MOCCA: Mirrored Convex/Concave Optimization for Nonconvex Composite Functions.

Area of Science:

• Optimization Theory
• High-Dimensional Statistics
• Convex and Non-Convex Analysis

Background:

• Many high-dimensional statistical problems involve composite objective functions that are sums of simpler terms.
• Existing optimization methods struggle with non-convexity and non-differentiability in these composite functions, potentially failing to converge or handle complexity.
• Problems in areas like computed tomography (CT) imaging present specific challenges for current optimization techniques.

Purpose of the Study:

• To develop a novel optimization algorithm capable of handling non-convex and non-differentiable terms in composite objective functions.
• To address the limitations of existing methods in solving complex optimization problems arising in high-dimensional statistics and signal recovery.
• To provide theoretical convergence guarantees for a class of approximately convex problems.

Main Methods:

• Proposal of the mirrored convex/concave (MOCCA) algorithm, a primal/dual optimization approach.
• MOCCA utilizes local convex approximations of each term in the objective function at every iteration.
• The algorithm is inspired by and tested on optimization problems encountered in computed tomography (CT) imaging.

Main Results:

• The MOCCA algorithm demonstrates the ability to handle a range of non-convex composite optimization problems.
• Theoretical guarantees for convergence are established for problems that are approximately convex.
• Empirical results show fast convergence rates for several structured signal recovery applications.

Conclusions:

• The MOCCA algorithm offers a robust and efficient solution for complex optimization problems in high-dimensional statistics.
• It overcomes limitations of existing methods by effectively managing non-convexity and non-differentiability.
• The algorithm shows significant promise for applications in signal recovery and related fields.