Wyniki wyszukiwania - Biblioteka Nauki

1

Obliczanie namiaru wsadu metalowego z wykorzystaniem algorytmu simpleks

100%

Piątkowski J. , Szymszal J. , Binczyk F.

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tom R. 2, nr 6

179-188

PL

Uniwersalną metodą rozwiązywania zagadnień optymalizacyjnych z zastosowaniem programowania liniowego jest algorytm simpleks. Jest to nowoczesny sposób badań operacyjnych, polegający na poszukiwaniu kolejnych rozwiązań bazowych programowania liniowego w postaci kanonicznej. W pracy przedstawiono metodę wyznaczania namiaru wsadu metalowego opartą o optymalizację simpleks, pozwalającą na określenie położenia ekstremum funkcji wieIu zmiennych. Obliczenia prowadzono na przykladzie nadeutektycznego siluminu AISi17 z dodatkiem Cu, Ni i Mg.

EN

Simplex algorithm is universal method of dissolving of questions optimisation with utilization of linear programmes. It is this modern way of operating investigations, depending on search of next base solutions of linear programme in canonical form. In work metal leaning of marking of bearing of batch method was introduced about optimisation simplex, permitting onto qualification of position of extreme variable function. Calculation hypereutectic AISi17 alloy with addition Cu, Ni and Mg alloys was led on example.

2

A linear programming based analysis of the CP-rank of completely positive matrices

100%

Li Y. , Kummert A. , Frommer A.

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nr 1

25-31

EN

A real matrix A is said to be completely positive (CP) if it can be decomposed as A = BB^T, where the real matrix B has exclusively non-negative entries. Let k be the rank of A and Φ_k the least possible number of columns of the matrix B, the so-called completely positive rank (cp-rank) of A. The present work is devoted to a study of a general upper bound for the cp-rank of an arbitrary completely positive matrix A and its dependence on the ordinary rank k. This general upper bound of the cp-rank has been proved to be at most k(k + 1)/2. In a recent pioneering work of Barioli and Berman it was slightly reduced by one, which means that Φ_k ≤ k(k + 1)/2 - 1 holds for k ≥ 2. An alternative constructive proof of the same result is given in the present paper based on the properties of the simplex algorithm known from linear programming. Our proof illuminates complete positivity from a different point of view. Discussions concerning dual cones are not needed here. In addition to that, the proof is of constructive nature, i.e. starting from an arbitrary decomposition A = B_1B^T_1 (B_1 ≥ 0) a new decomposition A = B_2B^T_2 (B_2 ≥ 0) can be generated in a constructive manner, where the number of column vectors of B_2 does not exceed k(k + 1)/2 − 1. This algorithm is based mainly on the well-known techniques stemming from linear programming, where the pivot step of the simplex algorithm plays a key role.

3

Zastosowanie symulowanego wyżarzania do optymalizacji aparatów i procesów

84%

Jeżowski J. , Poplewski G. , Słoma R.

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tom T. 22, z. 3C

573--578

PL

Opracowano metodę optymalizacji według strategii symulowanego wyżarzania (SW) opartą na propozycji z [1, 2]. Polega ona na wbudowaniu algorytmu simpleksów w ramy podstawowych koncepcji SW. Opracowana metoda umożliwia rozwiązywanie problemów nieliniowych z ograniczeniami równościowymi i nierównościowymi. W tej pracy przedstawiono przykłady zastosowania do rozwiązania dwóch zagadnień z zakresu inżynierii chemicznej.

EN

Simulated annealing (SA) optimisation method has developed based on the proposition from [1,2]. The approach relies on embedding simplex algorithm into basic concepts of SA. The optimisation method can be applied for solving non-linear problems with equality and inequality constraints. Examples of applications are presented for two chemical engineering problems.

4

A linear programming based analysis of the CP-rank of completely positive matrices

84%

Li Y. , Kummert A. , Frommer A.

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2004

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tom Vol. 14, no 1

25-31

EN

A real matrix A is said to be completely positive (CP) if it can be decomposed as A= B BT, where the real matrix B has exclusively non-negative entries. Let k be the rank of A and Phik the least possible number of columns of the matrix B, the so-called completely positive rank (cp-rank) of A. The present work is devoted to a study of a general upper bound for the cp-rank of an arbitrary completely positive matrix A and its dependence on the ordinary rank k. This general upper bound of the cp-rank has been proved to be at most k(k + 1)/2. In a recent pioneering work of Barioli and Berman it was slightly reduced by one, which means that Phik \leq k(k + 1)/2-1 holds for k \geq 2. An alternative constructive proof of the same result is given in the present paper based on the properties of the simplex algorithm known from linear programming. Our proof illuminates complete positivity from a different point of view. Discussions concerning dual cones are not needed here. In addition to that, the proof is of constructive nature, i.e. starting from an arbitrary decomposition A= B1 B1T (B1\geq 0) a new decomposition A= B2 B2T (B2\geq 0) can be generated in a constructive manner, where the number of column vectors of B2 does not exceed k(k + 1)/2-1. This algorithm is based mainly on the well-known techniques stemming from linear programming, where the pivot step of the simplex algorithm plays a key role.

5

A general iterative solver for unbalanced inconsistent transportation problems

84%

Carp D. , Popa C. , Serban C.

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tom Vol. 37, iss. 1

7--13

EN

The transportation problem, as a particular case of a linear programme, has probably the highest relative frequency with which appears in applications. At least in its classical formulation, it involves demands and supplies. When, for practical reasons, the total demand cannot satisfy the total supply, the problem becomes unbalanced and inconsistent, and must be reformulated as e.g. finding a least squares solution of an inconsistent system of linear inequalities. A general iterative solver for this class of problems has been proposed by S. P. Han in his 1980 original paper. The drawback of Han’s algorithm consists in the fact that it uses in each iteration the computation of the Moore-Penrose pseudoinverse numerical solution of a subsystem of the initial one, which for bigger dimensions can cause serious computational troubles. In order to overcome these difficulties we propose in this paper a general projection-based minimal norm solution approximant to be used within Han-type algorithms for approximating least squares solutions of inconsistent systems of linear inequalities. Numerical experiments and comparisons on some inconsistent transport model problems are presented.