Analytic center cutting plane methods for variational inequalities over convex bodies
Abstract
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.
Keywords
Variational inequality Quasimonotonicity Pseudomonotonicity Analytic center cutting plane method Convex bodyMSC
65K05 65K10 65K15 90C331 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, 2, 3, 4, 5, 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.
We denote by \(X ^{*}\) the set of solutions of \(\mathit{VI} (F, X )\), and by \(X_{D}^{*}\) the set of solutions of \(\mathit{VID} (F, X )\).
From Auslender [15] we have the following lemma.
Lemma 1
IfFis continuous, then a solution of\(\mathit{VID} (F, X )\)is a solution of\(\mathit{VI} (F, X )\); and ifFis continuous pseudomonotone, then\(x ^{*} \in X\)is a solution of\(\mathit{VI} (F, X )\)if and only if it is a solution of\(\textit{VID} (F, X )\).
Lemma 2
A point\(x ^{*} \in X\)is a solution of\(\mathit{VI}[F,X]\) (\(\mathit{VID}[F, X]\)) if and only if\(g_{X}(x^{*}) = 0\) (\(f_{X}(x^{*}) = 0\)).
A point \(x^{*} \in X\) is said to be a εsolution of the variational inequality (1) if \(g_{X}(x^{*}) < \varepsilon\).
2 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 \(F: X \rightarrow\mathbb{R}^{{n}}\) is pseudomonotone plus on a compact convex body X, then our Algorithm 1 either stops with a solution of the variational inequality \(\mathit{VI} (F, X )\) after a finite number of iterations, or there exists an infinite sequence \(\{x_{j}\}\) in X that converges to a solution of \(\mathit{VI} (F, X )\). If \(F: X \rightarrow\mathbb{R}^{{n}}\) is pseudomonotone plus on a compact convex body X, then our Algorithm 2 stops with an εsolution of the variational inequality \(\mathit{VI} (F, X )\) after a finite number of iterations. If \(F: X \rightarrow\mathbb{R}^{{n}}\) is Lipschitz continuous on a compact convex body X, then our Algorithm 3 either stops with a solution of the variational inequality \(\mathit{VI} (F, X )\) after a finite number of iterations, or there exists an infinite sequence \(\{x_{j}\}\) in X that converges to a solution of \(\mathit{VI} (F, X )\). If \(F: X \rightarrow\mathbb{R}^{{n}}\) is Lipschitz continuous on a compact convex body X, then our Algorithm 4 stops with an εsolution of \(\mathit{VI} (F, X )\) 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 ndimensional Euclidean space \(\mathbb{R}^{{n}}\) itself. If \(F: X \rightarrow\mathbb{R}^{{n}}\) is strongly monotone on X, then our Algorithm 5 either stops with a solution of the variational inequality \(\mathit{VI} (F, X )\) after a finite number of iterations, or there exists an infinite sequence \(\{x_{j}^{k}\}\) in X that converges to a solution of \(\mathit{VI} (F, X )\). Furthermore, the proof of the previous result also indicates that, if \(F: X \rightarrow\mathbb{R}^{{n}}\) is strongly pseudomonotone on X, then our Algorithm 5 either stops with a solution of \(\mathit{VI} (F, X )\) after a finite number of iterations, or there exists an infinite sequence \(\{x_{j}\}\) in X that converges to a solution of \(\mathit{VI} (F, X )\).
3 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, 18, 19].
4 Compact convex bodies
A polytope is a set \(P \subseteq\mathbb{R}^{{n}}\) which is the convex hull of a finite set.
Every polytope is a polyhedron, whereas not every polyhedron is a polytope.
Minkowski proved the following lemma in 1896.
Lemma 3
A set\(P \subseteq\mathbb{R}^{{n}}\)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 fulldimensional polytope and that includes \(0 \le x \le e\), where e is a vector of all ones.
We note that if a polytope has nonempty interior, then (a) can be met by rescaling.
A convex body\(X \subseteq\mathbb{R}^{{n}}\) is a convex and bounded subset with nonempty interior.
Theorem 1
Proof
The sufficiency is trivial. We only prove the necessity.
Since X is bounded, there exists a rectangle B such that \(X \subseteq B\).
It is easy to see that \((\bigcup_{j = 1}^{\infty} C_{j} )^{c} = X\). □
It is quite straightforward to prove the following Corollary 1, Proposition 1, and Proposition 2.
Corollary 1
Proposition 1
Let\(X \subseteq\mathbb{R}^{{n}}\)be a compact convex body and\(F: X \rightarrow\mathbb{R}^{{n}}\)be a continuous function, then the variational inequality\(\mathit{VI}[F, X]\)has solutions.
Proposition 2
Let\(X \subseteq\mathbb{R}^{{n}}\)be a compact convex body and\(F:X \rightarrow\mathbb{R}^{{n}}\)be a continuous and strictly pseudomonotone function, then the variational inequality\(\mathit{VI}[F, X]\)has a unique solution.
5 Generalized analytic center cutting plane algorithms for solving pseudomonotone variational inequalities
Now we modify Goffin, Marcotte, and Zhu’s [2] Algorithm 1 to solve \(\mathit{VI} (F, X )\). We propose an algorithm for solving variational inequalities, whose feasible sets are compact convex bodies.
Algorithm 1
 Step 1. (initialization)$$k = 0, \qquad j = 1,\qquad A^{k} = A_{j},\qquad b^{k} = b_{j},\qquad C_{j}^{k} = \bigl\{ x \in \mathbb{R}^{n};A_{j}^{k}x \le b_{j}^{k} \bigr\} ; $$

