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

• Autorzy: Li Y. , Kummert A. , Frommer A.
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

completely positive matrices; cp-rank; linear programming; simplex algorithm; basic feasible solution; pivot process

PL

macierz pozytywna; programowanie liniowe; algorytm simplex

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2004
• Tom: Vol. 14, no 1
• Strony: 25--31
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Li Y.

• Department of Electrical and Information Engineering, University of Wuppertal, Rainer-Gruenter-Street 21, 42119 Wuppertal, Germany

autor

Kummert A.

• Department of Electrical and Information Engineering, University of Wuppertal, Rainer-Gruenter-Street 21, 42119 Wuppertal, Germany

autor

Frommer A.

• Department of Mathematics, University of Wuppertal, Gauß Street 20, 42119 Wuppertal, Germany

Bibliografia

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