Analytic center cutting plane methods for variational inequalities over convex bodies.

An analytic center cutting plane method is an iterative algorithm based on the computation of analytic centers. In this paper, we propose some analytic center cutting plane methods for solving quasimonotone or pseudomonotone variational inequalities whose domains are bounded or unbounded convex bodies.

Introduction and preliminaries

Some recent developments in solving variational inequalities are analytic center cutting plane methods. An analytic center cutting plane method is an interior algorithm based on the computation of analytic centers. In order to work with analytic center cutting plane methods, some authors assume that the feasible sets of variational inequalities are polytopes, e.g., see [1-6], while others pay more attention to problems with infinitely many linear constraints, e.g., see [7, 8], etc. Analytic center cutting plane methods also can be used to other types of optimization problems, like mathematical programming with equilibrium constraints [9], convex programming [10, 11], conic programming [12], stochastic programming [13, 14], and combinatorial optimization [11]. In this paper, we propose some analytic center cutting plane methods for solving pseudomonotone or quasimonotone variational inequalities.

Let X be a nonempty subset of the n-dimensional Euclidean space , and let be a function. We say that a point is a solution of the variational inequality if

The point is a solution of the dual variational inequality if

We denote by the set of solutions of , and by the set of solutions of .

From Auslender [15] we have the following lemma.

Lemma 1

If F is continuous, then a solution of is a solution of ; and if F is continuous pseudomonotone, then is a solution of if and only if it is a solution of .

Given (), the gap function is defined as Since , , and we have the following lemma.

Lemma 2

A point is a solution of () if and only if ().

A point is said to be a ε-solution of the variational inequality (1) if .

A function is said to be monotone on X if strongly monotone if there exists a constant such that quasimonotone on X if pseudomonotone on X if pseudomonotone plus on X if it is pseudomonotone on X and if and strongly pseudomonotone on X if there exist constants , such that

Results and discussion

We proposed some analytic center cutting plane methods (ACCPM) for convex feasibility problems. Convex feasibility problem is a problem of finding a point in a convex set, which contains a full dimensional ball and is contained in a compact convex set described by matrix inequalities. There are many applications of these types of problems in nonsmooth optimization. The ACCPM is an efficient technique for nondifferentiable optimization. We employed some nonpolyhedral models into the ACCPM.

We present five analytic center cutting plane methods for solving variational inequalities whose domains are bounded or unbounded convex bodies.

First four algorithms are for the variational inequalities with compact and convex feasible sets. If is pseudomonotone plus on a compact convex body X, then our Algorithm 1 either stops with a solution of the variational inequality after a finite number of iterations, or there exists an infinite sequence in X that converges to a solution of . If is pseudomonotone plus on a compact convex body X, then our Algorithm 2 stops with an ε-solution of the variational inequality after a finite number of iterations. If is Lipschitz continuous on a compact convex body X, then our Algorithm 3 either stops with a solution of the variational inequality after a finite number of iterations, or there exists an infinite sequence in X that converges to a solution of . If is Lipschitz continuous on a compact convex body X, then our Algorithm 4 stops with an ε-solution of after a finite number of iterations.

Our fifth algorithm is for variational inequalities with unbounded compact convex feasible regions, and these feasible regions can be the n-dimensional Euclidean space itself. If is strongly monotone on X, then our Algorithm 5 either stops with a solution of the variational inequality after a finite number of iterations, or there exists an infinite sequence in X that converges to a solution of . Furthermore, the proof of the previous result also indicates that, if is strongly pseudomonotone on X, then our Algorithm 5 either stops with a solution of after a finite number of iterations, or there exists an infinite sequence in X that converges to a solution of .

Conclusions

This paper works with variational inequalities whose feasible sets are bounded or unbounded convex bodies. We present some analytic center cutting plane algorithms that extend the algorithms proposed in [1, 2, 16], from polytopes/polyhedron to convex regions, or from bounded convex region to unbounded convex regions. We should mention that our approach can be used to extend many interior methods which are associated with polyhedral feasible regions, e.g., the algorithms given by [3, 4]. We can also extend some other algorithms for variational inequalities over polyhedral feasible sets [17-19].

Compact convex bodies

A polytope is a set which is the convex hull of a finite set.

A polyhedron is a set where , and A is an matrix.

