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The steepest descent dynamical system with control. Applications to constrained minimization

Alexandre Cabot (2004)

ESAIM: Control, Optimisation and Calculus of Variations

Let H be a real Hilbert space, Φ 1 : H a convex function of class 𝒞 1 that we wish to minimize under the convex constraint S . A classical approach consists in following the trajectories of the generalized steepest descent system (cf. Brézis [5]) applied to the non-smooth function Φ 1 + δ S . Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function Φ 0 : H whose critical points coincide with S and a control...

The steepest descent dynamical system with control. Applications to constrained minimization

Alexandre Cabot (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let H be a real Hilbert space, Φ 1 : H a convex function of class 𝒞 1 that we wish to minimize under the convex constraint S. A classical approach consists in following the trajectories of the generalized steepest descent system (cf.   Brézis [CITE]) applied to the non-smooth function  Φ 1 + δ S . Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function  Φ 0 : H whose critical points coincide with S and...

Theorems of the alternative for cones and Lyapunov regularity of matrices

Bryan Cain, Daniel Hershkowitz, Hans Schneider (1997)

Czechoslovak Mathematical Journal

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C * and D * may overlap. When T V W is linear and K V and D W are cones, these results will be applied to C = T ( K ) and D , giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D . The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T ( X ) = A X + X * A yields new and known results about the existence of block diagonal...

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