Aviv Gibali

Braude College · Mathematics

In this paper we propose a new reformulation of the linear Pascoletti-Serafini problem as a Bi-Level Optimization problem. The Pascoletti-Serafini problem stands at the core of many Multi-Criteria Optimization problems, and in particular its linear version is used for navigation purposes on the Pareto frontier. The new reformulation is based on the...

The Split Feasibility Problem (SFP), which was introduced by Censor and Elfving, consists of finding a point in a set C in one space such that its image under a linear transformation belongs to another set Q in the other space. This problem was well studied both theoretically and practically as it was also used in practice in the area of Intensity-...

This paper is concerned with the variational inequality problem VIP(F,Fix(T)): find u∈Fix(T) such that 〈F(u),z-u〉≥0 for all z∈Fix(T), where T:ℝⁿ→ℝⁿ is quasi-nonexpansive, Fix(T) is its nonempty fixed point set, and F:ℝⁿ→ℝⁿ is monotone. We study the convergence properties of sequences generated by the following hybrid steepest descent method: $u^{k+...

Modifying von Neumann's alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert...

We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms 59 (2012), 301--323]. The SCNPP with...

In this paper we study the split minimization problem that consists of two constrained minimization problems in two separate spaces that are connected via a linear operator that maps one space into the other. To handle the data of such a problem we develop a superiorization approach that can reach a feasible point with reduced (not necessarily mini...

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted a...

Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related wo...

The notion of well-posedness has drawn the attention of many researchers in non-linear analysis in connection with problems where the exact solution is unknown or may be costly to compute. Well-posedness guarantees the convergence of a sequence of approximate solutions obtained by iterative methods to the exact solution of the given problem. Motiva...

We study the problem of finding a common element that solves the multiple-sets feasibility and equilibrium problems in real Hilbert spaces. We consider a general setting in which the involved sets are represented as level sets of given convex functions, and propose a constructible linear approximation scheme that involves the subgradient of the ass...

In the study of many real-world problems such as engineering design and industrial process control, one often needs to select certain elements/controls from a feasible set in order to optimize the design or system based on certain criteria [...]

In this paper we study the split minimization problem that consists of two constrained minimization problems in two separate spaces that are connected via a linear operator that maps one space into the other. To handle the data of such a problem we develop a superiorization approach that can reach a feasible point with reduced (not necessarily mini...

In this paper we are concerned with the Common Fixed Point Problem (CFPP) with demi-contractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordoñez et al. [35] and in general, the field of online/real-time algorithms, for example [19, 20, 30], in which the entire...

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted a...

In this paper, we define a generalized a-v-Meir-Keeler-type contraction mapping in the setting of modular extended b-metric spaces which generalized some results in classical metric spaces, modular b-metric spaces alike. We also give a supportive example to justify our claims. The results we establish in this paper extended, improved, generalized a...

In this paper, we define a generalized α-v^-Meir–Keeler-type contraction mapping in the setting of modular extended b-metric spaces which generalized some results in classical metric spaces, modular b-metric spaces alike. We also give a supportive example to justify our claims. The results we establish in this paper extended, improved, generalized...

Nonlinear operator theory is an important area of nonlinear functional analysis. This area encompasses diverse nonlinear problems in many areas of mathematics, the physical sciences and engineering such as monotone operator equations, fixed point problems and more. In this work we are concern with the problem of finding a common solution of a monot...

Symmetry plays an important role in solving practical problems of applied science, especially in algorithm innovation. In this paper, we propose what we call the self-adaptive inertial-like proximal point algorithms for solving the split common null point problem, which use a new inertial structure to avoid the traditional convergence condition in...

Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical gu...

In this paper, we introduce two simple inertial algorithms for solving the split variational inclusion problem in Banach spaces. Under mild and standard assumptions, we establish the weak and strong convergence of the proposed methods, respectively. As theoretical realization we study existence of solutions of the split common fixed point problem i...

