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On the convergence of an inertial proximal algorithm with a Tikhonov regularization term

Szilárd Csaba László

Abstract

This paper deals with an inertial proximal algorithm that contains a Tikhonov regularization term, in connection to the minimization problem of a convex lower semicontinuous function $f$. We show that for appropriate Tikhonov regularization parameters the value of the objective function in the sequences generated by our algorithm converges fast (with arbitrary rate) to the global minimum of the objective function and the generated sequences converges weakly to a minimizer of the objective function. We also obtain the fast convergence of the discrete velocities towards zero and some sum estimates. Nevertheless, our main goal is to obtain strong convergence results and also pointwise and sum estimates for the same constellation of the parameters involved. Our analysis reveals that the extrapolation coefficient and the Tikhonov regularization coefficient are strongly correlated and there is a critical setting of the parameters that separates the cases when strong respective weak convergence results can be obtained.

On the convergence of an inertial proximal algorithm with a Tikhonov regularization term

Abstract

This paper deals with an inertial proximal algorithm that contains a Tikhonov regularization term, in connection to the minimization problem of a convex lower semicontinuous function . We show that for appropriate Tikhonov regularization parameters the value of the objective function in the sequences generated by our algorithm converges fast (with arbitrary rate) to the global minimum of the objective function and the generated sequences converges weakly to a minimizer of the objective function. We also obtain the fast convergence of the discrete velocities towards zero and some sum estimates. Nevertheless, our main goal is to obtain strong convergence results and also pointwise and sum estimates for the same constellation of the parameters involved. Our analysis reveals that the extrapolation coefficient and the Tikhonov regularization coefficient are strongly correlated and there is a critical setting of the parameters that separates the cases when strong respective weak convergence results can be obtained.
Paper Structure (13 sections, 10 theorems, 201 equations)

This paper contains 13 sections, 10 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. The state of the art in continuous case
  3. The problem formulation, motivation and a model result
  4. The organization of the paper
  5. Convergence rates and weak convergence for the case $q+1\le p$
  6. Convergence rates
  7. On weak convergence and boundedness of the generated sequences
  8. Convergence rates and strong convergence results for the case $p\le q+1$
  9. Convergence rates
  10. Strong convergence results
  11. Full strong convergence for the case $\delta=0,\,\lambda=1$
  12. Conclusions, perspectives
  13. Auxiliary results

Key Result

Theorem 1.1

Assume that $0<q<1,\,\alpha,c,p>0$ and for some starting points $x_0,x_1\in\mathcal{H}$ let $(x_k)$ be a sequence generated by algo, that is, $x_{k+1}=\mathop{\mathrm{prox}}\limits\nolimits_{f}\left(x_k+\left(1-\frac{\alpha}{k^q}\right)(x_k - x_{k-1})-\frac{c}{k^p}\right),\hbox{for all}k\ge 1.$ For

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • ...and 11 more