Variable Smoothing for Weakly Convex Composite Functions.

Area of Science:

• Optimization Theory
• Applied Mathematics
• Image Processing

Background:

• Minimizing structured objective functions is crucial in many scientific fields.
• Existing methods often struggle with weakly convex functions, limiting applications like image reconstruction.
• Regularizers in image reconstruction can introduce bias, motivating research into less biased alternatives.

Purpose of the Study:

• To develop an efficient algorithm for minimizing a structured objective function.
• To address the challenge of minimizing functions involving weakly convex components.
• To improve upon existing complexity bounds for nonsmooth optimization problems.

Main Methods:

• A variable smoothing algorithm based on the Moreau envelope was developed.
• A decreasing sequence of smoothing parameters was employed.
• The algorithm's convergence complexity was rigorously analyzed.

Main Results:

• A novel complexity bound of O(1/epsilon^2) was established for achieving an epsilon-approximate solution.
• This bound interpolates between established bounds for smooth functions and subgradient methods.
• The proposed method is suitable for image reconstruction with less biased regularizers.

Conclusions:

• The variable smoothing algorithm offers an effective approach for structured nonsmooth optimization.
• The derived complexity bound advances the theoretical understanding of minimizing weakly convex functions.
• This work has direct implications for improving image reconstruction techniques.