Proximal Point Methods for Solving Mixed Variational Inequalities on the Hadamard Manifolds

Abstract

We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results.

1. Introduction

In recent years, much attention has been given to study the variational inequalities and related problems on the Riemannian manifold and the Hadamard manifold. This framework is a useful for the developments of various fields. Several ideas and techniques from the Euclidean space have been extended and generalized to this nonlinear framework. The Hadamard manifolds are examples of hyperbolic spaces and geodesics; see [1–7] and the references therein. Németh [8], Tang et al. [6], and Colao et al. [2] have considered the variational inequalities and equilibrium problems on the Hadamard manifolds. They have studied the existence of a solution of the equilibrium problems under some suitable conditions. To the best of our knowledge, no one has considered the auxiliary principle technique for solving the mixed variational inequalities on the Hadamard manifolds. In this paper, we use the auxiliary principle technique to suggest and analyze a proximal iterative method for solving the mixed variational inequalities. If the nonlinearity in the mixed variational inequalities is an indicator function, then the mixed variational inequalities are equivalent to the variational inequality on the Hadamard manifold. This shows that the results obtained in this paper continue to hold for variational inequalities on the Hadamard manifold, which is due to Tang et al. [6] and Németh [8]. We hope that the technique and idea of this paper may stimulate further research in this area.

2. Preliminaries

We now recall some fundamental and basic concepts needed for a reading of this paper. These results and concepts can be found in the books on the Riemannian geometry [2, 3, 5].

Let 𝑀 be a simply connected 𝑚-dimensional manifold. Given 𝑥∈𝑀, the tangent space of 𝑀 at 𝑥 is denoted by 𝑇_𝑥𝑀 and the tangent bundle of 𝑀 by 𝑇𝑀=∪_𝑥∈𝑀𝑇_𝑥𝑀, which is naturally a manifold. A vector field 𝐴 on 𝑀 is a mapping of 𝑀 into 𝑇𝑀 which associates to each point 𝑥∈𝑀, a vector 𝐴(𝑥)∈𝑇_𝑥𝑀. We always assume that 𝑀 can be endowed with a Riemannian metric to become a Riemannian manifold. We denote by ⟨,⋅,⟩ the scalar product on 𝑇_𝑥𝑀 with the associated norm ‖⋅‖_𝑥, where the subscript 𝑥 will be omitted. Given a piecewise smooth curve 𝛾∶[𝑎,𝑏]→𝑀 joining 𝑥 to 𝑦 (i.e., 𝛾(𝑎)=𝑥 and 𝛾(𝑏)=𝑦) by using the metric, we can define the length of 𝛾 as 𝐿(𝛾)=∫^𝑏𝑎‖𝛾^(𝑡)‖𝑑𝑡. Then, for any 𝑥,𝑦∈𝑀 the Riemannian distance 𝑑(𝑥,𝑦), which includes the original topology on 𝑀, is defined by minimizing this length over the set of all such curves joining 𝑥 to 𝑦.

Let Δ be the Levi-Civita connection with (𝑀,⟨⋅,⋅⟩). Let 𝛾 be a smooth curve in 𝑀. A vector field 𝐴 is said to be parallel along 𝛾 if Δ_𝛾′𝐴=0. If 𝛾^ itself is parallel along 𝛾, we say that 𝛾 is a geodesic and in this case ‖𝛾^‖ is constant. When ‖𝛾^‖=1,𝛾 is said to be normalized. A geodesic joining 𝑥 to 𝑦 in 𝑀 is said to be minimal if its length equals 𝑑(𝑥,𝑦).

A Riemannian manifold is complete, if for any 𝑥∈𝑀 all geodesics emanating from 𝑥 are defined for all 𝑡∈𝑅. By the Hopf-Rinow theorem, we know that if 𝑀 is complete, then any pair of points in 𝑀 can be joined by a minimal geodesic. Moreover, (𝑀,𝑑) is a complete metric space, and bounded closed subsets are compact.

Let 𝑀 be complete. Then the exponential map exp_𝑥∶𝑇_𝑥𝑀→𝑀 at 𝑥 is defined by exp_𝑥𝑣=𝛾_𝑣(1,𝑥) for each 𝑣∈𝑇_𝑥𝑀, where 𝛾(⋅)=𝛾_𝑣(⋅,𝑥) is the geodesic starting at 𝑥 with velocity 𝑣(i.e.,𝛾(0)=𝑥and𝛾^(0)=𝑣) Then exp_𝑥𝑡𝑣=𝛾_𝑣(𝑡,𝑥) for each real number 𝑡.

