Predictive Control (MPC) is increasing in popularity in industry as more efficient algorithms for solving the related optimization problem are developed. The main computational bottle-neck in on-line MPC is often the computation of the search step direction, i.e. the Newton step, which is often done using generic sparsity exploiting algorithms or Riccati recursions. However, as parallel hardware is becoming increasingly popular the demand for efficient parallel algorithms for solving the Newton step is increasing. In this paper a tailored, non-iterative parallel algorithm for computing the Riccati factorization is presented. The algorithm exploits the special structure in the MPC problem, and when sufficiently many processing units are available, the complexity of the algorithm scales logarithmically in the prediction horizon. Computing the Newton step is the main computational bottle-neck in many MPC algorithms and the algorithm can significantly reduce the computation cost for popular state-of-the-art MPC algorithms

Reduced Memory Footprint in Multiparametric Quadratic Programming by Exploiting Low Rank Structure

In multiparametric programming an optimization problem which is dependent on a parameter vector is solved parametrically. In control, multiparametric quadratic programming (mp-QP) problems have become increasingly important since the optimization problem arising in Model Predictive Control (MPC) can be cast as an mp-QP problem, which is referred to as explicit MPC. One of the main limitations with mp-QP and explicit MPC is the amount of memory required to store the parametric solution and the critical regions. In this paper, a method for exploiting low rank structure in the parametric solution of an mp-QP problem in order to reduce the required memory is introduced. The method is based on ideas similar to what is done to exploit low rank modifications in generic QP solvers, but is here applied to mp-QP problems to save memory. The proposed method has been evaluated experimentally, and for some examples of relevant problems the relative memory reduction is an order of magnitude compared to storing the full parametric solution and critical regions

Low-Rank Modifications of Riccati Factorizations for Model Predictive Control

In Model Predictive Control (MPC) the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem at each sample in the control loop. The main computational effort is often spent on computing the search directions, which in MPC corresponds to solving unconstrained finite-time optimal control (UFTOC) problems. This is commonly performed using Riccati recursions or generic sparsity exploiting algorithms. In this work the focus is efficient search direction computations for active-set (AS) type methods. The system of equations to be solved at each AS iteration is changed only by a low-rank modification of the previous one, and exploiting this structured change is important for the performance of AS type solvers. In this paper, theory for how to exploit these low-rank changes by modifying the Riccati factorization between AS iterations in a structured way is presented. A numerical evaluation of the proposed algorithm shows that the computation time can be significantly reduced by modifying, instead of re-computing, the Riccati factorization. This speed-up can be important for AS type solvers used for linear, nonlinear and hybrid MPC

Approximate transformations and robust manipulation of bipartite pure state entanglement

We analyze approximate transformations of pure entangled quantum states by local operations and classical communication, finding explicit conversion strategies which optimize the fidelity of transformation. These results allow us to determine the most faithful teleportation strategy via an initially shared partially entangled pure state. They also show that procedures for entanglement manipulation such as entanglement catalysis [Jonathan and Plenio, Phys. Rev. Lett. 83, 3566 (1999)] are robust against perturbation of the states involved, and motivate the notion of non-local fidelity, which quantifies the difference in the entangled properties of two quantum states.Comment: 11 pages, 4 figure

On the Optimality of Quantum Encryption Schemes

It is well known that n bits of entropy are necessary and sufficient to perfectly encrypt n bits (one-time pad). Even if we allow the encryption to be approximate, the amount of entropy needed doesn't asymptotically change. However, this is not the case when we are encrypting quantum bits. For the perfect encryption of n quantum bits, 2n bits of entropy are necessary and sufficient (quantum one-time pad), but for approximate encryption one asymptotically needs only n bits of entropy. In this paper, we provide the optimal trade-off between the approximation measure epsilon and the amount of classical entropy used in the encryption of single quantum bits. Then, we consider n-qubit encryption schemes which are a composition of independent single-qubit ones and provide the optimal schemes both in the 2- and the operator-norm. Moreover, we provide a counterexample to show that the encryption scheme of Ambainis-Smith based on small-bias sets does not work in the operator-norm.Comment: 15 page

