Global Optimality for Non-linear Constrained Restoration Problems via Invexity

Signal restoration is an important constrained optimization problem with significant applications in various domains. Although non-convex constrained optimization problems have been shown to perform better than convex counterparts in terms of reconstruction quality, convex constrained optimization problems have been preferred for their global optima guarantees. Despite the success of nonconvex methods in many applications, it is not an overstatement to say that there is little or no hope for non-convex problems to ensure global optima. In this paper, for the first time, we propose a family of invex functions for handling constrained inverse problems using the non-convex setting along with guarantees for their global optima -invex function is a mapping where any critical point is a global minimizer. We also develop relevant theories to extend the global optima guarantee to a family of quasi-invex functions -the largest set of optimizable mappings. Our theoretical results show that the proposed family of invex and quasi-invex functions can aid in extending existing convex optimization algorithms, such as the alternating direction of multipliers and accelerated proximal gradient methods. Moreover, our numerical tests show that the proposed family of invex/quasi-invex functions overcome the performance of convex mappings in terms of reconstruction quality across several relevant metrics and imaging tasks.