Let $H$ be a real Hilbert space, ${\Phi }_1:H\to ℝ$ 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 ${\Phi }_1+{\delta }_S$. 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 ${\Phi }_0:H\to ℝ$ whose critical points coincide with $S$ and a control...

Let H be a real Hilbert space, ${\Phi }_1:H\to$ 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 ${\Phi }_1+{\delta }_S$. 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 ${\Phi }_0:H\to$ whose critical points coincide with S and...

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 $TV\to W$ is linear and $K\subset V$ and $D\subset W$ are cones, these results will be applied to $C=T\left(K\right)$ 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 $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T\left(X\right)=AX+X^{*}A$ yields new and known results about the existence of block diagonal...