904 research outputs found
On the Convergence of (Stochastic) Gradient Descent with Extrapolation for Non-Convex Optimization
Extrapolation is a well-known technique for solving convex optimization and
variational inequalities and recently attracts some attention for non-convex
optimization. Several recent works have empirically shown its success in some
machine learning tasks. However, it has not been analyzed for non-convex
minimization and there still remains a gap between the theory and the practice.
In this paper, we analyze gradient descent and stochastic gradient descent with
extrapolation for finding an approximate first-order stationary point in smooth
non-convex optimization problems. Our convergence upper bounds show that the
algorithms with extrapolation can be accelerated than without extrapolation
A Nonlinear Projection Neural Network for Solving Interval Quadratic Programming Problems and Its Stability Analysis
This paper presents a nonlinear projection neural network for solving interval
quadratic programs subject to box-set constraints in engineering applications. Based on the Saddle point theorem, the equilibrium point of the proposed neural network is proved to be equivalent to the optimal solution of the interval quadratic optimization problems. By employing Lyapunov function approach, the global exponential stability of the proposed neural network is analyzed. Two illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper
Fixed-Time Stable Proximal Dynamical System for Solving MVIPs
In this paper, a novel modified proximal dynamical system is proposed to
compute the solution of a mixed variational inequality problem (MVIP) within a
fixed time, where the time of convergence is finite, and is uniformly bounded
for all initial conditions. Under the assumptions of strong monotonicity and
Lipschitz continuity, it is shown that a solution of the modified proximal
dynamical system exists, is uniquely determined and converges to the unique
solution of the associated MVIP within a fixed time. As a special case for
solving variational inequality problems, the modified proximal dynamical system
reduces to a fixed-time stable projected dynamical system. Furthermore, the
fixed-time stability of the modified projected dynamical system continues to
hold, even if the assumption of strong monotonicity is relaxed to that of
strong pseudomonotonicity. Connections to convex optimization problems are
discussed, and commonly studied dynamical systems in the continuous-time
optimization literature follow as special limiting cases of the modified
proximal dynamical system proposed in this paper. Finally, it is shown that the
solution obtained using the forward-Euler discretization of the proposed
modified proximal dynamical system converges to an arbitrarily small
neighborhood of the solution of the associated MVIP within a fixed number of
time steps, independent of the initial conditions. Two numerical examples are
presented to substantiate the theoretical convergence guarantees.Comment: 12 pages, 5 figure
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