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Singular Riemannian barrier methods and gradient-projection dynamical systems for constrained optimization. (English) Zbl 1153.34312

Summary: This work is devoted to the dynamical system (SRB): \[ \frac{d}{dt}\partial h(x(t))+\nabla \Phi(x(t))\ni 0,\tag{1} \] with \(h\) a proper lower semicontinuous convex function. Existence and uniqueness of solutions are examined. The systems (SRB) include the class of gradient systems with respect to a Hessian Riemannian metric induced by a convex Legendre function \(h:D x(t)+\nabla^2h(x(t))^{-1}\Phi(x(t))=0\). Moreover, the class (SRB) is closed in a variational sense: links are made with regularized Lotka-Volterra systems and the limit equations obtained by letting the barrier parameter go to 0. Of particular interest is the case \(h(x)= (1/2)|x|^2+\delta_C(x)\): this way, one obtains a new gradient-projection method. The system (SRB) bears a direct relation with the minimization of \(\Phi\) over the domain of \(\partial h\); the asymptotic behaviour of solutions, as time goes to infinity, is a real issue, which is addressed for a convex \(\Phi\) and a \(h\) of the form \(h=k+\delta_C\), with \(k\) convex \(\mathcal C^1\) and \(C\) a finite-dimensional polyhedron.

MSC:

34A60 Ordinary differential inclusions
90C30 Nonlinear programming
34G25 Evolution inclusions
37N35 Dynamical systems in control
37N40 Dynamical systems in optimization and economics
49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000)
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