Solving cone-constrained convex programs by differential inclusions
Let (P) be the problem of minimizing a continuous convex function \(f_ 0: X\to R\) subject to the constraint \(- F(x)\in K\), where \(X\) is a subset of a Hilbert space and \(K\) is a closed acute convex cone strictly contained in another Hilbert space. An exterior vector-valued penalization for the triple \((F, K, X)\) is a function \(f\) between the two spaces such that \(f(X)\subset K\) and \(f(x)= 0\) if and only if \(- F(x)\in K\); one assumes that the function \(\langle y, f(\cdot)\rangle\) is convex for all \(y\in K^*\), \(K^*\) being the positive polar cone of \(K\) and \(\langle\cdot,\cdot\rangle\) denoting inner product. The Lagrangian \(L\) associated to the penalization is given by \(L(x, y)= f_ 0(x)+ \langle y, f(x)\rangle\). To solve (P), the authors consider the differential inclusion \[ \dot x(t)\in -\partial_ x L(x(t), y(t))- N_ x(x(t)),\qquad \dot y(t)\in \partial_ y L(x(t), y(t))\tag{DI} \] together with the viability condition \(y(t)\in K^*\), where \(\partial_ x\) and \(\partial_ y\) are the partial subgradient operators and \(N_ X(x(t))\) stands for the normal cone to \(X\) at \(x(t)\). The existence and uniqueness of the trajectories of (DI) emanating from each point is established and their convergence to the set of saddle points of \(L\) is proved under suitable assumptions. An application to stochastic programming is given.
scientific article; zbMATH DE number 644746
scientific article