In this work we propose an accelerated algorithm that combines various techniques, such as inertial proximal algorithms, Tseng’s splitting algorithm, and more, for solving the common variational inclusion problem in real Hilbert spaces. We establish a strong convergence theorem of the algorithm under standard and suitable assumptions and illustrate...

In this paper, we introduce a relaxed CQ method with alternated inertial step for solving split feasibility problems. We give convergence of the sequence generated by our method under some suitable assumptions. Some numerical implementations from sparse signal and image deblurring are reported to show the efficiency of our method.

Dynamic user equilibrium (DUE) is a Nash-like solution concept describing an equilibrium in dynamic traffic systems over a fixed planning period. DUE is a challenging class of equilibrium problems, connecting network loading models and notions of system equilibrium in one concise mathematical framework. Recently, Friesz and Han introduced an integr...

In this paper, we are interested in the generalized variational inequality problem in real Hilbert spaces. We propose an explicit proximal method which requires only one proximal step and one mapping evaluation per iteration and also uses an adaptive step-size rule that enables to avoid the prior knowledge of the Lipschitz constant of the involved...

We propose a multi-time generalized Nash equilibrium problem and prove its equivalence with a multi-time quasi-variational inequality problem. Then, we establish the existence of equilibria. Furthermore, we demonstrate that our multi-time generalized Nash equilibrium problem can be applied to solving traffic network problems, the aim of which is to...

In this work we study various gap functions for the generalized multivalued mixed variational-hemivariational inequality problems by using the \((\tau _{\mathscr {M}},\sigma _{\mathscr {M}})\)-relaxed cocoercive mapping and Hausdorff Lipschitz continuity. Moreover, we establish global error bounds for such inequalities using the characteristic of t...

The ENGINO Toy System introduced two challenging problems. The first was to get bounds on the number of possible models/toys which can be constructed using a given package of building blocks. And the second is to generate automatically the assembly instructions for a given toy. In this report we summarize our insights and provide preliminary result...

Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical gu...

In this paper, we study a class of two level multiple sets split pseudomonotone equilibrium problem in real Hilbert spaces. A new parallel projection method for these class of problems is presented and a strong convergence theorem for solving the problem is established under mild and standard assumptions. A numerical example is given in support of...

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs e...

Constrained convex optimization problems arise naturally in many real-world applications. One strategy to solve them in an approximate way is to translate them into a sequence of convex feasibility problems via the recently developed level set scheme and then solve each feasibility problem using projection methods. However, if the problem is ill-co...

In this paper, we are concern with the classical equilibrium problem in real Hilbert spaces and introduce two new extragradient variants for it. By taking into account several fixed point theory techniques, we obtain simple structure methods that converge strongly and hence demonstrate the theoretical advantage of our methods. Moreover, our converg...

In this paper, we introduce a new algorithm by incorporating an inertial term with a subgradient extragradient algorithm to solve the equilibrium problems involving a pseudomonotone and Lipschitz-type continuous bifunction in real Hilbert spaces. A weak convergence theorem is well established under certain mild conditions for the bifunction and the...

Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in the fact that only one projection onto the...

In this paper, we study the split common fixed point problem for demicontractive mappings in real Hilbert spaces. We propose an alternative regularization scheme with self-adaptive step size for solving the problem and prove its strong convergence. Furthermore, we apply our main results to split feasibility problem, split variational inequality pro...

We study a feasibility‐seeking problem with percentage violation constraints (PVCs). These are additional constraints that are appended to an existing family of constraints, which single out certain subsets of the existing constraints and declare that up to a specified fraction of the number of constraints in each subset is allowed to be violated b...

In this paper we study a feasibility-seeking problem with percentage violation constraints. These are additional constraints, that are appended to an existing family of constraints, which single out certain subsets of the existing constraints and declare that up to a specified fraction of the number of constraints in each subset is allowed to be vi...

In this work we are concerned with variational inequalities in real Hilbert spaces and introduce a new double projection method for solving it. The algorithm is motivated by the Korpelevich extragradient method, the subgradient extragradient method of Gibali et al. and Popov’s method. The proposed scheme combines some of the advantages of the metho...

