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optimization 13 (1988) 5, 663-691 Some Properties of the Sormed Afternatin: Least Squares (AT,$) Q p r i t h m Rnmmary: I n mcst epplicatiens of rls.ra. .tnalysis quantitative \-ariables are used t,ogether wit,h q~iaiitat~ive variables (nominal, ordinal or even of a more complex type). One main task in d a t a anaiysis is tilerefore to End aii ailecpate methci! ?c aua!yze a variab!e set of si~cl; clifferent information levels. 1 well-known approach fbr tilis task is the optimal ,-. -I -. . ali~ig . method, which wiil be iiimtrated shorl.ly b v the linear regression model. The optima! sca!ing rn~t,horl leads t o a constrained noniinear optimization probiern in the W R , tvhich cannot be soloed efficiently by sta:lclaril methods became of the amount of variabies ant1 restrictions. Therefore, a so-called normed alternat,ing !east squares (ALSj aigorithm ir proposetl i n t h e literature t o solve these optimization problems. Because problems of conrrergence are only tangent by the literature ! d l , in this paper some con\-ergenoe properties of the normed ALS algorithm are given, AWS 1380 Subject Classificazlon: Primary: 65 K 10; Secondary: 62 J 0.5, 62 $1 25.Key words: A l t~r n a t i n g leasc squares, ALS, optimal sealing, 0 6 , projection methods, linear regression.

Monotone (or isotonic) regression plays an important role in data analysis and in other fields. In many cases the monotonicity is only defined for a partial instead of a total preorder. No efficient algorithm is known which solves the general problem in a finite number of steps. For an approximate solution of the optimum some error estimations are given.Moreover, some new results concerning monotone regression and the treatment of missing values are presented in this paper.