A complete simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. Throughout the remainder of this paper, we always assume that 𝑀 is an 𝑚-manifold Hadamard manifold.

We also recall the following well-known results, which are essential for our work.

Lemma 2.1 (see [5]). Let 𝑥∈𝑀. Then exp_𝑥∶𝑇_𝑥𝑀→𝑀 is a diffeomorphism, and for any two points 𝑥,𝑦∈𝑀, there exists a unique normalized geodesic joining 𝑥 to 𝑦,𝛾_𝑥,𝑦, which is minimal.

So from now on, when referring to the geodesic joining two points, we mean the unique minimal normalized one. Lemma 2.1 says that 𝑀 is diffeomorphic to the Euclidean space 𝑅^𝑚. Thus 𝑀 has the same topology and differential structure as 𝑅^𝑚. It is also known that the Hadamard manifolds and Euclidean spaces have similar geometrical properties. Recall that a geodesic triangle ▵(𝑥₁,𝑥₂,and𝑥₃) of a Riemannian manifold is a set consisting of three points 𝑥₁,𝑥₂,𝑥₃ and three minimal geodesics joining these points.

Lemma 2.2 (see (comparison Theorem for Triangles [2, 3, 5])). Let ▵(𝑥₁,𝑥₂,𝑥₃) be a geodesic triangle. Denote, for each 𝑖=1,2,3(mod3), by 𝛾_𝑖∶[0,𝑙_𝑖]→𝑀 the geodesic joining 𝑥_𝑖 to 𝑥_𝑖+1, and 𝛼_𝑖;=𝐿(𝛾^𝑖(0),−𝛾^𝑙(𝑖−1)(𝑙𝑖−1)), the angle between the vectors 𝛾^𝑖(0) and −𝛾^𝑖−1(𝑙_𝑖−1), and 𝑙_𝑖;=𝐿(𝛾_𝑖). Then 𝑙𝛼₁+𝛼₂+𝛼₃≤𝜋,(2.1)_2𝑙+𝑙^2𝑖+1−2𝐿_𝑖𝑙_𝑖+1cos𝛼_𝑖+1≤𝑙^2𝑖−1.(2.2) In terms of the distance and the exponential map, the inequality (2.2) can be rewritten as 𝑑²𝑥_𝑖,𝑥_𝑖+1+𝑑²𝑥_𝑖+1,𝑥_𝑖+2−2exp^𝑥−1_𝑖+1𝑥_𝑖,exp^𝑥−1_𝑖+1𝑥_𝑖+2≤𝑑²𝑥_𝑖−1,𝑥_𝑖,(2.3) since exp^𝑥−1_𝑖+1𝑥_𝑖,exp^𝑥−1_𝑖+1𝑥_𝑖+2=𝑑𝑥^𝑖,𝑥_𝑖+1𝑑𝑥_𝑖+1,𝑥_𝑖+2cos𝛼_𝑖+1.(2.4)

Lemma 2.3 (see [5]). Let ▵(𝑥,𝑦,𝑧) be a geodesic triangle in a Hadamard manifold 𝑀. Then, there exist 𝑥^,𝑦^,𝑧^∈𝑅² such that 𝑑(𝑥,𝑦)=‖‖𝑥^−𝑦^‖‖,𝑑(𝑦,𝑧)=‖‖𝑦^−𝑧^‖‖,𝑑(𝑧,𝑥)=‖‖𝑧^−𝑥^‖‖.(2.5) The triangle ▵(𝑥^,𝑦^,𝑧^) is called the comparison triangle of the geodesic triangle ▵(𝑥,𝑦,𝑧), which is unique up to isometry of 𝑀.

From the law of cosines in inequality (2.3), we have the following inequality, which is a general characteristic of the spaces with nonpositive curvature [5]:exp^𝑥−1𝑦,exp^𝑥−1𝑧+exp^𝑦−1𝑥,exp^𝑦−1𝑧≥𝑑²(𝑥,𝑦).(2.6) From the properties of the exponential map, we have the following known result.