Emergence and Adult Biology of \u3ci\u3eAgrilus Difficilis\u3c/i\u3e (Coleoptera: Buprestidae), a Pest of Honeylocust, \u3ci\u3eGleditsia Triacanthos\u3c/i\u3e

Emergence and adult biology of Agrilus difficilis were examined in relation to its host Gleditsia triacanthos. began as early as 5 June in 1982 and completed as late as 22 July in 1983. Females lived significantly longer, 48 days, than males, 29 days. Average fecundity was one egg per day during a 36-day oviposition period

The screen behind the field : a netnographic study of Manchester United Fans and their perceptions of the club

This dissertation aims to take a deeper look into the online interactions of Manchester United supporters. A netnographic approach is utilized in order to understand what perception these fans have with regards to the club. Some negative perceptions were observed to represent concerns to supporters. The dissertation also addresses who the main stakeholders of these concerns are. Data gathered online from two supporter forums were analyzed using a qualitative thematic approach. The results indicate that the main concerns of these supporters are related to their perceptions on ownership, performance, sponsorships and merchandise. This dissertation discusses the implications of researching the perceptions of football fans within their online communities. The results suggest that the fans show dismay with a current sponsor and with a lack of ownership on the fans’ side. Managers and stakeholders are encouraged to ‘listen’ to these supporters and engage with them in co-creative and collaborative roles in hopes that this behavior will profit the stakeholders in the long-term.Esta dissertação tem como objectivo principal analisar as interações online dos torcedores do Manchester United. Uma abordagem netnografica é utilizada de modo a perceber as percepções que os torcedores têm acerca do clube. Algumas percepções negativas foram observadas a gerar preocupações nos torcedores. Esta dissertação também abrange quem são os principais afectados por estas mesmas preocupações.Dados recolhidos de dois fóruns online de torcedores foram analisados usando uma abordagem temática qualitativa. Os resultados indicam que a principal preocupação dos torcedores está relacionada com suas próprias percepções acerca da posse, desempenho, patrocínios e merchandising do clube. Esta dissertação discute as implicações de pesquisar os torcedores de futebol dentro das comunidades online. Os resultados sugerem que os torcedores mostram uma desaprovação relativa ao patrocinador actual e relativa à falta de domínio que têm sobre o clube.Managers e stakeholders deveriam ser incentivados a “ouvir” os torcedores e interagir com eles em papéis co-criativos e colaborativos na esperança que este comportamento traga lucro para os stakeholders a longo prazo

The Groverian Measure of Entanglement for Mixed States

The Groverian entanglement measure introduced earlier for pure quantum states [O. Biham, M.A. Nielsen and T. Osborne, Phys. Rev. A 65, 062312 (2002)] is generalized to the case of mixed states, in a way that maintains its operational interpretation. The Groverian measure of a mixed state of n qubits is obtained by a purification procedure into a pure state of 2n qubits, followed by an optimization process based on Uhlmann's theorem, before the resulting state is fed into Grover's search algorithm. The Groverian measure, expressed in terms of the maximal success probability of the algorithm, provides an operational measure of entanglement of both pure and mixed quantum states of multiple qubits. These results may provide further insight into the role of entanglement in making quantum algorithms powerful.Comment: 6 pages, 2 figure

Vanishing quantum discord is necessary and sufficient for completely positive maps

Two long standing open problems in quantum theory are to characterize the class of initial system-bath states for which quantum dynamics is equivalent to (1) a map between the initial and final system states, and (2) a completely positive (CP) map. The CP map problem is especially important, due to the widespread use of such maps in quantum information processing and open quantum systems theory. Here we settle both these questions by showing that the answer to the first is "all", with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from the class of separable states with vanishing quantum discord.Comment: 4 pages, no figures. v2: Accepted for publication in Phys. Rev. Let