Steepest Descent with Curvature Dynamical System

F. Alvarez and A. Cabot
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F. Alvarez: Universidad de Chile
A. Cabot: Université de Limoges

Journal of Optimization Theory and Applications, 2004, vol. 120, issue 2, No 2, 247-273

Abstract: Abstract Let H be a real Hilbert space and let denote the corresponding scalar product. Given a $$\mathcal{C}^2$$ function $$\Phi :H \to \mathbb{R}$$ that is bounded from below, we consider the following dynamical system: $$( {\text{SDC) }}\dot x(t) + \lambda (x(t))\triangledown \Phi (x(t)) = 0,{\text{ }}t \geqslant 0,$$ where λ(x) corresponds to a quadratic approximation to a linear search technique in the direction −∇Φ(x). The term λ(x) is connected intimately with the normal curvature radius ρ(x) in the direction ∇Φ(x). The remarkable property of (SDC) lies in the fact that the gradient norm |∇Φ(x(t))| decreases exponentially to zero when t→+∞. When Φ is a convex function which is nonsmooth or lacks strong convexity, we consider a parametric family {Φε, ε>0} of smooth strongly convex approximations of Φ and we couple this approximation scheme with the (SDC) system. More precisely, we are interested in the following dynamical system: $$( {\text{ASDC) }}\dot x(t) + \lambda (t,x(t))\triangledown _x \Phi (t,x(t)) = 0,{\text{ }}t \geqslant 0,$$ where λ(t, x) is a time-dependent function involving a curvature term. We find conditions on the approximating family and on ε(⋅) ensuring the asymptotic convergence of the solution trajectories x(⋅) toward a particular solution of the problem min {Φ(x), x∈H}. Applications to barrier and penalty methods in linear programming and to viscosity methods are given.

Keywords: Gradient-like systems; asymptotic analysis; convex optimization; approximate methods; optimal trajectories (search for similar items in EconPapers)
Date: 2004
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Citations: View citations in EconPapers (1)

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DOI: 10.1023/B:JOTA.0000015684.50827.49

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