In this paper, we study the problem of finding a common solution to variational inequality and fixed point problems for a countable family of Bregman weak relatively nonexpansive mappings in real reflexive Banach spaces. Two inertial-type algorithms with adaptive step size rules for solving the problem are presented and their strong convergence the...

Van Quy (Optimization 68(4):753-771, 2018) established an extragradient-CQ algorithm for solving a class of bilevel split equilibrium problem. The step-size in the algorithm requires computation of a certain matrix norm which is costly. Moreover, bilevel problems often possess huge number of constraints and possibly require a robust and adaptive st...

In this paper, we propose a catalog of iterative methods for solving the Split Feasibility Problem in the non-convex setting. We study four different optimization formulations of the problem, where each model has advantageous in different settings of the problem. For each model, we study relevant iterative algorithms, some of which are well-known i...

In this paper, we propose a catalog of iterative methods for solving the Split Feasibility Problem in the non-convex setting. We study four different optimization formulations of the problem, where each model has advantageous in different settings of the problem. For each model, we study relevant iterative algorithms, some of which are well-known i...

In this paper, we introduce several inertial-like algorithms for solving equilibrium problems (EP) in real Hilbert spaces. The algorithms are constructed using the resolvent of the EP associated bifunction and combines the inertial and the Mann-type technique. Under mild and standard conditions imposed on the cost bifunction and control parameters...

The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a forward–backward–forward splitting method with reflection steps (symmetric) in real Hilbert spaces. Weak and strong convergence analyses of t...

In this paper, we study the split equality problem for systems of monotone variational inclusions and fixed point problems of set-valued demi-contractive mappings in real Hilbert spaces. A new viscosity algorithm for solving this problem is introduced along with its strong convergence theorem. Several known theoretical applications, such as, split...

Dynamic user equilibrium (DUE) is a Nash-like solution concept describing an equilibrium in dynamic traffic systems over a fixed planning period. DUE is a challenging class of equilibrium problems, connecting network loading models and notions of system equilibrium in one concise mathematical framework. Recently, Friesz and Han introduced an integr...

In this paper, we study the problem of finding a common solution to an equilibrium and split convex feasibility problems in real Hilbert spaces. Inspired and motivated by classical as well as recent works in this field, we introduce several simple inertial self-adaptive algorithms for solving this problem. Convergence of the algorithms is given und...

In this paper we propose a new generalized mixed equilibrium problem which includes many existing monotone equilibrium problems and their derivatives. Using the KKM technique, we establish the existence of solutions of the proposed problem under suitable conditions. Furthermore, introduce an iterative algorithm for solving the problem and prove its...

In this paper, we study the variational inclusion problem which consists of finding zeros of the sum of a single and multivalued mappings in real Hilbert spaces. Motivated by the viscosity approximation, projection and contraction and inertial forward–backward splitting methods, we introduce two new forward–backward splitting methods for solving th...

In this paper, we give a general inertial Krasnoselskii–Mann algorithm for solving inclusion problems in Banach Spaces. First, we establish a weak convergence in real uniformly convex and q-uniformly smooth Banach spaces for finding fixed points of nonexpansive mappings. Then, a strong convergence is obtained for the inertial generalized forward-ba...

In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical exp...

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\ma...

In this work we are concerned with variational inequalities in real Hilbert spaces and introduce a new double projection method for solving it. The algorithm is motivated by the Korpelevich extragradient method, the subgradient extragradient method of Gibali et al. and Popov’s method. The proposed scheme combines some of the advantages of the metho...

In this paper, we study a special instance of the split inverse problem (SIP), which is the split variational inclusion problem (SVIP). Three simple iterative methods for solving it are introduced and weak and strong convergence theorems are established under mild and standard assumptions. As an application, the problem of minimizing two proper, co...

In this paper we study a classical monotone and Lipschitz continuous variational inequality in real Hilbert spaces. Two projection type methods, Mann and its viscosity generalization are introduced with their strong convergence theorems. Our methods generalize and extend some related results in the literature and their main advantages are: the stro...