Lemma 2.4 (see [5]). Let 𝑥₀∈𝑀 and {𝑥_𝑛}⊂𝑀 such that 𝑥_𝑛→𝑥₀. Then the following assertions hold.(i)For any𝑦∈𝑀, exp^𝑥−1_𝑛𝑦⟶exp^𝑥−1₀𝑦,exp^𝑦−1𝑥_𝑛⟶exp^𝑦−1𝑥₀.(2.7)(ii)If {𝑣_𝑛} is a sequence such that 𝑣_𝑛∈𝑇_𝑥𝑛𝑀 and 𝑣_𝑛→𝑣₀, then 𝑣₀∈𝑇_𝑥0𝑀.(iii)Given the sequences {𝑢_𝑛} and {𝑣_𝑛} satisfying 𝑢_𝑛,𝑣_𝑛∈𝑇_𝑥𝑛𝑀, if 𝑢_𝑛→𝑢₀ and 𝑣_𝑛→𝑣₀, with 𝑢₀,𝑣₀∈𝑇_𝑥0𝑀, then ⟨𝑢_𝑛,𝑣_𝑛⟩⟶⟨𝑢₀,𝑣₀⟩.(2.8)

A subset 𝐾⊆𝑀 is said to be convex if for any two points 𝑥,𝑦∈𝐾, the geodesic joining 𝑥 and 𝑦 is contained in 𝐾,𝐾; that is, if 𝛾∶[𝑎,𝑏]→𝑀 is a geodesic such that 𝑥=𝛾(𝑎) and 𝑦=𝛾(𝑏), then 𝛾((1−𝑡)𝑎+𝑡𝑏)∈𝐾,forall𝑡∈[0,1]. From now on 𝐾⊆𝑀 will denote a nonempty, closed and convex set, unless explicitly stated otherwise.

A real-valued function 𝑓 defined on 𝐾 is said to be convex if, for any geodesic 𝛾 of 𝑀, the composition function 𝑓∘𝛾∶𝑅→𝑅 is convex; that is, (𝑓∘𝛾)(𝑡𝑎+(1−𝑡)𝑏)≤𝑡(𝑓∘𝛾)(𝑎)+(1−𝑡)(𝑓∘𝛾)(𝑏),∀𝑎,𝑏∈𝑅,𝑡∈[0,1].(2.9)

The subdifferential of a function 𝑓∶𝑀→𝑅 is the set-valued mapping 𝜕𝑓∶𝑀→2^𝑇𝑀 defined as 𝜕𝑓(𝑥)=𝑢∈𝑇_𝑥𝑀∶𝑢,exp^𝑥−1𝑦≤𝑓(𝑦)−𝑓(𝑥),∀𝑦∈𝑀,∀𝑥∈𝑀,(2.10) and its elements are called subgradients. The subdifferential 𝜕𝑓(𝑥) at a point 𝑥∈𝑀 is a closed and convex (possibly empty) set. Let 𝐷(𝜕𝑓) denote the domain of 𝜕𝑓 defined by𝐷(𝜕𝑓)={𝑥∈𝑀∶𝜕𝑓(𝑥)≠∅}.(2.11)

The existence of subgradients for convex functions is guaranteed by the following proposition; see [7].

Lemma 2.5 (see [5, 7]). Let 𝑀 be a Hadamard manifold and 𝑓∶𝑀→𝑅 convex. Then, for any 𝑥∈𝑀, the subdifferential 𝜕𝑓(𝑥) of 𝑓 at 𝑥 is nonempty; that is, 𝐷(𝜕𝑓)=𝑀.

For a given single-valued vector field 𝑇∶𝑀→𝑇𝑀 and a real-valued function 𝑓∶𝑀→𝑅, we consider the problem of finding 𝑢∈𝑀 such that𝑇𝑢,exp^𝑢−1𝑣+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑣∈𝑀,(2.12) which is called the mixed variational inequality. This problem was considered by Colao et al. [2]. They proved the existence of a solution of problem (2.12) using the KKM maps. For the applications, formulation, and other aspects of the mixed variational inequalities in the linear setting, see [8–16].

