Optimization methods for regularization-based ill-posed problems: a survey and a multi-objective framework

Ill-posed problems are widely existed in signal processing. In this paper, we review popular regularization models such as truncated singular value decomposition regularization, iterative regularization, variational regularization. Meanwhile, we also retrospect popular optimization approaches and regularization parameter choice methods. In fact, the regularization problem is inherently a multi-objective problem. The traditional methods usually combine the fidelity term and the regularization term into a single-objective with regularization parameters, which are difficult to tune. Therefore, we propose a multi-objective framework for ill-posed problems, which can handle complex features of problem such as non-convexity, discontinuity. In this framework, the fidelity term and regularization term are optimized simultaneously to gain more insights into the ill-posed problems. A case study on signal recovery shows the effectiveness of the multi-objective framework for ill-posed problems.