Step 2. (computation of an approximate analytic center)
Find an approximate analytic center \(x_{j}^{k}\) of \(C_{j}^{k}\);

Step 3. (stop criterion)
Compute \(g_{X}(x_{j}^{k}) = \max_{x \in X}F(x_{j}^{k})^{T}(x_{j}^{k}  x)\),
if \(g_{X}(x_{j}^{k}) = 0\), then STOP,
else GO TO step 4;

Step 4. (find an εsolution for \(\varepsilon = \frac{1}{2^{j}}\))
Compute \(g_{C_{j}}(x_{j}^{k}) = \max_{x \in C_{j}}F(x_{j}^{k})^{T}(x_{j}^{k}  x)\),
if \(g_{C_{j}}(x_{j}^{k}) < \frac{1}{2^{j}}\), then increase j by one RETURN TO Step 1,
else GO TO Step 5;

Step 5. (cut generation)
Set\(H_{j}^{k} = \{ x \in \mathbb{R}^{n};F(x_{j}^{k})^{T}(x  x_{j}^{k}) = 0\}\) is the new cutting plane for \(\mathit{VI}(F, C_{j}^{k})\).$$A_{j}^{k + 1} = \left [ \textstyle\begin{array}{l} A_{j}^{k} \\ F(x_{j}^{k})^{T} \end{array}\displaystyle \right ],\qquad b_{j}^{k + 1} = \left [ \textstyle\begin{array}{l} b_{j}^{k} \\ F(x_{j}^{k})^{T}x_{j}^{k} \end{array}\displaystyle \right ], $$Increase k by one GO TO Step 2.
Theorem 2
Let\(F: X \rightarrow\mathbb{R}^{{n}}\)be pseudomonotone plus on a compact convex bodyX, then Algorithm 1either stops with a solution of\(\mathit{VI} (F, X )\)after a finite number of iterations, or there exists a subsequence of the infinite sequence\(\{x_{j}^{k}\}\)that converges to a point\(x ^{*} \in X ^{*}\).
Proof
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 3. (stop criterion)
Compute \(g_{X}(x_{j}^{k}) = \max_{x \in X}F(x_{j}^{k})^{T}(x_{j}^{k}  x)\),
if \(g_{X}(x_{j}^{k}) < \varepsilon\), then STOP,
else GO TO step 4.
From Theorem 2 we have the following.
Theorem 3
Let\(F: X \rightarrow\mathbb{R}^{{n}}\)be pseudomonotone plus on a compact convex bodyX, then Algorithm 2stops with anεsolution of\(\mathit{VI} (F, X )\)after a finite number of iterations.
6 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 \(\mathit{VI} (F, X )\). We assume that the feasible sets are compact convex bodies.
From Theorem 1 there is a sequence of variational inequalities \(\mathit{VI} [F, C_{{j}}]\) (\(j = 1, \ldots\)) induced by the original variational inequality \(\mathit{VI}[F, X]\).
According to [1], the following are the conditions that are required in the construction of algorithms for solving quasimonotone variational inequalities.
It is known that \(w _{j}(x)\) are continuous [21], and that x is a solution of \(\mathit{VI}[F,C _{j}]\) if and only if it is a fixed point of w.
Algorithm 3