We remark that if the function 𝑓 is an indicator of a closed and convex set 𝐾 in 𝑀, then problem (2.12) is equivalent to finding 𝑢∈𝐾 such that𝑇𝑢,exp^𝑥−1𝑣≥0,∀𝑣∈𝐾,(2.13) which is called the variational inequality on the Hadamard manifolds. Németh [8], Colao et al. [2] and Udrişte [7] studied variational inequalities on the Hadamard manifold from different point of views. In the linear setting, variational inequalities have been studied extensively; see [8–25] and the references therein.

Definition 2.6. An operator 𝑇 is said to be speudomonotone with respect a mapping 𝑓, if and only if 𝑇(𝑢),exp^𝑢−1𝑣+𝑓(𝑣)−𝑓(𝑢)≥0⟹𝑇(𝑣),exp^𝑣−1𝑢+𝑓(𝑣)−𝑓(𝑢)≥0,∀𝑢,𝑣∈𝑀.(2.14)

3. Main Results

We now use the auxiliary principle technique of Glowinski et al. [9] to suggest and analyze an implicit iterative method for solving the mixed variational inequality (2.12) on the Hadamard manifold.

For a given 𝑢∈𝑀 satisfying (2.12), consider the problem of finding 𝑤∈𝑀 such that𝜌𝑇𝑤+exp^𝑢−1𝑤,exp^𝑤−1𝑣+𝑓(𝑣)−𝑓(𝑤)≥0,∀𝑣∈𝑀,(3.1) which is called the auxiliary mixed variational inequality on the Hadamard manifolds. We note that if 𝑤=𝑢, then 𝑤 is a solution of the mixed variational inequality (2.12). This observation enable to suggest and analyzes the following proximal point method for solving the mixed variational inequality (2.12).

Algorithm 3.1. For a given 𝑢₀, compute the approximate solution by the iterative scheme: 𝜌𝑇𝑢^𝑛+1+exp^𝑢−1_𝑛𝑢_𝑛+1,exp^𝑢−1_𝑛+1𝑣+𝑓(𝑣)−𝑓𝑢_𝑛+1≥0,∀𝑣∈𝑀,(3.2) Algorithm 3.1 is called the implicit (proximal point) iterative method for solving the mixed variational inequality on the Hadamard manifold.

If 𝑀=𝑅^𝑛, then Algorithm 3.1 collapses to the following algorithm:

Algorithm 3.2. For a given 𝑢₀∈𝑅^𝑛, find the approximate solution 𝑢_𝑛+1 by the iterative scheme. 𝜌𝑇𝑢_𝑛+1+𝑢_𝑛+1−𝑢_𝑛,𝑣−𝑢_𝑛+1+𝜌𝑓(𝑣)−𝑓𝑢_𝑛+1≥0,∀𝑣∈𝑅^𝑛,(3.3) which is known as the proximal pint method for solving the mixed variational inequalities. For the convergence analysis of Algorithm 3.2, see [11, 12].

If 𝑓 is the indicator function of a closed and convex set 𝐾 in 𝑀, then Algorithm 3.1 reduces to the following method, which is due to Tang et al. [6].

Algorithm 3.3. For a given 𝑢₀∈𝐾, compute the approximate solution by the iterative scheme 𝜌𝑇𝑢^𝑛+1+exp^𝑢−1_𝑛𝑢_𝑛+1,exp^𝑢−1_𝑛+1𝑣≥0,∀𝑣∈𝐾.(3.4)

We would like to mention that Algorithm 3.1 can be rewritten in the following equivalent form.

Algorithm 3.4. For a given 𝑢₀∈𝑀, compute the approximate solution by the iterative scheme: 𝜌𝑇𝑇𝑦𝑢_𝑛𝑛+exp^𝑢−1_𝑛𝑦_𝑛,exp^𝑦−1_𝑛𝑣+𝜌𝑓(𝑣)−𝜌𝑓𝑦_𝑛≥0∀𝑣∈𝑀,+exp^𝑢−1_𝑛𝑢_𝑛+1,exp^𝑢−1_𝑛+1𝑣+𝜌𝑓(𝑣)−𝜌𝑓𝑢_𝑛+1,∀𝑣∈𝑀,(3.5) which is called the extraresolvent method for solving the mixed variational inequalities on the Hadamard manifolds.

In a similar way, one can obtain several iterative methods for solving the variational inequalities on the Hadamard manifold.

We now consider the convergence analysis of Algorithm 3.1, and this is the main motivation of our next result.

