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Approximation analysis for the minimization problem of difference-of-convex functions with Moreau envelopes

Yan Tang, Shiqing Zhang

Abstract

In this work the minimization problem for the difference of convex (DC) functions is studied by using Moreau envelopes and the descent method with Moreau gradient is employed to approximate the numerical solution. The main regularization idea in this work is inspired by Hiriart-Urruty [14], Moudafi[17], regularize the components of the DC problem by adapting the different parameters and strategic matrices flexibly to evaluate the whole DC problem. It is shown that the inertial gradient method as well as the classic gradient descent scheme tend towards an approximation stationary point of the original problem.

Approximation analysis for the minimization problem of difference-of-convex functions with Moreau envelopes

Abstract

In this work the minimization problem for the difference of convex (DC) functions is studied by using Moreau envelopes and the descent method with Moreau gradient is employed to approximate the numerical solution. The main regularization idea in this work is inspired by Hiriart-Urruty [14], Moudafi[17], regularize the components of the DC problem by adapting the different parameters and strategic matrices flexibly to evaluate the whole DC problem. It is shown that the inertial gradient method as well as the classic gradient descent scheme tend towards an approximation stationary point of the original problem.
Paper Structure (7 sections, 12 theorems, 99 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 7 sections, 12 theorems, 99 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Some definitions and lemmas
  3. Main result
  4. Some properties on the difference of Moreua Envelopes with different parameters
  5. Algorithms and approximation analysis
  6. Numerical Example
  7. Conclusion

Key Result

Lemma 2.5

Let $g, f$ be proper convex functions on $X$, and $\Phi=g-f$ attains its minimum at $\tilde{x}$, then $\partial g(\tilde{x})\cap \partial f(\tilde{x})\neq\emptyset$.

Figures (5)

  • Figure 1: $\Phi(x)=|x|^3-|x|,\phi(x)=\frac{|x|^3}{2}- \frac{2}{3}.$
  • Figure 2: Left two:$\lambda=0.01,\mu=0.001$, Right two:$\lambda=\mu=0.001$ for Algo.1 and Algo. 2
  • Figure 3: Left two:$\lambda=0.2,\mu=0.1$, Right two:$\lambda=\mu=0.1$ for Algo.1 and Algo. 2
  • Figure 4: Left two:$\lambda=0.2,\mu=0.01$, Right two:$\lambda=\mu=0.2$ for Algo.1 and Algo.2
  • Figure 5: Left two:$\mu=0.05,\lambda=0.02$,Right two$\lambda=\mu=0.02$ for Algo.1 and Algo.2

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 17 more