We consider the problem of optimizing a criterion subject to several equality constraints. Lagrangian Theory requires that, at an optimum, all partial derivatives be exactly linear in a set of Lagrange Multipliers. Thus the partial derivatives, viewed as response variables, must exactly satisfy a Linear Model with the Lagrange Multipliers as parameters. This then is a model 'without' errors, implying a 'fitted model' with zero residuals. The residuals appear to play the role of directional derivatives, as defined in the optimal approximate design arena when A = 1

Further we extend a class of multiplicative algorithms, designed to find the optimum in the latter case, to our general problem. This algorithm has two steps:

(i) a multiplicative one, multiplying the current values of the components of p by an increasing function of partial or directional derivatives;

(ii) a scaling step under which the products formed in (i) are scaled to meet the summation to one equality constraint.

Step (i) readily extends to our more general problem, while the more challenging

step (ii) has been surmounted.

Results in two main areas will be reported:

(a) constraints on multinomial models, given data from multidimensional contingency tables, constraints being defined by fixed marginal distributions or, when the tables are 'square', hypotheses of marginal homogeneity;

(b) optimal approximate designs subject to cost constraints, or, in the case of two dimensional design spaces, subject to given marginal approximate designs; in these cases an optimal p can have zero components.

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