Every polytope is a polyhedron, whereas not every polyhedron is a polytope.

Minkowski proved the following lemma in 1896.

Lemma 3

A set is a polytope if and only if it is a bounded polyhedron.

We make the following assumptions for polytopes throughout this paper.

(a) Interior assumption: A polytope is always a full-dimensional polytope and that includes , where e is a vector of all ones.

We note that if a polytope has nonempty interior, then (a) can be met by re-scaling.

A convex body is a convex and bounded subset with nonempty interior.

A rectangle is defined by where .

A rectangle can also be given by some inequalities where is a finite set of hyperplanes, H is an matrix. And, if we denote by V the finite set of all vertices of B, then

Theorem 1

A bounded subset is a compact convex body if and only if there exists a sequence of polytopes satisfying () such that

Proof

The sufficiency is trivial. We only prove the necessity.

Since X is bounded, there exists a rectangle B such that .

Take a partition of B. Then B is divided into a set of finite sub-rectangles by a finite set of hyperplanes. Let , where () are all the sub-rectangles that lie entirely within X. Let be the set of all vertices of (), then is a finite set. So, is a polytope, and it obviously satisfies (For the case of a 2-dimensional Euclidean space, see Fig. 1.)

Figure 1

Take a finer partition of B. Similarly, we have a set , where () are all the sub-rectangles which correspond to and lie entirely within X; and we have a polytope , where is the set of all vertices of () such that

By mathematical induction, there exists a sequence of polytopes which satisfies

It is easy to see that . □

It is quite straightforward to prove the following Corollary 1, Proposition 1, and Proposition 2.

Corollary 1

A subset is a compact convex body if there is a uniformly bounded sequence of polytopes , i.e., for a given rectangle B, such that

Proposition 1

Let be a compact convex body and be a continuous function, then the variational inequality has solutions.

Proposition 2

Let be a compact convex body and be a continuous and strictly pseudomonotone function, then the variational inequality has a unique solution.

Generalized analytic center cutting plane algorithms for solving pseudomonotone variational inequalities

For any polytope , is associated with the potential function

It is known that an analytic center is the maximizer of the potential function φ, and the unique solution of the system where y is a positive dual vector, and Y the diagonal matrix built upon y.

An approximate analytic center [20] is the maximizer of the potential function φ, and the unique solution of the system where z is a dual vector, and Z is the diagonal matrix built upon z.

Now we modify Goffin, Marcotte, and Zhu’s [2] Algorithm 1 to solve . We propose an algorithm for solving variational inequalities, whose feasible sets are compact convex bodies.

From Theorem 1, there exists a sequence of variational inequalities () induced by the original variational inequality , where the polytope is given by the linear inequalities , , and is an matrix. So, we may apply the algorithm in [2] to each . Algorithm 1 uses this idea, but the algorithm in [2] is applied to for only a certain number of iterations until we get by use of Theorem 1 of [2].

Algorithm 1

Step 1. (initialization)

Step 2. (computation of an approximate analytic center)

Find an approximate analytic center of ;

Step 3. (stop criterion)

Compute ,

if , then STOP,

else GO TO step 4;

Step 4. (find an ε-solution for )

Compute ,

if , then increase j by one RETURN TO Step 1,

else GO TO Step 5;

Step 5. (cut generation)

Set is the new cutting plane for .

Increase k by one GO TO Step 2.

Theorem 2

Let be pseudomonotone plus on a compact convex body X, then Algorithm 1 either stops with a solution of after a finite number of iterations, or there exists a subsequence of the infinite sequence that converges to a point .

According to Algorithm 1 and Theorem 1 of [2], for any given j, such that after a finite number of iterations, Since X is compact, there exists a subsequence of and a point such that , we have

On the other hand, due to the compactness of X, such that , . Since for , By the continuities of and , is a continuous function on X.

Consequently, ∀p

Then we have

On the other hand,, such that Because we have Therefore, which deduces that is a solution of . □

Algorithm 1 usually generates an infinite sequence. In order to terminate at a finite number of iterations, we change the stop criterion, Step 3 in Algorithm 1, to get the following algorithm.

Algorithm 2

Step 1, Step 2, Step 4, and Step 5 are the same as those of Algorithm 1.

Step 3. (stop criterion)

Compute ,

if , then STOP,

else GO TO step 4.

From Theorem 2 we have the following.