Step 1. (initialization)
Let \(\beta_{j} > 0\) be the strong monotonicity constant for \(\Gamma_{j}(y,x):\mathbb{R}^{n} \to \mathbb{R}^{n}\), with respect to y, and let \(\alpha _{j} \in (0, \beta_{j})\).$$k = 0,\qquad j = 1,\qquad A^{k} = A_{j},\qquad b^{k} = b_{j},\qquad C_{j}^{k} = \bigl\{ x \in \mathbb{R}^{n};A_{j}^{k}x \le b_{j}^{k} \bigr\} ; $$ 
Step 2. (computation of an approximate analytic center)
Find an approximate analytic center \(x_{j}^{k}\) of \(C_{j}^{k}\);

Step 3. (stop criterion)
Compute \(g_{X}(x_{j}^{k}) = \max_{x \in X}F(x_{j}^{k})^{T}(x_{j}^{k}  x)\),
if \(g_{X}(x_{j}^{k}) = 0\), then STOP,
else GO TO step 4;

Step 4. (find an εsolution for \(\varepsilon = \frac{1}{2^{j}}\))
Compute \(g_{C_{j}}(x_{j}^{k}) = \max_{x \in C_{j}}F(x_{j}^{k})^{T}(x_{j}^{k}  x)\),
if \(g_{C_{j}}(x_{j}^{k}) < \frac{1}{2^{j}}\), then increase j by one RETURN TO Step 1,
else GO TO Step 5;

Step 5. (auxiliary variational inequality)
Let \(w_{j}(x_{j}^{k})\) satisfy the variational inequalityLet$$\bigl\langle F\bigl(x_{j}^{k}\bigr) + \Gamma_{j} \bigl(w_{j}\bigl(x_{j}^{k}\bigr),x_{j}^{k} \bigr)  \Gamma_{j}\bigl(x_{j}^{k},x_{j}^{k} \bigr),y  w_{j}\bigl(x_{j}^{k}\bigr) \bigr\rangle \ge 0,\quad\forall y \in C_{j}. $$where \(l(k, j)\) is the smallest integer which satisfies$$y_{j}^{k} = x_{j}^{k} + \rho_{j}^{l(k,j)}\bigl(w_{j}\bigl(x_{j}^{k} \bigr)  x_{j}^{k}\bigr) \quad \mbox{and}\quad G_{j} \bigl(x_{j}^{k}\bigr) = F\bigl(y_{j}^{k} \bigr), $$$$\bigl\langle F\bigl(x_{j}^{k} + \rho_{j}^{l(k,j)} \bigl(w_{j}\bigl(x_{j}^{k}\bigr)  x_{j}^{k}\bigr)\bigr),x_{j}^{k}  w_{j}\bigl(x_{j}^{k}\bigr) \bigr\rangle \ge \alpha_{j}\big\ w_{j}\bigl(x_{j}^{k}\bigr)  x_{j}^{k}\big\ ^{2}; $$ 
Step 6. (cutting plane generation)
Set\(H_{j}^{k} = \{ x \in \mathbb{R}^{n};G(x_{j}^{k})^{T}(x  x_{j}^{k}) = 0\}\) is the new cutting plane for \(\mathit{VI}(F, C_{j}^{k})\).$$A_{j}^{k + 1} = \left [ \textstyle\begin{array}{l} A_{j}^{k} \\ G(x_{j}^{k})^{T} \end{array}\displaystyle \right ],\qquad b_{j}^{k + 1} = \left [ \textstyle\begin{array}{l} b_{j}^{k} \\ G(x_{j}^{k})^{T}x_{j}^{k} \end{array}\displaystyle \right ], $$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
Algorithm 4

