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Adaptive optimization and D-optimum experimental design
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DOI：10.1214/aos/1015957479.png)
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摘要
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We consider the situation where one has to maximize a function eta(theta, x) with respect to x is an element of R-q, when a is unknown and estimated by least squares through observations y(k) = f(inverted perpendicular)(x(k))theta + epsilon (k), with epsilon (k) some random error. Classical applications are regulation and extremum control problems. The approach we adopt corresponds to maximizing the sum of the current estimated objective and a penalization for poor estimation: x(k+1) maximizes eta(<()over cap>(k), x) + (alpha (k)/k), d(k)(x), with <()over cap>(k) the estimated value of theta at step k and dk the penalization function. Sufficient conditions for strong consistency of <()over cap>(k) and for almost sure convergence of(1/k) Sigma (k)(i=1) eta(theta, x(i)) to the maximum value of eta(theta, x) are derived in the case where d(k)(.) is the variance function used in the sequential construction of D-optimum designs. A classical sequential scheme from adaptive control is shown not to satisfy these conditions, and numerical simulations confirm that it indeed has convergence problems.
Keyword:
adaptive control
least-squares estimation
D-optimum design
sequential design
strong consistency
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