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Yair Censor

Publications and source records attributed to Yair Censor.

At least 19 recordsLinked to original sources
arXiv2026

The best approximation tuple: an extension of the Cheney-Goldstein algorithm and results to the multiple sets case

In this paper we extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is the consideration of more than two sets. The other is the ability to handle each set as an intersections of a finite family of sets. We call the resulting problem the "Best Approximation Tuple (BAT) problem". The fundamental observation that leads to this generalizations is to recognize and handle one set (the "pivot set") as different from the remaining sets (the "satellite sets") instead of seeking cycles as the minimizers of a target functional. This enable us to overcome a certain theoretical obstacle related to cycles and minimizers of general functionals. We prove the convergence of the algorithm to the unique solution of the problem in the Euclidean case with strictly convex and compact satellite sets. Because of the lack of Fej\'er monotonicity, our convergence analysis is not standard, and is based on almost unknown properties of orthogonal projections regarding equality and inequality in the definition of nonexpansiveness.

math.OC↗
arXiv2026

A new theorem of alternatives leading to sufficient conditions for the superiorization guarantee question of Dynamic String-Averaging in the inconsistent case

We study the Superiorization Methodology (SM) in the context of the General Dynamic String-Averaging (GDSA) method in the inconsistent case (that is, where the input operators don't have a common fixed point) which primarily aims at achieving convex feasibility while simultaneously reducing an objective function. In many scientific and real-world problems modeled as constrained minimization tasks, striving for the exact constrained optimum can be costly in terms of time, energy, and resources. Therefore, applying the SM can offer a practical and efficient alternative. In particular, we present a new "theorem of alternatives" for the superiorization method which leads to investigation of theoretical conditions under which the superiorized version of the GDSA algorithm converges to a "superior" feasible point, i.e., one with an objective function value that is smaller or equal to that produced by the unperturbed feasibility-seeking algorithm. While this question has only been partially addressed in the existing literature, we present new sufficient conditions that guarantee that the SM attains such a superior outcome.

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arXiv2026

On the boundedness of infinite products of relaxed projections: perturbations resilience and dynamic string-averaging

Very recently (2026), Bauschke and Tung extended from finite- to infinite-dimensional Hilbert spaces a result published by Meshulam in 1996 (following an earlier result of Aharoni-Duchet-Wajnryb from 1984) regarding the boundedness of infinite products of relaxed projections onto a finite family of closed affine subspaces. In the present note we extend in various ways the result of Bauschke and Tung by allowing certain perturbations and proving perturbation resilience, by considering a mixture of closed half-spaces and closed hyperplanes, and by using dynamic weighted sums of dynamic strings (of dynamic lengths) of relaxed projections in the iterative process. We also discuss the limitation to generalize the Bauschke-Tung result to arbitrary closed and convex sets by presenting a large family of counterexamples in which the associated control is not cyclic and not even almost cyclic. Along the way we establish a general theorem of independent interest regarding the uniform boundedness and uniform unboundedness of infinite products of nonexpansive mappings in a normed space setting.

math.OC↗
arXiv2026

GPU-accelerated superiorization on constrained physical problems with SupPy

The superiorization method (SM) is situated between feasibility-seeking and constrained optimization. Instead of aiming at the minimum of a given objective function over a constraint set, it seeks a feasible point at which the objective function value is reduced - though not necessarily minimal - rather than hard targets, or in which a mathematically optimal solution is not strictly required. While the method has been investigated for several applications in physics, its broader use has been limited, in part due to the lack of openly available software for researchers wishing to explore it. In this work we apply superiorization to three problems from applied physics: seismic image reconstruction, low-dose CT reconstruction and intensity-modulated radiotherapy treatment planning. These experiments are conducted with SupPy, an open-source modularized Python toolbox developed for this work, which supports execution of feasibility-seeking algorithms and their superiorized version on both the CPU and the GPU. In all three cases the superiorized algorithms achieve favorable results compared to feasibility-seeking alone, with reduced noise in the imaging examples and lowered body dose in the radiotherapy plans. For the radiotherapy case we further observe that superiorization produces clinically viable plans on infeasible constraint sets.

physics.comp-ph↗
arXiv2026

Strong convergence, perturbation resilience and superiorization of Generalized Modular String-Averaging with infinitely many input operators

