scholarly journals Topology of Pareto Sets of Strongly Convex Problems

2020 ◽  
Vol 30 (3) ◽  
pp. 2659-2686
Author(s):  
Naoki Hamada ◽  
Kenta Hayano ◽  
Shunsuke Ichiki ◽  
Yutaro Kabata ◽  
Hiroshi Teramoto
Keyword(s):  
Strongly Convex ◽  
Convex Problems ◽  
Pareto Sets

scholarly journals Accelerated Incremental Gradient Descent using Momentum Acceleration with Scaling Factor

2019 ◽  
Author(s):  
Yuanyuan Liu ◽  
Fanhua Shang ◽  
Licheng Jiao
Keyword(s):  
Gradient Descent ◽  
Linear Convergence ◽  
Strongly Convex ◽  
Convex Problems ◽  
Accelerated Methods ◽  
Gradient Descent Methods ◽  
Gradient Descent Algorithm ◽  
Oracle Complexity ◽  
Incremental Gradient Descent ◽  
Theoretical Results

Recently, research on variance reduced incremental gradient descent methods (e.g., SAGA) has made exciting progress (e.g., linear convergence for strongly convex (SC) problems). However, existing accelerated methods (e.g., point-SAGA) suffer from drawbacks such as inflexibility. In this paper, we design a novel and simple momentum to accelerate the classical SAGA algorithm, and propose a direct accelerated incremental gradient descent algorithm. In particular, our theoretical result shows that our algorithm attains a best known oracle complexity for strongly convex problems and an improved convergence rate for the case of n>=L/\mu. We also give experimental results justifying our theoretical results and showing the effectiveness of our algorithm.

Convergence of a Greedy Algorithm on Nonlinear Convex Problems and Application to Uncertainty Quantification on Obstacle Problems

2010 ◽  
Author(s):  
Virginie Ehrlacher
Keyword(s):  
Greedy Algorithm ◽  
General Theorem ◽  
Uncertainty Propagation ◽  
Obstacle Problems ◽  
Convex Functional ◽  
One Dimensional ◽  
Strongly Convex ◽  
Convex Problems ◽  
Membrane Problem ◽  
Bounded Sets

In this article, a method is proposed to study uncertainty propagation on high-dimensional obstacle problems. A greedy algorithm, based on variable decomposition, is used to approximate the solution of regularized problems obtained by penalization of the initial problem. The convergence of this algorithm is a consequence of a more general theorem. Indeed, the algorithm converges for the minimization of a strongly convex functional whose derivative is Lipschitz on bounded sets. We describe how this algorithm was numerically implemented and present the results which were obtained with a one-dimensional membrane problem.

scholarly journals CONVERGENCE OF A GREEDY ALGORITHM FOR HIGH-DIMENSIONAL CONVEX NONLINEAR PROBLEMS

2011 ◽  
Vol 21 (12) ◽  
pp. 2433-2467 ◽  
Author(s):  
ERIC CANCÈS ◽  
VIRGINIE EHRLACHER ◽  
TONY LELIÈVRE
Keyword(s):  
Greedy Algorithm ◽  
Global Minimum ◽  
Uncertainty Propagation ◽  
Nonlinear Problems ◽  
Energy Functional ◽  
High Dimensional ◽  
Product Decomposition ◽  
Strongly Convex ◽  
Convex Problems ◽  
Bounded Sets

In this paper, we present a greedy algorithm based on a tensor product decomposition, whose aim is to compute the global minimum of a strongly convex energy functional. We prove the convergence of our method provided that the gradient of the energy is Lipschitz on bounded sets. The main interest of this method is that it can be used for high-dimensional nonlinear convex problems. We illustrate this method on a prototypical example for uncertainty propagation on the obstacle problem.

Computable bounds on parametric solutions of convex problems

1988 ◽  
Vol 40-40 (1-3) ◽  
pp. 213-221 ◽  
Author(s):  
Anthony V. Fiacco ◽  
Jerzy Kyparisis
Keyword(s):  
Parametric Solutions ◽  
Convex Problems
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