We consider a broad class of regularized structured total least squares (RSTLS) problems encompassing many scenarios in image processing. This class of problems results in a nonconvex and often nonsmooth model in large dimension. To tackle this difficult class of problems we introduce a novel algorithm which blends proximal and alternating minimization methods by beneficially exploiting data information and structures inherently present in RSTLS. The proposed algorithm, which can also be

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... to more general problems, is proven to globally converge to critical points and is amenable to efficient and simple computational steps. We illustrate our theoretical findings by presenting numerical experiments on deblurring large scale images, which demonstrate the viability and effectiveness of the proposed method. A key difficulty in the RSTLS problem is the nonconvexity in the variables (x, y) due to the coupling term in the squared norm of the objective function in (1). Another difficulty is the large scale nature of the problem which naturally arises in many applications of interest, as well as the possible nonsmoothness of the regularizer F (·). Thus, we face a problem sharing the three most difficult properties an optimization problem can have-nonconvexity, nonsmoothness, and large size-precluding the direct use of any standard optimization schemes in its solution. However, the RSTLS problem also shares some particular structures and data information, such as convexity in separate arguments (x, y) and smoothness of the least squares term, that can be beneficially exploited. In the present work we will strongly exploit the aforementioned structures and properties; our main objective is to devise a simple and efficient algorithm proven to globally converge to a critical point of the nonconvex objective function of (1) and capable of handling large scale instances. To achieve this goal we blend alternating minimization and proximal methods. These two very well known paradigms have recently attracted intensive research activity in many disparate applications due to their simplicity and remarkable theoretical and practical performance, mainly in the convex setting; see, e.g., [3, 6, 13, 39] and references therein for a small representative sample of this activity. However, here the RSTLS problem under consideration is nonconvex and nonsmooth. Motivated by the recent algorithmic and convergence analysis framework developed in [12] , which builds on the powerful Kurdyka-Lojasiewicz property [22, 24] to handle genuine nonconvex and nonsmooth minimization problems, we address the inherent nonconvex difficulty present in the RSTLS problem by further exploiting the problem's data information. This leads us to introduce a novel algorithm which relies on alternating minimization and on semiproximal regularization, which for ease of reference is called SPA. While the focus of this paper is on RSTLS problems, our analysis is developed for broader class of problems which captures the class of RSTLS problems and the corresponding algorithm as a particular case. Thus, our results can also be applied to other applications and contexts sharing this proposed broader formulation. For the RSTLS problem, the resulting algorithm involves two simple computational steps. One asks for the solution of a small scale (p × p) linear system, while the other step, depending on the choice of the regularizer F (·), either admits a closed-form solution or can be efficiently computed via a fast dual proximal method [7] . We prove that the proposed algorithm SPA globally converges to a critical point of the problem at hand. Finally, we illustrate our theoretical findings by presenting some numerical examples on image deblurring which demonstrate the effectiveness of the proposed method. Outline of the paper. The paper is organized as follows. In the next section we describe various TLS models which have motivated the proposed RSTLS approach and make our setting more precise. While the focus of this paper is on the RSTLS problem, our analysis is developed for a broader class of problems which captures the class of RSTLS problems and the corresponding algorithm as a particular case; this is developed in section 3. In section 4, we present a general analysis framework, and we state and prove the main convergence results. Finally, in section 5, numerical results on image deblurring problems illustrate our theoretical findings. To make the paper self-contained, a short appendix includes some relevant additional technical material. Our notation is quite standard and will be defined throughout the text if and when necessary.