Optimal Design, Langrangian, Linear Model Theories. A Fusion.
Ben Torsney (University of Glasgow)
Friday 6th December, 2013 15:00-16:00 Maths 204
The general approximate optimal design problem (P1) aims to maximise a criterion of several variables, subject to them being nonnegative and summing to 1. If all variables must be positive, a necessary condition of optimality is that all vertex directional derivatives be zero; equivalently all partial derivatives should share a common value, a Lagrange Multiplier value.
We consider the problem of optimizing a criterion of several variables, subject to them satisfying several (non-linear) equality constraints. Lagrangian Theory requires that at an optimum all partial derivatives be exactly linear in a set of Lagrange Multipliers. It seems we can argue that 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.
Further, if we must have all variables nonnegative, we might exploit the multiplicative algorithm for (P1).
Strictly speaking this has two steps:
- a multiplicative step, under which we multiply each variable by a positive function, say g(.), of its vertex directional derivative or of its partial derivative; and
- a scaling step to ensure the variables sum to 1.
The multiplicative step naturally extends to our more general problem, but some deeper consideration is needed to devise a scaling step to scale the resultant products to (approximately) satisfy the required set of equality constraints.
We will explore this idea in the case when the variables form a matrix which must satisfy fixed row and column sum constraints. It can be seen that partial derivatives must exactly satisfy a linear model, additive in two sets of main effect parameters.