Theorem 3.5. Let 𝑇 be a pseudomonotone vector field. Let 𝑢_𝑛 be the approximate solution of the mixed variational inequality (2.12) obtained from Algorithm 3.1; then 𝑑²𝑢_𝑛+1,𝑢+𝑑²𝑢_𝑛+1,𝑢_𝑛≤𝑑²𝑢_𝑛,𝑢,(3.6) where 𝑢∈𝑀 is the solution of the mixed variational inequality (2.12).

Proof. Let 𝑢∈𝑀 be a solution of the mixed variational inequality (). Then, by using the pseudomonotonicity of the vector filed, 𝑇(𝑢), we have 𝜌𝑇(𝑣),exp^𝑢−1𝑣+𝜌𝑓(𝑣)−𝜌𝑓(𝑢)≤0,∀𝑣∈𝑀.(3.7) Taking 𝑣=𝑢_𝑛+1 in (3.7), we have 𝜌𝑇𝑢_𝑛+1,exp^𝑢−1𝑢_𝑛+1+𝜌𝑓𝑢_𝑛+1−𝜌𝑓(𝑢)≤0.(3.8) Taking 𝑣=𝑢 in (3.2), we have 𝜌𝑇𝑢^𝑛+1+exp^𝑢−1_𝑛𝑢_𝑛+1,exp^𝑢−1_𝑛+1𝑢+𝑓(𝑢)−𝑓𝑢_𝑛+1≥0.(3.9) From (3.8) and (3.9), we have exp^𝑢−1_𝑛+1𝑢_𝑛,exp^𝑢−1_𝑛+1𝑢≤0.(3.10) For the geodesic triangle ▵(𝑢_𝑛,𝑢_𝑛+1,𝑢) the inequality (3.10) can be written as, 𝑑²𝑢_𝑛+1,𝑢+𝑑²𝑢_𝑛+1,𝑢_𝑛−exp^𝑢−1_𝑛+1𝑢_𝑛,exp^𝑢−1_𝑛+1𝑢≤𝑑²𝑢_𝑛,𝑢.(3.11) Thus, from (3.10) and (3.11), we obtained inequality (3.6), the required result.

Theorem 3.6. Let 𝑢∈𝑀 be solution of (2.12), and let 𝑢_𝑛+1 be the approximate solution obtained from Algorithm 3.1; then lim_𝑛→∞(𝑢_𝑛+1)=𝑢.

Proof. Let 𝑢̂∈𝑀 be a solution of (2.12). Then, from (3.6), it follows that the sequence {𝑢_𝑛} is bounded and ^∞_𝑛=0𝑑²𝑢_𝑛+1,𝑢_𝑛≤𝑑²𝑢₀,𝑢;(3.12) it follows that lim_𝑛→∞𝑑𝑢_𝑛+1,𝑢_𝑛=0.(3.13) Let 𝑢̂ be a cluster point of {𝑢_𝑛}. Then there exists a subsequence {𝑢_𝑛𝑖} such that {𝑢_𝑢𝑖} converges to 𝑢̂. Replacing 𝑢_𝑛+1 by 𝑢_𝑛𝑖 in (3.2), taking the limit, and using (3.13), we have 𝑇𝑢,̂exp𝑢⁻¹̂𝑣+𝑓(𝑣)−𝑓(𝑢)̂≥0,∀𝑣∈𝑀.(3.14) This shows that 𝑢̂∈𝑀 solves (2.12) and 𝑑²𝑢_𝑛+1𝑢̂≤𝑑²𝑢_𝑛,𝑢̂,(3.15) which implies that the sequence {𝑢_𝑛} has unique cluster point and lim_𝑛→∞𝑢_𝑛=𝑢̂ is a solution of (2.12), the required result.

4. Conclusion

We have used the auxiliary principle technique to suggest and analyzed a proximal point iterative method for solving the mixed quasi-variational inequalities on the Hadamard manifolds. Some special cases are also discussed. Convergence analysis of the new proximal point method is proved under weaker conditions. Results obtained in this paper may stimulate further research in this area. The implementation of the new method and its comparison with other methods is an open problem. The ideas and techniques of this paper may be extended for other related optimization problems.

Acknowledgment

The authors would like to thank Dr. S. M. Junaid Zaidi, Rector, the COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities.