Regularization of certain linear discrete ill-posed problems, as well as of certain regression problems, can be formulated as large-scale, possibly nonconvex, minimization problems, whose objective function is the sum of the $p$-th power of the $\ell^p$-norm of a fidelity term and the $q$-th power of the $\ell^q$-norm of a regularization term, with $0< p,q \leq 2$. We describe new restarted iterative solution methods that require less computer storage and execution time than the methods described by [1]. The reduction in computer storage and execution time is achieved by periodic restarts of the method. Computed examples illustrate that restarting does not reduce the quality of the computed solutions.

[1] G.-X. Huang, A. Lanza, S. Morigi, L. Reichel and F. Sgallari. Majorization–minimization generalized Krylov subspace methods for $\ell_p-\ell_q$ optimization applied to image restoration, BIT Numerical Mathematics 57: 351-378, 2017.

In the present work, we investigate the inverse problem of identifying simultaneously the denoised image and the weighting parameter that controls the balance between two diffusion operators for an evolutionary partial differential equation (PDE). The problem is formulated as a non-smooth PDE-constrained optimization model. This PDE is constructed by second- and fourth-order diffusive tensors that combines the benefits from the diffusion model of Perona-Malik in the homogeneous regions, the Weickert model near sharp edges and the fourth-order term in reducing staircasing. The existence and uniqueness of solutions for the proposed PDE-constrained optimization system are provided in a suitable Sobolev space. Also, an optimization problem for the determination of the weighting parameter is introduced based on the Primal-Dual algorithm. Finally, simulation results show that the obtained parameter usually coincides with the better choice related to the best restoration quality of the image.

We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with partial differential equations (PDEs). By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg method using a block triangular constraint preconditioner. Numerical experiments show reliable and competitive performance for an optimal control problem in elasticity.

In this talk, we develop a new trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations. We prove global convergence of our method in Hilbert space and analyze the worst-case complexity to reach a prescribed tolerance. Our method employs the proximal mapping of the nonsmooth objective function and is simple to implement. Moreover, when using a quadratic Taylor model, our algorithm represents a matrix-free proximal Newton-type method that permits indefinite Hessians. We additionally elaborate on potential trust-region sub-problem solvers and discuss local convergence guarantees. We demonstrate the efficacy of our algorithm on various examples from PDE-constrained optimization.