Self-adaptive Projection Method Research Articles

Inertial projection and contraction methods for solving variational inequalities with applications to image restoration problems

In this paper, we introduce two inertial self-adaptive projection and contraction methods for solving the pseudomonotone variational inequality problem with a Lipschitz-continuous mapping in real Hilbert spaces. The adaptive stepsizes provided by the algorithms are simple to update and their computations are more efficient and flexible. Also we prove some weak and strong convergence theorems without prior knowledge of the Lipschitz constant of the mapping. Finally, we present some numerical experiments to demonstrate the effectiveness of the proposed algorithms by comparisons with related methods and some applications of the proposed algorithms to the image deblurring problem.

A self-adaptive projection method for nonlinear monotone equations with convex constraints

In this paper, we propose a self-adaptive derivative-free projection method for solving large-scale nonlinear monotone equations with convex constraints. The search direction satisfies the sufficient descent property, which is independent of any line search. Based on the Lipschitz continuity and monotonicity property, the proposed method is shown to be globally convergent. Moreover, the numerical results are reported to show the effectiveness of the proposed method by comparing with the existing current methods.

A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces

Abstract In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.

A Self-Adaptive Shrinking Projection Method with an Inertial Technique for Split Common Null Point Problems in Banach Spaces

In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm of the bounded operator and a strong convergence result is proved for the sequence generated by our algorithm under mild conditions. Moreover, we give some applications of our result to split convex minimization and split equilibrium problems in real Banach spaces. This result improves and extends several other results in this direction in the literature.

Strong convergence inertial projection and contraction method with self adaptive stepsize for pseudomonotone variational inequalities and fixed point problems

In this paper, we introduce a new inertial self-adaptive projection method for finding a common element in the set of solution of pseudomonotone variational inequality problem and set of fixed point of a pseudocontractive mapping in real Hilbert spaces. The self-adaptive technique ensures the convergence of the algorithm without any prior estimate of the Lipschitz constant. With the aid of Moudafi’s viscosity approximation method, we prove a strong convergence result for the sequence generated by our algorithm under some mild conditions. We also provide some numerical examples to illustrate the accuracy and efficiency of the algorithm by comparing with other recent methods in the literature.

A projection method based on self-adaptive rules for Stokes equations with nonlinear slip boundary conditions

In this work we propose and study a self-adaptive projection method for the numerical solution of Stokes equations under slip boundary conditions. We introduce the projection operator to formulate the slip constraint as a fixed point equation. To solve this problem we suggest a projection method by reformulating the nonlinear slip boundary conditions into an iterative form. To make the method more efficient, we find a self-adaptive rule that uses iterative functions to adjust the penalty parameter automatically. We show the convergence of the method in function space and give its application in detail. Finally, the numerical results are given to support our theoretical results.

Ridesharing user equilibrium problem under OD-based surge pricing strategy

Ridesharing is one of the effective urban traffic supply and demand management policies to reduce car ownership and mitigate traffic congestion. The origin-destination (OD) based surge pricing strategy is widely adopted by ridesharing service operators in practice due to its fairness and effectiveness. In this study, we aim to investigate the ridesharing user equilibrium (RUE) problem for an urban transportation network under the OD-based surge pricing strategy. We first build a variational inequality (VI) model for the proposed RUE problem. In particular, we explicitly formulate the necessary ride-matching constraints for the participants of multiple ridesharing services and rigorously demonstrate the existence and uniqueness of the RUE solution under some mild conditions. A parallel self-adaptive projection method (PSPM) incorporating column generation is developed to find an RUE solution for the large-scale problems. Finally, numerical experiments are conducted to provide valuable insights and examine the effectiveness of the proposed solution method. The results quantitatively show that the ridesharing under the OD-based surge pricing strategy reduces not only the travel cost for travelers but also the deliberate detours. Traffic congestion is significantly mitigated by ridesharing. Moreover, the proposed solution method has satisfactory computational efficiency for solving the large-scale problems.

Comparison of two projection methods for the solution of frictional contact problems

Frictional contact problems in linear elasticity are considered in this paper. The contact constraint is imposed in the weak sense using the fixed point method, which leads to a variational equation problem. For solving such a nonlinear variational problem, we study two projection methods using different self-adaptive rules. Based on the self-adaptive projection method, we propose a modified self-adaptive rule which is more effective to update the parameter. The methods can be implemented easily in conjunction with the boundary element method for the solution. Numerical experiments are reported to illustrate theoretical results.

A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems

In this work, we study the split feasibility problem (SFP) in the framework of p-uniformly convex and uniformly smooth Banach spaces. We propose an iterative scheme with inertial terms for seeking the solution of SFP and then prove a strong convergence theorem for the sequences generated by our iterative scheme under implemented conditions on the step size which do not require the computation of the norm of the bounded linear operator. We finally provide some numerical examples which involve image restoration problems and demonstrate the efficiency of the proposed algorithm. The obtained result of this paper complements many recent results in this direction and seems to be the first one to investigate the SFP outside Hilbert spaces involving the inertial technique.

Self-adaptive projection and boundary element methods for contact problems with Tresca friction

A self-adaptive projection method is considered for the numerical simulation of contact problems with Tresca friction. The contact constraints are imposed in the fixed point formulation using projection operators. It leads to a projection method where each iterative step consists of a linear elasticity problem with Robin conditions on the contact boundary. To ensure the efficiency of the method, a self-adaptive rule is given to adjust the parameter automatically. We prove the convergence of the method in function space and demonstrate its application with the boundary element method. The numerical experiments are presented to show the performance of the proposed method.

Two projection methods for the solution of Signorini problems

Two iterative methods, based on projection and boundary element methods, are considered for Signorini problems. The regularized problem with the projection boundary condition is first deduced from the Signorini problem. By using the equivalence between the Signorini boundary condition and the projection fixed point problem, our methods formulate the Signorini boundary condition into a sequence of Robin boundary conditions. In the new boundary condition there is a penalty parameter ρ. The convergence speed of the first method is greatly influenced by the value of ρ which is difficult to choose for individual problems. To improve the performance of this method, we present a self-adaptive projection method which adjusts the parameter ρ automatically per iteration based on the iterative data. The main result of this work is to provide the convergence of the methods under mild assumptions. As the iteration process is given by the potential and its derivative on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments indicate efficiency of the methods proposed.

A self-adaptive projection method for contact problems with the BEM

• A new and simple computational method is developed for the contact problem. • We use the variational method to analyze the convergence of proposed method. • We give the self-adaptive rule and the boundary element approximation for the method. • The performance of the method is demonstrated by the numerical results. A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.

Projection and self-adaptive projection methods for the Signorini problem with the BEM

We propose two new projection methods for solving the Signorini problem and study the relationship between them. Since the Signorini boundary condition is equivalent to a projection fixed point problem, the methods formulate the Signorini boundary condition into a sequence of Robin boundary conditions and only require solving a variational problem with simple boundary conditions at each iterative step. The convergence speed of the first method depends on the parameter ρ heavily, and it is difficult to choose a proper ρ for individual problems. To improve the efficiency of the method, we propose an adaptive projection method which adjusts the parameter ρ automatically per iteration. We establish the convergence analysis of the methods. Because the Signorini boundary condition is given in an iterative form by the potential and its derivative on the boundary of the domain, we can easily provide the boundary element approximation of the problem. Finally, we present some numerical simulations which illustrate the performance of the methods.

Elastic demand dynamic network user equilibrium: Formulation, existence and computation

This paper is concerned with dynamic user equilibrium with elastic travel demand (E-DUE) when the trip demand matrix is determined endogenously. We present an infinite-dimensional variational inequality (VI) formulation that is equivalent to the conditions defining a continuous-time E-DUE problem. An existence result for this VI is established by applying a fixed-point existence theorem (Browder, 1968) in an extended Hilbert space. We present three computational algorithms based on the aforementioned VI and its re-expression as a differential variational inequality (DVI): a projection method, a self-adaptive projection method, and a proximal point method. Rigorous convergence results are provided for these methods, which rely on increasingly relaxed notions of generalized monotonicity, namely mixed strongly-weakly monotonicity for the projection method; pseudomonotonicity for the self-adaptive projection method, and quasimonotonicity for the proximal point method. These three algorithms are tested and their solution quality, convergence, and computational efficiency are compared. Our convergence results, which transcend the transportation applications studied here, apply to a broad family of VIs and DVIs, and are the weakest reported to date.

