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Title: Optimal design, lagrangian and linear models theories: A fusion Authors:  Ben Torsney - University of Glasgow (United Kingdom) [presenting]
Abstract: The purpose is optimizing criterion $f(p)$ ($p$ nonneg vector) subject to several equality constraints: $Ap=b$. (wlog $b$ is nonneg.) Lagrangian Theory requires that, at optima, partial derivatives be exactly linear in Lagrange Multipliers (LMs). So partial derivatives, viewed as response variables, must exactly satisfy a Linear Model with LMs as parameters. This is a model without errors, implying zero residuals. Residuals appear to play the role of directional derivatives, as defined for optimal designs when $A = (1,1, .. ,1)$, $b=1$. Further we extend a class of multiplicative algorithms, designed for the latter case, to our problem. The 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 constraint. Step (i) readily extends to our problem, while the more challenging step (ii) has been surmounted in some examples, but needs further development. Results in two main areas will be reported: (a) constraints on multinomial models, given data from multidimensional contingency tables, defined by fixed marginal distributions or, for square tables, hypotheses of marginal homogeneity; (b) optimal approximate designs subject to cost constraints, or, subject to given marginal approximate designs; in these cases an optimal $p$ can have zero components.