We study the strong convergence and bounded perturbation resilience of iterative algorithms based on the Generalized Modular String-Averaging (GMSA) procedure for infinite sequences of input operators under a general admissible control. These methods address a variety of feasibility-seeking problems in real Hilbert spaces, including the common fixed point problem and the convex feasibility problem. In addition to the general case, involving certain strongly quasi-nonexpansive input operators, we consider a specific subclass of their corresponding relaxed firmly nonexpansive operators. This subclass proves useful for establishing bounded perturbation resilience. We further demonstrate the applicability of our strong convergence results, within the GMSA framework, to the Superiorization Methodology and to Dynamic String-Averaging, analyzing the behavior of a superiorized version of our main algorithm. The novelty and significance of this work is that it not only includes a variety of earlier algorithms as special cases but, more importantly, it allows the use of modular options of string-averaging that give rise to new, hitherto unavailable, algorithmic schemes with emphasis on infinitely many input operators. The strong convergence guarantees and the applications for superiorization and dynamic string-averaging are also important facets.

math.OC↗
arXiv2026

Multi-Criteria Inverse Robustness in Radiotherapy Planning Using Semidefinite Programming

Radiotherapy planning naturally leads to a multi-criteria optimization problem which is subject to different sources of uncertainty. In order to find the desired treatment plan, a decision maker must balance these objectives as well as the level of robustness towards uncertainty against each other. This paper showcases a quantitative approach to do so, which combines the theoretical model with the ability to deal with practical challenges. To this end, the uncertainty, which can be expressed via the so-called dose-influence matrix, is modelled using interval matrices. We use inverse robustness to introduce an additional objective, which aims to maximize the volume of the uncertainty set. A multi-criteria approach allows to handle the uncertainty while keeping appropriate values of the other objective functions. We solve the resulting quadratically constrained quadratic optimization problem (QCQP) by first relaxing it to a convex semidefinite problem (SDP) and then reconstructing optimal solutions of the QCQP from solutions of the SDP.

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arXiv2025

GPU-accelerated FREDopt package for simultaneous dose and LETd proton radiotherapy plan optimization via superiorization methods

This study presents FREDopt, a newly developed GPU-accelerated open-source optimization software for simultaneous proton dose and dose-averaged LET (LETd) optimization in IMPT treatment planning. FREDopt was implemented entirely in Python, leveraging CuPy for GPU acceleration and incorporating fast Monte Carlo (MC) simulations from the FRED code. The treatment plan optimization workflow includes pre-optimization and optimization, the latter equipped with a novel superiorization of feasibility-seeking algorithms. Feasibility-seeking requires finding a point that satisfies prescribed constraints. Superiorization interlaces computational perturbations into iterative feasibility-seeking steps to steer them toward a superior feasible point, replacing the need for costly full-fledged constrained optimization. The method was validated on two treatment plans of patients treated in a clinical proton therapy center, with dose and LETd distributions compared before and after reoptimization. Simultaneous dose and LETd optimization using FREDopt led to a substantial reduction of LETd and (dose)x(LETd) in organs at risk (OARs) while preserving target dose conformity. Computational performance evaluation showed execution times of 14-50 minutes, depending on the algorithm and target volume size-satisfactory for clinical and research applications while enabling further development of the well-tested, documented open-source software.

physics.med-ph↗
arXiv2025

General Perturbation Resilient Dynamic String-Averaging for Inconsistent Problems with Superiorization

In this paper we introduce a General Dynamic String-Averaging (GDSA) iterative scheme and investigate its convergence properties in the inconsistent case, that is, when the input operators don't have a common fixed point. The Dynamic String-Averaging Projection (DSAP) algorithm itself was introduced in an 2013 paper, where its strong convergence and bounded perturbation resilience were studied in the consistent case (that is, when the sets under consideration had a nonempty intersection). Results involving combination of the DSAP method with superiorization, were presented in 2015. The proof of the weak convergence of our GDSA method is based on the notion of "strong coherence" of sequences of operators that was introduced in 2019. This is an improvement of the property of "coherence" of sequences of operators introduced in 2001 by Bauschke and Combettes. Strong coherence provides a more convenient sufficient convergence condition for methods that employ infinite sequences of operators and it turns out to be a useful general tool when applied to proving the convergence of many iterative methods. In this paper we combine the ideas of both dynamic string-averaging and strong coherence, in order to analyze our GDSA method for a general class of operators and its bounded perturbation resilience in the inconsistent case with weak and strong convergence. We then discuss an application of the GDSA method to the Superiorization Methodology, developing results on the behavior of its superiorized version.

math.OC↗
arXiv2025

A necessary condition for the guarantee of the superiorization method

We study a method that involves principally convex feasibility-seeking and makes secondary efforts of objective function value reduction. This is the well-known superiorization method (SM), where the iterates of an asymptotically convergent iterative feasibility-seeking algorithm are perturbed by objective function nonascent steps. We investigate the question under what conditions a sequence generated by an SM algorithm asymptotically converges to a feasible point whose objective function value is superior (meaning smaller or equal) to that of a feasible point reached by the corresponding unperturbed one (i.e., the exactly same feasibility-seeking algorithm that the SM algorithm employs.) This question is yet only partially answered in the literature. We present a condition under which an SM algorithm that uses negative gradient descent steps in its perturbations fails to yield such a superior outcome. The significance of the discovery of this negative condition is that it necessitates that the inverse of this condition will have to be assumed to hold in any future guarantee result for the SM. The condition is important for practitioners who use the SM because it is avoidable in experimental work with the SM, thus increasing the success rate of the method in real-world applications.

