Download PDF by F. Bethuel: Calculus of variations and geometric evolution problems:

By F. Bethuel

The overseas summer time institution on Calculus of diversifications and Geometric Evolution difficulties used to be held at Cetraro, Italy, 1996. The contributions to this quantity replicate particularly heavily the lectures given at Cetraro that have supplied a picture of a pretty large box in research the place lately we've seen many very important contributions. one of the issues handled within the classes have been variational tools for Ginzburg-Landau equations, variational types for microstructure and part transitions, a variational remedy of the Plateau challenge for surfaces of prescribed suggest curvature in Riemannian manifolds - either from the classical perspective and within the environment of geometric degree conception.

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Additional info for Calculus of variations and geometric evolution problems: lectures given at the 2nd session of the Centro Internazionale Matematico Estivo

Example text

In [147, 148, 186] this method is used for solving multiobjective optimization problems via evolutionary algorithms. For an arbitrary k ∈ {1, . . , m} and parameters εi ∈ R, i ∈ {1, . . 1)): min fk (x) subject to the constraints fi (x) ≤ εi , i ∈ {1, . . , m} \ {k}, x ∈ Ω. 24) It is easy to see that this is just a special case of the Pascoletti-Serafini scalarization for the ordering cone K = Rm + . We even get a connection w. r. t. 27. 25 hold and let K = Rm + , C = R+ , and Sˆ = S = Rn . A point x ¯ is a minimal solution of (Pk (ε)) with Lagrange multipliers μ ¯i ∈ R+ for i ∈ {1, .

This is a suitable restriction of the parameter set H as with H the following lemma shows. 20. Let x ¯ be a K-minimal solution of the multiobjective optimization problem (MOP). Let r ∈ K \ {0m }. Then there is a pa¯) is a minimal solution of rameter a ¯ ∈ H 0 and some t¯ ∈ R so that (t¯, x (SP(¯ a, r)). Proof. 11 the point (t¯, x ¯) with x) − β b f (¯ t¯ := b r is a minimal solution of (SP(¯ a, r)) for a ¯ := f (¯ x) − t¯r ∈ H. Because of 0 0 ¯ ∈ H . 20) a point s¯ ∈ R with m−1 s¯i v i . 22). Thus it is smin,i i i 0 i = 1, .

1 Pascoletti-Serafini Scalarization 23 relationship to other scalarization problems are examined in the last section of this chapter. 1 Pascoletti-Serafini Scalarization Pascoletti and Serafini propose the following scalar optimization problem with parameters a ∈ Rm and r ∈ Rm for determining minimal solutions of (MOP) w. r. t. the cone K: (SP(a,r)) min t subject to the constraints a + t r − f (x) ∈ K, g(x) ∈ C, h(x) = 0q , t ∈ R, x ∈ S. This problem has the parameter dependent constraint set Σ(a, r) := {(t, x) ∈ Rn+1 | a + t r − f (x) ∈ K, x ∈ Ω}.

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