Attouch, H.; Bolte, J.; Redont, P.; Teboulle, M. Singular Riemannian barrier methods and gradient-projection dynamical systems for constrained optimization. (English) Zbl 1153.34312 Optimization 53, No. 5-6, 435-454 (2004). 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. Cited in 15 Documents 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) Keywords:Dynamical systems; Continuous gradient method; Asymptotic analysis; Barrier methods in constrained optimization; Singular Hessian Riemannian metric; Convex Legendre functions; Bregman distances PDFBibTeX XMLCite \textit{H. Attouch} et al., Optimization 53, No. 5--6, 435--454 (2004; Zbl 1153.34312) Full Text: DOI References: [1] DOI: 10.1137/S0363012902419977 · Zbl 1077.34050 · doi:10.1137/S0363012902419977 [2] Attouch H, Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program. Pitman Publishing Ltd. (1984) [3] DOI: 10.1023/B:JOTA.0000037603.51578.45 · Zbl 1076.90053 · doi:10.1023/B:JOTA.0000037603.51578.45 [4] Attouch H, J. Convex Anal. 3 pp 1– (1996) [5] Bayer DA, Trans. of AMS 314 pp 499, 527– (1989) [6] DOI: 10.1137/S0363012902410861 · Zbl 1051.49010 · doi:10.1137/S0363012902410861 [7] Brézis H, Mathematics Studies 5, Elsevier (1973) [8] DOI: 10.1137/0803026 · Zbl 0808.90103 · doi:10.1137/0803026 [9] Goudou X oral communication [10] DOI: 10.1137/S0363012995290744 · Zbl 0918.90113 · doi:10.1137/S0363012995290744 [11] DOI: 10.1016/0362-546X(86)90058-1 · Zbl 0635.35043 · doi:10.1016/0362-546X(86)90058-1 [12] DOI: 10.1090/S0002-9904-1967-11761-0 · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0 [13] Rockafellar RT, Convex Analysis, Princeton University Press (1970) [14] Rockafellar RT, Variational Analysis, Springer Verlag (1998) · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.