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Approximation of set-valued functions by single-valued one

100%

Ginchev I., Hoffmann A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2002

tom 22

nr 1

33-66

Let $Σ: M → 2^Y\{∅}$ be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y,||·||). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a single-valued function g from a given closed convex set V ⊂ C(M,Y). In an abstract setting this problem is posed as the extremal problem $sup_{t ∈ M} ρ(g(t), (t)) → inf$, g ∈ V. Here ρ is a functional whose values ρ(q,S) can be interpreted as some distance from the point q to the set S ⊂ Y. In the paper, we are confined to two natural distance functionals ρ = H and ρ = D. H(q,S) is the Hausdorff distance (the excess) from the point q to the set cl S, and D(q,S) is referred to as the oriented distance from the point q to set cl conv S. We prove that both these problems are convex optimization problems. While distinguishing between the so called regular and irregular case problems, in particular the case V = C(M,Y) is studied to show that the solutions in the irregular case are obtained as continuous selections of certain set-valued maps. In the general case, optimality conditions in terms of directional derivatives are obtained of both primal and dual type.

Penalty/barrier path-following in linearly constrained optimization

100%

Grossmann C.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2000

tom 20

nr 1

7-26

In the present paper rather general penalty/barrier path-following methods (e.g. with p-th power penalties, logarithmic barriers, SUMT, exponential penalties) applied to linearly constrained convex optimization problems are studied. In particular, unlike in previous studies [1,11], here simultaneously different types of penalty/barrier embeddings are included. Together with the assumed 2nd order sufficient optimality conditions this required a significant change in proving the local existence of some continuously differentiable primal and dual path related to these methods. In contrast to standard penalty/barrier investigations in the considered path-following algorithms only one Newton step is applied to the generated auxiliary problems. As a foundation of convergence analysis the radius of convergence of Newton's method depending on the penalty/barrier parameter is estimated. There are established parameter selection rules which guarantee the overall convergence of the considered path-following penalty/barrier techniques.

Ograniczanie wyników

2 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Ginchev I.

1 Grossmann C.

1 Hoffmann A.

1 2002

1 2000

Wyniki wyszukiwania

Approximation of set-valued functions by single-valued one

Penalty/barrier path-following in linearly constrained optimization