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The heavy ball with friction dynamical system for convex constrained minimization problems. (English) Zbl 0980.90062
Nguyen, Van Hien (ed.) et al., Optimization. Proceedings of the 9th Belgian-French-German conference, Namur, Belgium, September 7-11, 1998. Berlin: Springer. Lect. Notes Econ. Math. Syst. 481, 25-35 (2000).
The article under review is a valuable contribution to the minimization of a convex $$C^1$$ function $$\Phi$$ on a Hilbert space $$H$$, which theoretically prepares numerical methods in future. A mechanical and geometrical illustration of the basic idea consists in pathfollowing of a trajectory described within a potential by a heavy ball with friction. While former research of $$F$$. Alvarez referred to the unconstrained case, now, the minimization is subject to a closed convex feasible set $$C\subset H$$. Incorporating $$C$$ by its indicator function, (sub) differentiation delivers the optimality condition $$0\in\nabla \Phi(u) +N_C(u)$$, where $$N_C(u)$$ denotes the outwards normal cone. Following A. S. Antipin, this condition becomes rewritten in a gradient-projective, single-valued way by $$u-\text{proj}_C(u-\mu\nabla \Phi(u)) =0$$ $$(\mu$$ being a parameter). As the authors can prove that for $$0<\mu \leq {2\over L}$$ the latter projector term is a contraction $$T$$ and, herewith, our left-hand side $$A=I-T$$ is a maximal monotone, cocoercive operator, from here they derive the dynamical system $$\ddot u+\gamma\dot u+u-\text{proj}_C(u-\mu \nabla \Phi(u))=0$$ $$(\gamma >0)$$ with data $$u(0)= u_0$$, $$\dot u(0)= v_0$$.
Provided additionally that $$\nabla\Phi$$ is Lipschitzian with Lipschitz constant $$L$$, and $$\gamma >\sqrt 2$$, now, the main result says that the above second-order equation with its Cauchy initial data has a unique solution $$u\in C^2$$, its derivatives $$\dot u,\ddot u$$ being $$L^2$$ and tending to 0, while $$u$$ itself weakly tends to a desired global minimizer $$\overline u$$. The refined analytical proof bases, e.g., on Opial’s Lemma, and it works in a manner which allows a generalized result by referring to a (general) contraction $$T$$ and, then, $$\overline u$$ becoming a fixed point of $$T$$.
This careful investigation is considered in the context of various further research, e.g., by H. Attouch et al., and it asks for discretization from both the analytical and algorithmical viewpoint. Furthermore, a version with $$C$$ being defined by equality and inequality constraints may also be expected, e.g., by Morse theoretical methods developed by H. Th. Jongen, P. Jonker and F. Twilt.
For the entire collection see [Zbl 0935.00054].

##### MSC:
 90C25 Convex programming 70F40 Problems involving a system of particles with friction