Optimisation in space of measures and optimal design

Abstract

Abstract. The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes it possible to use the whole arsenal of descent algorithms from the general optimisation literature for finding optimal designs. The corresponding steepest descent involves adding a signed measure at every step and converges faster than the conventional sequential algorithms used to construct optimal designs. We study a new class of design problems when the observation points are distributed according to a Poisson point process arising in the situation when the total control on the placement of measurements is impossible.Mathematics Subject Classification. 62K05, 49K45, 60G55.

Approximate optimal designs for multivariate polynomial regression

“…which proves thatŷ is an optimal solution to (15), and lim δ→∞ ρ δ = ρ. b) As the optimal solution to (15) is unique, we have y =ŷ d withŷ d defined in the proof of a) and the whole sequence (y d,δ ) δ∈N converges to y , that is, for α ∈ N n with |α| ≤ 2d fixed lim The last point follows directly observing that, in this case, the two Programs (15) and (17) satisfy the same KKT conditions.…”

“…The method is highly versatile as it can be used for all classical functionals of the information matrix. Furthermore, it can easily be tailored to incorporate prior knowledge on some multidimensional moments of the targeted optimal measure (as proposed in [15]). In future works, we will extend the method to multi-response polynomial regression problems and to general smooth parametric regression models by linearization.…”

“…Since the ideal moment Problem (15) involves the moment cone M 2d (X ) which is not SDP representable, we use the hierarchy (8) of outer approximations of the moment cone to relax Problem (15) to an SDP problem. So for a fixed integer δ ≥ 1 we consider the problem…”

“…, let y d,δ be an optimal solution to (17) and p d,δ ∈ R[x] 2d the Christoffel polynomial associated with y d,δ defined in Theorem 3. Then, a) ρ δ → ρ as δ → ∞, where ρ is the supremum in (15). b) For every α ∈ N n with |α| ≤ 2d, we have lim δ→∞ y d,δ,α = y α , where y = (y α ) |α|≤2d ∈ M 2d (X ) is the unique optimal solution to (15).…”

“…First, let us prove that Problem (15) has an optimal solution. The feasible set is nonempty with finite associated objective value-take as feasible point the vector y ∈ M 2d (X ) associated with the Lebesgue measure on the compact set X , scaled to be a probability measure.…”