Scalable Proximal Jacobian Iteration Method With Global Convergence Analysis for Nonconvex Unconstrained Composite...

Area of Science:

• Optimization Theory
• Numerical Analysis
• Applied Mathematics

Background:

• Nonconvex relaxation functions often yield superior performance in l0-norm and rank minimization compared to convex alternatives.
• The lack of convexity in these problems presents significant challenges for developing algorithms with guaranteed convergence.

Purpose of the Study:

• To extend proximal gradient methods (PGMs) to proximal Jacobian iteration methods (PJIMs) for a class of nonconvex composite optimization problems with two splitting variables.
• To develop an accelerated version of PJIMs using Nesterov's acceleration strategy and extend these methods to multivariable cases.

Main Methods:

• Extension of proximal gradient methods (PGMs) to proximal Jacobian iteration methods (PJIMs).
• Incorporation of Nesterov's acceleration strategy for improved iteration efficiency.
• Rigorous convergence analysis utilizing the Kurdyka-Łojasiewica (KŁ) property.

Main Results:

• Global convergence of the generated variable sequence to a critical point is proven for a broad class of functions.
• Establishment of linear and sublinear convergence rates for the objective function.
• Numerical experiments demonstrate fast convergence and high precision in nonconvex sparse and low-rank recovery problems.

Conclusions:

• The proposed proximal Jacobian iteration methods (PJIMs) effectively address nonconvex composite optimization problems.
• The accelerated PJIMs offer computational advantages by reducing the number of iterations.
• These methods show significant promise for applications in sparse and low-rank recovery.