In this paper we study a feasibility-seeking problem with percentage violation constraints. These are additional constraints, that are appended to an existing family of constraints, which single out certain subsets of the existing constraints and declare that up to a specified fraction of the number of constraints in each subset is allowed to be vi...

In this paper, we introduce a new algorithm of inertial form for solving monotone variational inequalities (VI) in real Hilbert spaces. Motivated by the subgradient extragradient method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and Lipschitz contin...

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\ma...

In this paper we study the convergence of continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on a recent progress in the geometric theory of spirallike functions. We prove convergence theorems and illustrate them by numerical simulations.

In this paper, we study strongly pseudo-monotone equilibrium problems in real Hilbert and introduce two simple subgradient-type methods for solving it. The advantages of our schemes are the simplicity of their algorithmic structure which consists of only one projection onto the feasible set and there is no need to solve any strongly convex programm...

Inspired by the works of L\'{o}pez et al. \cite{lopez} and the recent paper of Dang et al. \cite{dangxu}, we devise a new inertial relaxation of the CQ algorithm for solving Split Feasibility Problems (SFP) in real Hilbert spaces. Under mild and standard conditions we establish weak convergence of the proposed algorithm. We also propose a Mann-type...

The purpose of this paper is to study and analyze two new extragradient methods for solving non-Lipschitzian and pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, weak and strong convergence theorems of the proposed methods are established. We present academic and numerical examples for illustrating the beh...

In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-6...

In this paper we are focused on solving monotone and Lipschitz continuous variational inequalities in real Hilbert spaces. Motivated by several recent results related to the subgradient extragradient method (SEM), we propose two SEM extensions which do not require the knowledge of the Lipschitz constant associated with the variational inequality op...

In this paper, we are concerned with the Common Fixed Point Problem (CFPP) with demicontractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordon˜ ez et al. [35] and in general, the field of online/real-time algorithms, e.g., [20, 21, 30], in which the entire inpu...

The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. \cite{CGR}, replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors p...

In this paper, we introduce two modifications of the forward–backward splitting method with a new step size rule for inclusion problems in real Hilbert spaces. The modifications are based on Mann and viscosity-ideas. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish strong convergenc...

The Douglas-Rachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this paper we propose a structural generalization that allows to use $r$-sets-DR operators in a cyclic fashion. We prove convergence and present numerical illustrations of t...

Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable, possibly composed with linear operators and that in turn can be transformed to split feasibility problems (SFP)...

In this paper, we study the variational inequalities involving monotone and Lipschitz continues mapping in Banach spaces. A new and simple iterative method, which combines Halpern's technique and the subgradient extragradient idea is given. Under mild and standard assumptions, we establish the strong convergence of our algorithm in a uniformly smoo...

In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility problems by iteratively constraining the objective function from above until the feasibility problem is inconsistent....

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient ite...

In this paper we are concerned with the Split Feasibility Problem (SFP) in which there are given two Hilbert spaces \(H_1\) and \(H_2\), nonempty, closed and convex sets \(C\subseteq H_1\) and \(Q\subseteq H_2\), and a bounded and linear operator \(A:H_1 \rightarrow H_2\). The SFP is then to find a point \(x^*\in C\) such that its image under A bel...

In this paper we are concern with solving variational inequalities for monotone and Lipschitz mappings in real Hilbert spaces. Motivated by the works of Popov, Malitsky and Semenov and Semenov, we propose an extension of the subgradient extragradient method (Censor et al \cite{CGR,CGR1,CGR2}) with Bregman projections which calls for only one evalua...

In this paper we study the bounded perturbation resilience of projection and contraction algorithms for solving variational inequality (VI) problems in real Hilbert spaces. Under typical and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed projection and contraction algorithm...

In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the met...

In this paper we characterize classes of orthogonal polynomials, which brings us naturally to present a new proof to Bochner theorem. Our main concern is the classifications of the weight functions in the situation when a system of orthogonal polynomials satisfies the Sturm-Liouville equation. Moreover, we show that the Jacobi, Laguerre and Hermite...