math.OC↗
arXiv2024

New results related to cutters and to an extrapolated block-iterative method for finding a common fixed point of a collection of them

Given a Hilbert space and a finite family of operators defined on the space, the common fixed point problem (CFPP) is to find a point in the intersection of the fixed point sets of these operators. Instances of the problem have numerous applications in science and engineering. We consider an extrapolated block-iterative method with dynamic weights for solving the CFPP assuming the operators belong to a wide class of operators called cutters. Global convergence is proved in two different scenarios, one of them is under a seemingly new condition on the weights which is less restrictive than a condition suggested in previous works. In order to establish convergence, we derive various new results of independent interest related to cutters, some of them extend, generalize and clarify previously published results.

math.OC↗
arXiv2024

Ionization detail parameters and cluster dose: A mathematical model for selection of nanodosimetric quantities for use in treatment planning in charged particle radiotherapy

Objective: To propose a mathematical model for applying Ionization Detail (ID), the detailed spatial distribution of ionization along a particle track, to proton and ion beam radiotherapy treatment planning (RTP). Approach: Our model provides for selection of preferred ID parameters (I_p) for RTP, that associate closest to biological effects. Cluster dose is proposed to bridge the large gap between nanoscopic I_p and macroscopic RTP. Selection of I_p is demonstrated using published cell survival measurements for protons through argon, comparing results for nineteen Ip: N_k; k = 2,3,...,10, the number of ionizations in clusters of k or more per particle, and F_k; k = 1,2,...,10, the number of clusters of k or more per particle. We then describe application of the model to ID-based RTP and propose a path to clinical translation. Main results: The preferred I_p were N_4 and F_5 for aerobic cells, N_5 and F_7 for hypoxic cells. Signifcant differences were found in cell survival for beams having the same LET or the preferred N_k. Conversely, there was no signi?cant difference for F_5 for aerobic cells and F_7 for hypoxic cells, regardless of ion beam atomic number or energy. Further, cells irradiated with the same cluster dose for these I_p had the same cell survival. Based on these preliminary results and other compelling results in nanodosimetry, it is reasonable to assert that I_p exist that are more closely associated with biological effects than current LET-based approaches and microdosimetric RBE-based models used in particle RTP. However, more biological variables such as cell line and cycle phase, as well as ion beam pulse structure and rate still need investigation. Signifcance: Our model provides a practical means to select preferred I_p from radiobiological data, and to convert I_p to the macroscopic cluster dose for particle RTP.

physics.med-ph↗
arXiv2024

Immunity to Increasing Condition Numbers of Linear Superiorization versus Linear Programming

Given a family of linear constraints and a linear objective function one can consider whether to apply a Linear Programming (LP) algorithm or use a Linear Superiorization (LinSup) algorithm on this data. In the LP methodology one aims at finding an optimal point, i.e., a point that fulfills the constraints and has the minimal value of the objective function over these constraints. The Linear Superiorization approach considers the same data as in linear programming problems but instead of attempting to solve with linear programming methods it employs perturbation resilient feasibility-seeking algorithms that steer the iterations toward a feasible point with reduced (not necessarily minimal) objective function value. This aim of the superiorization method (SM) is less demanding than aiming to reach full-fledged constrained optimality and it places more importance on reaching feasibility than on reaching optimality. Previous studies (e.g., [1]) compared LP and LinSup in terms of their respective outputs and the resources they use. Here, we investigate classical LP approaches and LinSup in terms of their sensitivity to condition numbers of the system of linear constraints. Condition numbers are a measure for the impact of deviations in the input data on the output of a problem and, in particular, they describe the factor of error propagation when given wrong or erroneous data. Therefore, the ability of LP and LinSup to cope with increased condition numbers, thus with illposed problems, is an important matter to consider which was not studied until now. We investigate experimentally the advantages and disadvantages of both LP and LinSup on exemplary sets of data of problems of linear programming with multiple condition numbers and different problem dimensions.

math.OC↗
arXiv2024

Floorplanning with I/O assignment via feasibility-seeking and superiorization methods

