Continuity Properties of Performance Functions

Perturbed optimization problems of the following general form https://www.w3.org/1998/Math/MathML"> P W   maximize    f W x over   A W ⊂ X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0130.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where w ε W is a parameter, f: W × X → https://www.w3.org/1998/Math/MathML"> ℝ ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/ieq0234.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> = f_w(x) = f(w,x) can studied from different points of view. Here we focus our attention on the continuity properties of the performance functional p: W → R̅ given by https://www.w3.org/1998/Math/MathML"> p w = sup f W x ∣ x ∈ A w , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0131.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where W and X are topological spaces and A: W→X given by A(w) = A_W is a given relation defining the admissible (or feasible) set. The related w but more complex problem of establishing continuity properties of the set S_W of solutions of (P_W) will be studied elsewhere. The other aspects of parametric programming (measurability, lipschitzian properties, (generalized) differentiability…) are not considered in this paper although our lecture dealt with the marginal interpretation of multipliers. Let us observe that the generalized subdifferentiability properties of p cannot be deduced from the related properties of the marginal function m: W → R̅ given by https://www.w3.org/1998/Math/MathML"> m w = inf f W x ∣ x ∈ A W https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0132.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if one uses the subdifferential calculus of [14] or [18] (in contrast 78with the lipschitzian case studied in [9]). Here of course the studies of m and p are equivalent, changing f into -f. We choose p rather than m for the sake of symmetry of the statements, although m is more closely linked with convex optimization.