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Motion with friction of a heavy particle on a manifold - applications to optimization

Alexandre Cabot (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Let Φ : H → R be a C2 function on a real Hilbert space and ∑ ⊂ H x R the manifold defined by ∑ := Graph (Φ). We study the motion of a material point with unit mass, subjected to stay on Σ and which moves under the action of the gravity force (characterized by g>0), the reaction force and the friction force ( γ > 0 is the friction parameter). For any initial conditions at time t=0, we prove the existence of a trajectory x(.) defined on R+. We are then interested in the asymptotic behaviour of...

Motion with friction of a heavy particle on a manifold. Applications to optimization

Alexandre Cabot (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Let Φ : H → ℝ be a 𝒞 2 function on a real Hilbert space and Σ ⊂ H × ℝ the manifold defined by Σ : = Graph ( Φ ) . We study the motion of a material point with unit mass, subjected to stay on Σ and which moves under the action of the gravity force (characterized by g > 0 ), the reaction force and the friction force ( γ > 0 is the friction parameter). For any initial conditions at time t = 0 , we prove the existence of a trajectory x ( . ) defined on ℝ + . We are then interested in the asymptotic behaviour of the trajectories when t → + ∞ . More precisely,...

Optimal investment under stochastic volatility and power type utility function

Benchaabane, Abbes, Benchettah, Azzedine (2011)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 37F21, 70H20, 37L40, 37C40, 91G80, 93E20.In this work we will study a problem of optimal investment in financial markets with stochastic volatility with small parameter. We used the averaging method of Bogoliubov for limited development for the optimal strategies when the small parameter of the model tends to zero and the limit for the optimal strategy and demonstrated the convergence of these optimal strategies.

Solving a class of non-convex quadratic problems based on generalized KKT conditions and neurodynamic optimization technique

Alaeddin Malek, Najmeh Hosseinipour-Mahani (2015)

Kybernetika

In this paper, based on a generalized Karush-Kuhn-Tucker (KKT) method a modified recurrent neural network model for a class of non-convex quadratic programming problems involving a so-called Z -matrix is proposed. The basic idea is to express the optimality condition as a mixed nonlinear complementarity problem. Then one may specify conditions for guaranteeing the global solutions of the original problem by using results from the S-lemma. This process is proved by building up a dynamic system from...

The steepest descent dynamical system with control. Applications to constrained minimization

Alexandre Cabot (2004)

ESAIM: Control, Optimisation and Calculus of Variations

Let H be a real Hilbert space, Φ 1 : H → ℝ a convex function of class 𝒞 1 that we wish to minimize under the convex constraint S . A classical approach consists in following the trajectories of the generalized steepest descent system (cf. Brézis [5]) applied to the non-smooth function Φ 1 + δ S . Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function Φ 0 : H → ℝ whose critical points coincide with S and a control...

The steepest descent dynamical system with control. Applications to constrained minimization

Alexandre Cabot (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let H be a real Hilbert space, Φ 1 : H → a convex function of class 𝒞 1 that we wish to minimize under the convex constraint S. A classical approach consists in following the trajectories of the generalized steepest descent system (cf.   Brézis [CITE]) applied to the non-smooth function  Φ 1 + δ S . Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function  Φ 0 : H → whose critical points coincide with S and...

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