Special issue on superiorization: theory and applications

Many problems in science and engineering involve, as part of their solution process, the consideration of a separable function which is the sum of two convex functions, one of them possibly non-smooth. Recently a few works have discussed inexact versions of several accelerated proximal methods aiming at solving this minimization problem. This paper shows that inexact versions of a method of Beck and Teboulle (fast iterative shrinkable tresholding algorithm) preserve, in a Hilbert space setting, the same (non-asymptotic) rate of convergence under some assumptions on the decay rate of the error terms The notion of inexactness discussed here seems to be rather simple, but, interestingly, when comparing to related works, closely related decay rates of the errors terms yield closely related convergence rates. The derivation sheds some light on the somewhat mysterious origin of some parameters which appear in various accelerated methods. A consequence of the analysis is that the accelerated method is perturbation resilient, making it suitable, in principle, for the superiorization methodology. By taking this into account, we re-examine the superiorization methodology and significantly extend its scope.

In this paper, we introduce a subclass of strictly quasi-nonexpansive operators which consists of well-known operators as paracontracting operators (e.g., strictly nonexpansive operators, metric projections, Newton and gradient operators), subgradient projections, a useful part of cutter operators, strictly relaxed cutter operators and locally strongly Féjer operators. The members of this subclass, which can be discontinuous, may be employed by fixed point iteration methods; in particular, iterative methods used in convex feasibility problems. The closedness of this subclass, with respect to composition and convex combination of operators, makes it useful and remarkable. Another advantage with members of this subclass is the possibility to adapt them to handle convex constraints. We give convergence result, under mild conditions, for a perturbation resilient iterative method which is based on an infinite pool of operators in this subclass. The perturbation resilient iterative methods are relevant and important for their possible use in the framework of the recently developed superiorization methodology for constrained minimization problems. To assess the convergence result, the class of operators and the assumed conditions, we illustrate some extensions of existence research works and some new results.

Hierarchical convex optimization concerns two-stage optimization problems: the first stage problem is a convex optimization; the second stage problem is the minimization of a convex function over the solution set of the first stage problem. For the hierarchical convex optimization, the hybrid steepest descent method (HSDM) can be applied, where the solution set of the first stage problem must be expressed as the fixed point set of a certain nonexpansive operator.

In this paper, we propose a nonexpansive operator that yields a computationally efficient update when it is plugged into the HSDM. The proposed operator is inspired by the update of the linearized augmented Lagrangian method. It is applicable to characterize the solution set of recent sophisticated convex optimization problems found in the context of inverse problems, where the sum of multiple proximable convex functions involving linear operators must be minimized to incorporate preferable properties into the minimizers. For such a problem formulation, there has not yet been reported any nonexpansive operator that yields an update free from the inversions of linear operators in cases where it is utilized in the HSDM. Unlike previously known nonexpansive operators, the proposed operator yields an inversion-free update in such cases. As an application of the proposed operator plugged into the HSDM, we also present, in the context of the so-called superiorization, an algorithmic solution to a convex optimization problem over the generalized convex feasible set where the intersection of the hard constraints is not necessarily simple.

The common fixed point problem is to find a common fixed point of a finite family of mappings. In the present paper our goal is to obtain its approximate solution using two perturbed algorithms. The first algorithm is an iterative method for problems in a metric space while the second one is a dynamic string-averaging algorithms for problems in a Hilbert space.

We first consider nonexpansive self-mappings of a metric space and study the asymptotic behavior of their inexact orbits. We then apply our results to the analysis of iterative methods for finding approximate fixed points of nonexpansive mappings and approximate zeros of monotone operators.

Linear superiorization (LinSup) considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not necessarily minimal) target function values. The two questions that we set out to explore experimentally are: (i) does LinSup provide a feasible point whose linear target function value is lower than that obtained by running the same feasibility-seeking algorithm without superiorization under identical conditions? (ii) How does LinSup fare in comparison with the Simplex method for solving linear programming problems? Based on our computational experiments presented here, the answers to these two questions are: 'yes' and 'very well', respectively.

We first prove the bounded perturbation resilience for the successive fixed point algorithm of averaged mappings, which extends the string-averaging projection and block-iterative projection methods. We then apply the superiorization methodology to a constrained convex minimization problem where the constraint set is the intersection of fixed point sets of a finite family of averaged mappings.

Bounded perturbation resilience and superiorization techniques for the projected scaled gradient (PSG) method are studied under the general Hilbert space setting. Weak convergence results of the (superiorized) PSG method and its relaxed version are proved under the assumption that the errors be summable. It is also shown that the PSG method converges in a sublinear rate and can be accelerated to the convergence rate $O\left(\tfrac{1}{{n}^{2}}\right)$. Applications to linear inverse problems and split feasibility problems are discussed.

The convex feasibility problem is to find a common point of a finite family of closed convex subsets. In many applications one requires something more, namely finding a common point of closed convex subsets which minimizes a continuous convex function. The latter requirement leads to an application of the superiorization methodology which is actually settled between methods for convex feasibility problem and the convex constrained minimization. Inspired by the superiorization idea we introduce a method which sequentially applies a long-step algorithm for a sequence of convex feasibility problems; the method employs quasi-nonexpansive operators as well as subgradient projections with level control and does not require evaluation of the metric projection. We replace a perturbation of the iterations (applied in the superiorization methodology) by a perturbation of the current level in minimizing the objective function. We consider the method in the Euclidean space in order to guarantee the strong convergence, although the method is well defined in a Hilbert space.