Step 3. (stop criterion)
Compute \(g_{X}(x_{j}^{k}) = \max_{x \in X}F(x_{j}^{k})^{T}(x_{j}^{k}  x)\),
if \(g_{X}(x_{j}^{k}) < \varepsilon\), then STOP,
else GO TO step 4.
By Theorem 4 we have the following.
Theorem 5
Let\(F: X \rightarrow\mathbb{R}^{{n}}\)be Lipschitz continuous on a compact convex bodyXand\(X_{D}^{*}\)be nonempty. Then Algorithm 4stops with anεsolution of\(\mathit{VI} (F, X )\)after a finite number of iterations.
7 Generalized analytic center cutting plane algorithms for variational inequalities with unbounded domains
The following algorithm is proposed here to find \(x ^{*}\).
Algorithm 5
 Step 1. (initialization)$$k = 0,\qquad j = 1,\qquad A^{k} = A_{j},\qquad b^{k} = b_{j},\qquad C_{j}^{k} = \bigl\{ x \in \mathbb{R}^{n};A_{j}^{k}x \le b_{j}^{k} \bigr\} ; $$

Step 2. (find an εsolution for \(\varepsilon = \frac{1}{2^{j}}\))
Find an approximate analytic center \(x_{j}^{k}\) of \(C_{j}^{k}\).
Compute \(g_{C_{j}}(x_{j}^{k}) = \max_{x \in C_{j}}F(x_{j}^{k})^{T}(x_{j}^{k}  x)\),
if \(g_{C_{j}}(x_{j}^{k}) < \frac{1}{2^{j}}\), then increase j by one RETURN TO Step 1,
else GO TO Step 3;

Step 3. (cut generation)
Set\(H_{j}^{k} = \{ x \in \mathbb{R}^{n};F(x_{j}^{k})^{T}(x  x_{j}^{k}) = 0\}\) is the new cutting plane for \(\mathit{VI}(F, C_{j}^{k})\).$$A_{j}^{k + 1} = \left [ \textstyle\begin{array}{l} A_{j}^{k} \\ F(x_{j}^{k})^{T} \end{array}\displaystyle \right ],\qquad b_{j}^{k + 1} = \left [ \textstyle\begin{array}{l} b_{j}^{k} \\ F(x_{j}^{k})^{T}x_{j}^{k} \end{array}\displaystyle \right ], $$Increase k by one GO TO Step 2.
Theorem 6
Let\(F: X \rightarrow\mathbb{R}^{{n}}\)be strongly monotone onX, then Algorithm 5either stops with a solution of\(\mathit{VI} (F, X )\)after a finite number of iterations, or there exists a subsequence of the infinite sequence\(\{x_{j}^{k}\}\)that converges to a point\(x ^{*} \in X ^{*}\).
Proof
We notice that, in the proof of Theorem 6, the key condition is that \(\{x_{j}\}\) in X is a bounded subsequence. Therefore, similarly we have the following theorem.
Theorem 7
Let\(F: X \rightarrow\mathbb{R}^{{n}}\)be strongly pseudomonotone onX, then Algorithm 5either stops with a solution of\(\mathit{VI} (F, X )\)after a finite number of iterations, or there exists a subsequence of the infinite sequence inXthat converges to a point\(x ^{*} \in X\).
Theorems 6 and 7 state that Algorithm 5 can always stop and output an approximate solution after a finite number of iterations.
Notes
Acknowledgements
The author would like to thank the reviewers of this paper for their valuable comments on the earlier version of the paper.
Authors’ contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Competing interests
The author declares that they have no competing interests.
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