New step lengths in projection method for variational inequality problems

In this paper, we consider projection method for variational inequality problems. First, we give a new modification of recently proposed self-adaptive step length rules, with possibly minimal parameters. Then, the resulting self-adaptive projection method is proven to converge globally at a R-linear rate provided that the underlying mapping is strongly monotone and Lipschitz continuous. Preliminary numerical results confirm its flexibility and effectiveness.

Modified self-adaptive projection method for solving pseudomonotone variational inequalities

In this paper, a self-adaptive projection method with a new search direction for solving pseudomonotone variational inequality (VI) problems is proposed, which can be viewed as an extension of the methods in [B.S. He, X.M. Yuan, J.Z. Zhang, Comparison of two kinds of prediction-correction methods for monotone variational inequalities, Computational Optimization and Applications 27 (2004) 247–267] and [X.H. Yan, D.R. Han, W.Y. Sun, A self-adaptive projection method with improved step-size for solving variational inequalities, Computers & Mathematics with Applications 55 (2008) 819–832]. The descent property of the new search direction is proved, which is useful to guarantee the convergence. Under the relatively relaxed condition that F is continuous and pseudomonotone, the global convergence of the proposed method is proved. Numerical experiments are provided to illustrate the efficiency of the proposed method.

Self-adaptive projection methods for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point closest to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space. This problem arises in many practical fields, and it can be a model for many inverse problems. Noting that some existing algorithms require estimating the Lipschitz constant or calculating the largest eigenvalue of the matrix, in this paper, we first introduce a self-adaptive projection method by adopting Armijo-like searches to solve the MSFP, then we focus on a special case of the MSFP and propose a relaxed self-adaptive method by using projections onto half-spaces instead of those onto the original convex sets, which is much more practical. Convergence results for both methods are analyzed. Preliminary numerical results show that our methods are practical and promising for solving larger scale MSFPs.

A self-adaptive projection method for a class of variant variational inequalities

In this paper, we consider the general variant variational inequality of the type: Find a vector u ∗ ∈ R n , such that Q(u ∗ ) ∈ Ω, � v −Q(u ∗ ),Tu ∗ � 0, ∀v ∈ Ω, where T,Q are operators. We suggest and analyze a very simple self-adaptive iterative method for solving this class of general variational inequalities. Under certain conditions, the global con- vergence of the proposed method is proved. An example is given to illustrate the efficiency and implementation of the proposed method. Preliminary numerical results show that the proposed method is applicable.

A self-adaptive projection method for solving the multiple-sets split feasibility problem

The multiple-sets split feasibility problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. Censor et al (2005 Inverse Problems 21 2171–84) proposed a method for solving the multiple-sets split feasibility problem, whose efficiency depends heavily on step size, a fixed constant related to the Lipschitz constant of ∇p(x) (see the definition in section 1). To estimate the Lipschitz constant is a very difficult, if not an impossible task. On the other hand, even if we know the Lipschitz constant, a method with fixed step size may be slow. In this paper, we propose a new method for solving the multiple-sets split feasibility problem by adopting variable step sizes, which chooses suitable step sizes self-adaptively, based on the information from the current iterate. It thus avoids the difficult task of estimating the Lipschitz constant, while the efficiency is enhanced greatly. We prove the global convergence of the new method and report our numerical results, which are promising.

Self-adaptive projection method for co-coercive variational inequalities

In some real-world problems, the mapping of the variational inequalities does not have any explicit forms and only the function value can be evaluated or observed for given variables. In this case, if the mapping is co-coercive, the basic projection method is applicable. However, in order to determine the step size, the existing basic projection method needs to know the co-coercive modulus in advance. In practice, usually even if the mapping can be characterized co-coercive, it is difficult to evaluate the modulus, and a conservative estimation will lead an extremely slow convergence. In view of this point, this paper presents a self-adaptive projection method without knowing the co-coercive modulus. We also give a real-life example to demonstrate the practicability of the proposed method.