We propose the superiorization of incremental algorithms for tomographic image reconstruction. The resulting methods follow a better path in its way to finding the optimal solution for the maximum likelihood problem in the sense that they are closer to the Pareto optimal curve than the non-superiorized techniques. A new scaled gradient iteration is proposed and three superiorization schemes are evaluated. Theoretical analysis of the methods as well as computational experiments with both synthetic and real data are provided.

To reduce the x-ray dose in computerized tomography (CT), many constrained optimization approaches have been proposed aiming at minimizing a regularizing function that measures a lack of consistency with some prior knowledge about the object that is being imaged, subject to a (predetermined) level of consistency with the detected attenuation of x-rays. One commonly investigated regularizing function is total variation (TV), while other publications advocate the use of some type of multiscale geometric transform in the definition of the regularizing function, a particular recent choice for this is the shearlet transform. Proponents of the shearlet transform in the regularizing function claim that the reconstructions so obtained are better than those produced using TV for texture preservation (but may be worse for noise reduction). In this paper we report results related to this claim. In our reported experiments using simulated CT data collection of the head, reconstructions whose shearlet transform has a small ℓ₁-norm are not more efficacious than reconstructions that have a small TV value. Our experiments for making such comparisons use the recently-developed superiorization methodology for both regularizing functions. Superiorization is an automated procedure for turning an iterative algorithm for producing images that satisfy a primary criterion (such as consistency with the observed measurements) into its superiorized version that will produce results that, according to the primary criterion are as good as those produced by the original algorithm, but in addition are superior to them according to a secondary (regularizing) criterion. The method presented for superiorization involving the ℓ₁-norm of the shearlet transform is novel and is quite general: It can be used for any regularizing function that is defined as the ℓ₁-norm of a transform specified by the application of a matrix. Because in the previous literature the split Bregman algorithm is used for similar purposes, a section is included comparing the results of the superiorization algorithm with the split Bregman algorithm.

Multicriteria optimization problems occur in many real life applications, for example in cancer radiotherapy treatment and in particular in intensity modulated radiation therapy (IMRT). In this work we focus on optimization problems with multiple objectives that are ranked according to their importance. We solve these problems numerically by combining lexicographic optimization with our recently proposed level set scheme, which yields a sequence of auxiliary convex feasibility problems; solved here via projection methods. The projection enables us to combine the newly introduced superiorization methodology with multicriteria optimization methods to speed up computation while guaranteeing convergence of the optimization. We demonstrate our scheme with a simple 2D academic example (used in the literature) and also present results from calculations on four real head neck cases in IMRT (Radiation Oncology of the Ludwig-Maximilians University, Munich, Germany) for two different choices of superiorization parameter sets suited to yield fast convergence for each case individually or robust behavior for all four cases.

Proton therapy is a precise form of radiotherapy in which the range of an energetic beam of protons within a patient must be accurately known. The current approach based on single-energy computed tomography (SECT) can lead to uncertainties in the proton range of approximately 3%. This range of uncertainty may lead to under-dosing of the tumour or over-dosing of healthy tissues. Dual-energy CT (DECT) theoretically has the potential to reduce these range uncertainties by quantifying electron density and the effective atomic number. In practice, however, DECT images reconstructed with filtered backprojection (FBP) tend to suffer from high levels of noise. The objective of the current work was to examine the effect of total variation superiorization (TVS) on proton therapy planning accuracy when compared with FBP. A virtual CT scanner was created with the Monte Carlo toolkit Geant4. Tomographic images were reconstructed with FBP and TVS combined with diagonally relaxed orthogonal projections (TVS-DROP). A total variation minimization (TVM) filter was also applied to the image reconstructed with FBP (FBP-TVM). Quantitative accuracy and variance of proton relative stopping power (RSP) derived from each image set was assessed. Mean RSPs were comparable with each image; however, the standard deviation of pixel values with TVS-DROP was reduced by a factor of 0.44 compared with the FBP image and a factor of 0.66 when compared with the FBP-TVM image. Proton doses calculated with the TVS-DROP image set were also better able to predict a reference dose distribution when compared with the FBP and FBP-TVM image sets. The study demonstrated the potential advantages of TVS-DROP as an image reconstruction method for DECT applied to proton therapy treatment planning.

The recently-developed superiorization approach is efficient and robust for solving various constrained optimization problems. This methodology can be applied to multi-energy CT image reconstruction with the regularization in terms of the prior rank, intensity and sparsity model (PRISM). In this paper, we propose a superiorized version of the simultaneous algebraic reconstruction technique (SART) based on the PRISM model. Then, we compare the proposed superiorized algorithm with the Split-Bregman algorithm in numerical experiments. The results show that both the Superiorized-SART and the Split-Bregman algorithms generate good results with weak noise and reduced artefacts.