The feasibility-seeking approach offers a systematic framework for managing and resolving intricate constraints in continuous problems, making it a promising avenue to explore in the context of floorplanning problems with increasingly heterogeneous constraints. The classic legality constraints can be expressed as the union of convex sets. In implementation, we introduce a resetting strategy aimed at effectively reducing the problem of algorithmic divergence in the projection-based method used for the feasibility-seeking formulation. Furthermore, we introduce the novel application of the superiorization method (SM) to floorplanning, which bridges the gap between feasibility-seeking and constrained optimization. The SM employs perturbations to steer the iterations of the feasibility-seeking algorithm towards feasible solutions with reduced (not necessarily minimal) total wirelength. To evaluate the performance of Per-RMAP, we conduct comprehensive experiments on the MCNC benchmarks and GSRC benchmarks. The results demonstrate that we can obtain legal floorplanning results 166 times faster than the branch-and-bound (B&B) method while incurring only a 5% wirelength increase compared to the optimal results. Furthermore, we evaluate the effectiveness of the algorithmic flow that considers the I/O assignment constraints, which achieves an 6% improvement in wirelength. Besides, considering the soft modules with a larger feasible solution space, we obtain 15% improved runtime compared with PeF, the state-of-the-art analytical method. Moreover, we compared our method with Parquet-4 and Fast-SA on GSRC benchmarks which include larger-scale instances. The results highlight the ability of our approach to maintain a balance between floorplanning quality and efficiency.

math.OC↗
arXiv2024

The best approximation pair problem relative to two subsets in a normed space

In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. Motivated by our recent algorithm for solving the BAP problem [Censor, Mansour, Reem, J. Approx. Theory (2024)], we discuss the problem in more general normed spaces and with possibly non-convex subsets, and focus our attention on the fundamental issues of uniqueness and existence of the solution to the problem. We present several sufficient geometric conditions for the (at most) uniqueness of a BAP. These conditions are related to the structure and the relative orientation of the boundaries of the subsets and to the norm. We also present many sufficient conditions for the existence of a BAP. In general, the paper re-examines several aspects related to the BAP problem, including the historical one, and shows, probably for the first time, how wide is the scope of the BAP problem in terms of the scientific communities which are involved in it (frequently independently) and in terms of its applications.

math.OC↗
arXiv2023

The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets

Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of points for two polyhedra'', Computational Optimization and Applications 71 (2018), 509--523] who considered the case of finite-dimensional polyhedra.

math.OC↗
arXiv2023

Per-RMAP: Feasibility-Seeking and Superiorization Methods for Floorplanning with I/O Assignment

The feasibility-seeking approach provides a systematic scheme to manage and solve complex constraints for continuous problems, and we explore it for the floorplanning problems with increasingly heterogeneous constraints. The classic legality constraints can be formulated as the union of convex sets. However, the convergence of conventional projection-based algorithms is not guaranteed as the constrain sets are non-convex. In this work, we propose a resetting strategy to greatly eliminate the the divergence issue of the projection-based algorithm for the feasibility-seeking formulation. Furthermore, the superiorization methodology (SM), which lies between feasibility-seeking and constrained optimization, is firstly applied to floorplanning. The SM uses perturbations to steer the feasibility-seeking algorithm to a feasible solution with shorter total wirelength. The proposed flow is extendable to tackle various constraints and variants of floorplanning problems, e.g., floorplanning with I/O assignment problems. We have evaluated the proposed algorithm on the MCNC benchmarks. We can obtain legal floorplans only two times slower than the branch-and-bound method in its current prototype using MATLAB, with only 3% wirelength inferior to the optimal results. We evaluate the effectiveness of the flow by considering the constraints of I/O assignment, and our algorithm achieve 8% improvement on wirelength.

math.OC↗
arXiv2022

Superiorization: The asymmetric roles of feasibility-seeking and objective function reduction

The superiorization methodology can be thought of as lying conceptually between feasibility-seeking and constrained minimization. It is not trying to solve the full-fledged constrained minimization problem composed from the modeling constraints and the chosen objective function. Rather, the task is to find a feasible point which is "superior" (in a well-defined manner) with respect to the objective function, to one returned by a feasibility-seeking only algorithm. We telegraphically review the superiorization methodology and where it stands today and propose a rigorous formulation of its, yet only partially resolved, guarantee problem. The real-world situation in an application field is commonly represented by constraints defined by the modeling process and the data, obtained from measurements or otherwise dictated by the model-user. The feasibility-seeking problem requires to find a point in the intersection of all constraints without using any objective function to aim at any specific feasible point. At the heart of the superiorization methodology lies the modeler desire to use an objective function, that is exogenous to the constraints, in order to seek a feasible solution that will have lower (not necessarily minimal) objective function value. This aim is less demanding than full-fledged constrained minimization but more demanding than plain feasibility-seeking. Putting emphasis on the need to satisfy the constraints, because they represent the real-world situation, one recognizes the "asymmetric roles of feasibility-seeking and objective function reduction", namely, that fulfilling the constraints is the main task while reduction of the exogenous objective function plays only a secondary role. There are two research directions in the superiorization methodology: Weak superiorization and strong superiorization.

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