Reconstruction in inverse problems is commonly performed by minimizing a data-fidelity term, such as the least-squares objective. However, when the data are noisy, this minimizer may not represent the true solution. The superiorization methodology attempts to improve the structure of the solution by perturbing iterative schemes, but such methods still target the noisy data-fit minimizer. This work proposes a paradigm shift: rather than improving convergence to a flawed target, we redefine the reconstruction goal itself. We introduce the Superiorized Biased Steepest Descent (S-BSD-LS) algorithm, which combines two key components: (i) a deliberately biased residual update that decouples the method from the noisy least-squares solution and (ii) a total variation (TV) perturbation that promotes a desirable structure. We rigorously analyze convergence of the algorithm using an auxiliary sequence and characterize the systematic bias induced by both the residual update and the data noise. This approach offers a new perspective on solving ill-posed noisy problems. We demonstrate the performance of our proposed method, S-BSD-LS, for tomographic imaging examples using total variation regularization as a specific instance of the general framework and compare it with state-of-the-art methods, including the standard steepest descent method, the conjugate gradient method for least-squares (CGLS) , the superiorized conjugate gradient method with conjugate descent (S-CG-CD), and FISTA, a well-known proximal gradient algorithm. In each iteration, S-BSD-LS requires two matrix-vector multiplications—similar to CGLS—and one additional function evaluation compared to CGLS while still maintaining competitive performance. Additional experiments confirm that the superiorization mechanism is primarily responsible for the improved stability and a continuous error reduction while the biased residual update mainly enhances computational efficiency.
Nikazad et al. (2026) studied this question.