52 research outputs found
Nonlinear conjugate gradient method for vector optimization on Riemannian manifolds with retraction and vector transport
In this paper, we propose nonlinear conjugate gradient methods for vector
optimization on Riemannian manifolds. The concepts of Wolfe and Zoutendjik
conditions are extended for Riemannian manifolds. Specifically, we establish
the existence of intervals of step sizes that satisfy the Wolfe conditions. The
convergence analysis covers the vector extensions of the Fletcher--Reeves,
conjugate descent, and Dai--Yuan parameters. Under some assumptions, we prove
that the sequence obtained by the algorithm can converge to a Pareto stationary
point. Moreover, we also discuss several other choices of the parameter.
Numerical experiments illustrating the practical behavior of the methods are
presented
Monotonicity for Multiobjective Accelerated Proximal Gradient Methods
Accelerated proximal gradient methods, which are also called fast iterative
shrinkage-thresholding algorithms (FISTA) are known to be efficient for many
applications. Recently, Tanabe et al. proposed an extension of FISTA for
multiobjective optimization problems. However, similarly to the
single-objective minimization case, the objective functions values may increase
in some iterations, and inexact computations of subproblems can also lead to
divergence. Motivated by this, here we propose a variant of the FISTA for
multiobjective optimization, that imposes some monotonicity of the objective
functions values. In the single-objective case, we retrieve the so-called
MFISTA, proposed by Beck and Teboulle. We also prove that our method has global
convergence with rate , where is the number of iterations, and
show some numerical advantages in requiring monotonicity.Comment: - Added new numerical experiment
An accelerated proximal gradient method for multiobjective optimization
This paper presents an accelerated proximal gradient method for
multiobjective optimization, in which each objective function is the sum of a
continuously differentiable, convex function and a closed, proper, convex
function. Extending first-order methods for multiobjective problems without
scalarization has been widely studied, but providing accelerated methods with
accurate proofs of convergence rates remains an open problem. Our proposed
method is a multiobjective generalization of the accelerated proximal gradient
method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm
(FISTA), for scalar optimization. The key to this successful extension is
solving a subproblem with terms exclusive to the multiobjective case. This
approach allows us to demonstrate the global convergence rate of the proposed
method (), using a merit function to measure the complexity.
Furthermore, we present an efficient way to solve the subproblem via its dual
representation, and we confirm the validity of the proposed method through some
numerical experiments
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Publisher Correction: Genetic tool development in marine protists: emerging model organisms for experimental cell biology.
An amendment to this paper has been published and can be accessed via a link at the top of the paper
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