Theorem 3

Let be pseudomonotone plus on a compact convex body X, then Algorithm 2 stops with an ε-solution of after a finite number of iterations.

Generalized analytic center cutting plane algorithms for solving quasimonotone variational inequalities

In this section, we are going to modify Marcotte and Zhu’s [1] approach to solve quasimonotone variational inequalities . We assume that the feasible sets are compact convex bodies.

From Theorem 1 there is a sequence of variational inequalities () induced by the original variational inequality .

According to [1], the following are the conditions that are required in the construction of algorithms for solving quasimonotone variational inequalities.

For any given j, let the auxiliary function be continuous in x and strongly monotone in y, i.e., for . is said to be the strong monotonicity constant for . The function is associated with the variational inequality whose solution satisfies

It is known that are continuous [21], and that x is a solution of if and only if it is a fixed point of w.

Assume and . Let (which depends on x) be the smallest nonnegative integer for which

Define If is a solution of , then , , and .

Algorithm 3

Step 1. (initialization)

Let be the strong monotonicity constant for , with respect to y, and let .

Step 2. (computation of an approximate analytic center)

Find an approximate analytic center of ;

Step 3. (stop criterion)

Compute ,

if , then STOP,

else GO TO step 4;

Step 4. (find an ε-solution for )

Compute ,

if , then increase j by one RETURN TO Step 1,

else GO TO Step 5;

Step 5. (auxiliary variational inequality)

Let satisfy the variational inequality Let where is the smallest integer which satisfies

Step 6. (cutting plane generation)

Set is the new cutting plane for .

Increase k by one GO TO Step 2.

By Theorem 1 of [1], similar to the proof of Theorem 2, we have the following theorem.

Theorem 4

Let be Lipschitz continuous, i.e., there exists a constant such that on a compact convex body X, and be nonempty. Then Algorithm 3 either stops with a solution of after a finite number of iterations, or there exists a subsequence of the infinite sequence that converges to a point .

Algorithm 4

Step 1, Step 2, Step 4, Step 5, and Step 6 are the same as those in Algorithm 3.

Step 3. (stop criterion)

Compute ,

if , then STOP,

else GO TO step 4.

By Theorem 4 we have the following.

Theorem 5

Let be Lipschitz continuous on a compact convex body X and be nonempty. Then Algorithm 4 stops with an ε-solution of after a finite number of iterations.

Generalized analytic center cutting plane algorithms for variational inequalities with unbounded domains

This section presents analytic center cutting plane algorithms for solving a strongly pseudomonotone variational inequality whose domain is an unbounded convex body. By use of Propositions 1 and 2, due to the pseudomonotonicity, has a unique solution over X. Let be a sequence of polytopes that satisfies then has a unique solution over (). We can always assume that contains all boundary points of X (if there are any). Since the solution of is a fixed point, lies in if j is large enough (say ), therefore by Lemma 2 ().

The following algorithm is proposed here to find .

Algorithm 5

Step 1. (initialization)

Step 2. (find an ε-solution for )

Find an approximate analytic center of .

Compute ,

if , then increase j by one RETURN TO Step 1,

else GO TO Step 3;

Step 3. (cut generation)

Set is the new cutting plane for .

Increase k by one GO TO Step 2.

Theorem 6

Let be strongly monotone on X, then Algorithm 5 either stops with a solution of after a finite number of iterations, or there exists a subsequence of the infinite sequence that converges to a point .

F is strongly monotone on X implies that there exists a constant such that [22]

Let be the unique solution of over (). Suppose all boundary points of X (if there are any) are in (). Then, if j is large enough (say ), by Lemma 2 we have If Algorithm 5 does not stop after a finite number of iterations, then exists an infinite sequence with such that Hence which implies that is a bounded sequence. Therefore, ∃subsequence of , which is convergent to in X. Similar to the proof of Theorem 2, is a solution for , and so . □

We notice that, in the proof of Theorem 6, the key condition is that in X is a bounded subsequence. Therefore, similarly we have the following theorem.

Theorem 7

Let be strongly pseudomonotone on X, then Algorithm 5 either stops with a solution of after a finite number of iterations, or there exists a subsequence of the infinite sequence in X that converges to a point .

Theorems 6 and 7 state that Algorithm 5 can always stop and output an approximate solution after a finite number of iterations.