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1

Tikhonov regularization and constrained quadratic programming for magnetic coil design problems

Garda B., Galias Z.

International Journal of Applied Mathematics and Computer Science

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2014

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Vol. 24, no. 2

249--257

EN

In this work, the problem of coil design is studied. It is assumed that the structure of the coil is known (i.e., the positions of simple circular coils are fixed) and the problem is to find current distribution to obtain the required magnetic field in a given region. The unconstrained version of the problem (arbitrary currents are allowed) can be formulated as a Least-SQuares (LSQ) problem. However, the results obtained by solving the LSQ problem are usually useless from the application point of view. Moreover, for higher dimensions the problem is ill-conditioned. To overcome these difficulties, a regularization term is sometimes added to the cost function, in order to make the solution smoother. The regularization technique, however, produces suboptimal solutions. In this work, we propose to solve the problem under study using the constrained Quadratic Programming (QP) method. The methods are compared in terms of the quality of the magnetic field obtained, and the power of the designed coil. Several 1D and 2D examples are considered. It is shown that for the same value of the maximum current the QP method provides solutions with a higher quality magnetic field than the regularization method.

2

Ocena szybkości działania metod punktu wewnętrznego dla zadań programowania liniowego i kwadratowego

Horla D.

Studia z Automatyki i Informatyki

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2012

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T. 37

47--78

PL

W artykule poruszono zagadnienie szybkości działania metod punktu wewnętrznego dla losowo generowanych zadań programowania liniowego i kwadratowego z ograniczeniami. Wskazano struktury zadań, w których konkretna metoda działa szybciej niż pozostałe oraz wyciągnięto wnioski odnośnie takiego stanu rzeczy.

EN

Performance evaluation of interior-point methods for LQ and QP problems. The paper considers performance issues of a class of interior-point methods for randomly-generated con strained LP and QP problems. Problem structures that assuresuperior performance of a specific method have been stipulated with appropriate conclusions drawn.

3

An SQP trust region method for solving the discrete-time linear quadratic control problem

Mostafa E. M. E.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 2

353-363

EN

In this paper, a sequential quadratic programming method combined with a trust region globalization strategy is analyzed and studied for solving a certain nonlinear constrained optimization problem with matrix variables. The optimization problem is derived from the infinite-horizon linear quadratic control problem for discrete-time systems when a complete set of state variables is not available. Moreover, a parametrization approach is introduced that does not require starting a feasible solution to initiate the proposed SQP trust region method. To demonstrate the effectiveness of the method, some numerical results are presented in detail.

4

A family of model predictive control algorithms with artificial neural networks

Ławryńczuk M.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 2

217-232

EN

This paper details nonlinear Model-based Predictive Control (MPC) algorithms for MIMO processes modelled by means of neural networks of a feedforward structure. Two general MPC techniques are considered: the one with Nonlinear Optimisation (MPC-NO) and the one with Nonlinear Prediction and Linearisation (MPC-NPL). In the first case a nonlinear optimisation problem is solved in real time on-line. In order to reduce the computational burden, in the second case a neural model of the process is used on-line to determine local linearisation and a nonlinear free trajectory. Single-point and multi-point linearisation methods are discussed. The MPC-NPL structure is far more reliable and less computationally demanding in comparison with the MPC-NO one because it solves a quadratic programming problem, which can be done efficiently within a foreseeable time frame. At the same time, closed-loop performance of both algorithm classes is similar. Finally, a hybrid MPC algorithm with Nonlinear Prediction, Linearisation and Nonlinear optimisation (MPC-NPL-NO) is discussed.

5

Projektowanie topografii systemów VLSI. Cz. 3. Metody analityczne

Nagórny Z., Kos A.

Kwartalnik Elektroniki i Telekomunikacji

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2006

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Vol. 52, z. 4

669-695

PL

Niniejsza praca jest trzecią częścią przeglądu metod rozmieszczania modułów, stosowanych podczas projektowania topografii układów VLSI. W pracy szczegółowo został opisany algorytm zamiany parami oraz metody analityczne. Przedstawiono liczne modyfikacje algorytmu zamiany parami, łącznie z algorytmami wykorzystującymi metody relaksacyjne. Modyfikacje algorytmu zamiany parami oraz metody relaksacyjne są stosowane w programach rozmieszczania opartych na metodach analitycznych. Następnie, opisano podstawy zastosowania programowania kwadratowego i liniowego w rozmieszczaniu modułów. Ze względu na dużą liczbę rozwiązań stosowanych w metodach analitycznych, poszczególne rozwiązania szczegółowo przedstawiono na przykładzie wybranych programów rozmieszczania. W tym celu scharakteryzowano następujące programy rozmieszczania: GORDIAN / DOMINO, KraftWerk, FastPlace, mPL, PROUD, ATLAS, FAR, mFAR, BloBB, APlace. Przedstawiono również sposób zastosowania metody relaksacyjnej w układach o topografii swobodnej oraz możliwość optymalizacji topografii układu ze względu na aspekt termiczny.

EN

The design process of the VLSI circuits requires the use of computer aided design tools. This paper is the third part of the survey of the cell placement techniques for digital VLSI circuits. In this part of the survey, the pairwise interchange algorithm and some analytical methods are presented. The force-directed placement algorithm and some modifications of the pairwise interchange algorithm, which are used in analytical algorithms are described. Then, the nonlinear programming, quadratic programming and linear programming techniques are presented. An application of these techniques to the cell placement problem is described. Nowadays the tools used for the cell placement, which utilize the presented algorithms are characterized: GORDIAN, DOMINO, KraftWerk, FastPlace, mPL, PROUD, ATLAS, FAR, mFAR, BloBB, APlace. A force-directed placer for a building block design style is described. The principles of the multilevel optimization for the cell placement problem are presented. Applications of the flow network and branch and bound algorithm to the cell placement are characterized. Some conclusions concerning described techniques and tools are presented.

6

Optimization of elastic-plastic beam structures with hardening using finite element method

Grigusevicius A., Kalanta S.

Foundations of Civil and Environmental Engineering

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2005

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No. 6

31-52

EN

Elastic-plastic beam structures, subjected by distributed loads, optimization problems are considered in this paper. Mixed method is suggested to form static and geometrical equations by setting the finite element interpolation functions of internal forces and displacements. That allows forming optimization problems with restricted middle cross-sections of beams. General expressions of static and geometrical equations are presented for a beam subjected by a distributed load. The optimization mathematical models of elastic-plastic beam structures with linear hardening are formulated as quadratic programming problems. Results of numerical example are presented and briefly discussed.

PL

W rozprawach optymalizacji prętowych elastyczno-plastycznych konstrukcji rozpatruje się najczęściej systemy obciążane siłami skupionymi. Praca jest poświęcona udoskonaleniu algorytmów optymalizacji prętowych konstrukcji obciążonych siłami ciągłymi. Równania statyczne oraz geometryczne elementu skończonego i całej j konstrukcji proponuje się tworzyć metodą kombinowaną, zadając funkcje interpolacyjne przemieszczeń oraz sit. To sprzyja sformułowaniu zadania optymalizacji, w którym można ograniczyć ugięcia środkowego przekroju w elementach zginanych. Przedstawiono ogólne formy równań statycznych oraz geometrycznych dla pręta obciążanego ciągłym obciążeniem. Zadania optymalizacji prętowych elastyczno-plastycznych konstrukcji wyprodukowanych z liniowo wzmacniającego się materiału są przedstawione jak zadania kwadratowego programowania. Przedstawione są rezultaty analizy obliczeniowej.

7

An infinite horizon predictive control algorithm based on multivariable input-output models

Ławryńczuk M., Tatjewski P.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 2

167-180

EN

In this paper an infinite horizon predictive control algorithm, for which closed loop stability is guaranteed, is developed in the framework of multivariable linear input-output models. The original infinite dimensional optimisation problem is transformed into a finite dimensional one with a penalty term. In the unconstrained case the stabilising control law, using a numerically reliable SVD decomposition, is derived as an analytical formula, calculated off-line. Considering constraints needs solving on-line a quadratic programming problem. Additionally, it is shown how free and forced responses can be calculated without the necessity of solving a matrix Diophantine equation.