Under the condition that the values of mapping F are evaluated approximately, we propose a proximal analytic center cutting plane algorithm for solving variational inequalities. It can be considered as an approximation of the earlier cutting plane method, and the conditions we impose on the corresponding mappings are more relaxed. The convergence analysis for the proposed algorithm is also given at the end of this paper.

1. Introduction

According to [1], the history of algorithms for solving finite-dimensional variational inequalities is relatively short. A recent development of such methods is the analytic center method based on cutting plane methods. It combines the feature of the newly developed interior point methods with the classical cutting plane scheme to achieve polynomial complexity in theory and quick convergence in practice. More details can be found in [2, 3]. Specifically, Goffin et al. [4] developed a convergent framework for finding a solution x^* of the variational inequality VI _P(F, X) associated with the continuous mapping F from X to Rⁿ and the polyhedron X = {x ∈ Rⁿ | Ax ≤ b} under an assumption slightly stronger than pseudomonotonicity. Again, Marcotte and Zhu [5] extended this algorithm to quasimonotone variational inequalities that satisfy a weak additional assumption. Such methods are effective in practice.

Note that the facts in optimization problems, see [6–8], some functions from Rⁿ to R are themselves defined through other minimization problems. For example, consider the Lagrangian relaxation, see [9–12], the primal problem is

()

where P is a compact subset of R^m and q : R^m → R, h : R^m → Rⁿ are two functions. Lagrangian relaxation in this problem leads to the problem min {f(x) | x ∈ Rⁿ}, where

()

is the dual function. Trying to solve problem (1.1) by means of solving its dual problem min {f(x) | x ∈ Rⁿ} becomes more difficult since in this case evaluating the function value f(x) requires solving exactly another optimization problem (1.2). Let us see another example: consider the problem

()

where f is convex (not necessarily differentiable), C ⊂ Rⁿ is a nonempty closed convex set, F is called the Moreau-Yosida regularization of f on C, that is,

()

where α is a positive parameter. A point x ∈ C is a solution to (1.3) if and only if it is a solution to the problem:

()

The problem (1.5) is easier to deal with than (1.3), see [13]. But in this case, computing the exact function value of F at an arbitrary point x is difficult or even impossible since F itself is defined through a minimization problem involving another function f. Intuitively, we consider the approximate computation of function F.

The above-mentioned phenomenon also exists for mappings from X to Y, where X and Y are two subspaces of any two finite-dimensional spaces, respectively. Once a mapping, and more specifically, a continuous mapping is defined implicitly rather than explicitly, the approximation of the mapping becomes inevitable, see [14]. In this paper we try to solve VI _P(F, X) by assuming the values of the mapping F from X to Rⁿ can be only computed approximately. Under the assumption, we construct an algorithm for solving the approximate variational inequality problem AVI_P(F, X) and we also prove that there exists a cluster point of the iteration points generated by the proposed algorithm, it is a solution to the original problem VI _P(F, X).

This paper is organized as follows. Some basic concepts and results are introduced in Section 2. In Section 3, a proximal analytic center cutting plane algorithm for solving the variational inequality problems is given. The convergence analysis of the proposed algorithm is addressed in Section 4. In the last section, we give some conclusions.

2. Basic Concepts and Results

Let X = {x ∈ Rⁿ∣Ax ≤ b} be a polyhedron and F a continuous mapping from X to Rⁿ. A vector x^* is a solution to the variational inequality VI _P(F, X) if and only if it satisfies the system of nonlinear inequalities:

()

The vector x^* is a solution to the dual variational inequality VI _D(F, X) of VI _P(F, X) if and only if it satisfies

()

We denote by

the solution set of VI _P(F, X), and

the solution set of VI _D(F, X), respectively. Whenever F is continuous, we have

, see [15]. If F is pseudomonotone on X, then

, see [15]. If F is quasimonotone at

and F(x^*) is not normal to X at x^*, then

is nonempty, see Proposition 1 in [5]. For the definitions of monotone, pseudomonotone and quasimonotone, see [5, 15].

Definition 2.1. The gap functions g_P(x) and g_D(x) of VI _P(F, X) and VI _D(F, X) are, respectively, defined by

()

Note that g_P(x) ≥ 0, g_D(x) ≥ 0, and g_P(x^*) = 0 if and only if x^* is a solution to VI _P(F, X), g_D(x^*) = 0 if and only if x^* is a solution to VI _D(F, X). Thus,

Definition 2.2. A point is called an ε-solution to VI _P(F, X) if for given ε > 0.

Definition 2.3. For x, y ∈ X, we say F(x)⪯F(y) if and only if F_i(x) ≤ F_i(y), for i = 1,2, …, n, where F(x) = (F₁(x), F₂(x), …, F_n(x)) ^T.

Assumptions. Throughout this paper, we make the following assumptions: for each x, y ∈ X, given any , , where ε, δ ∈ (0,1), we can always find a and a such that

()

These assumptions are realistic in practice, see [16, 17]. By using the given architecture in [16, 17], we can approximate the mapping F arbitrary well since neural networks are capable of approximating any function from one finite-dimensional real vector space to another one arbitrary well, see [18]. Specifically, let us consider the case of univariate function. If f is a min-type function of the form

()

where each N_z(x) is convex and Z is an infinite set, then it may be impossible to calculate f(x). However, we may still consider two cases. In the first case of controllable accuracy, for each positive ε > 0 one can find an ε-minimizer of (2.5), that is, an element z_x ∈ Z satisfying

; in the second case, this may be possible only for some fixed (any possibly unknown) ε < ∞. In both cases, we may set

. A special case of (2.5) arises from Lagrangian relaxation [15], where the problem max {f(x)∣x ∈ S} with

is the Lagrangian dual of the primal problem

()

with N_z(x) = ψ₀(z)+〈x, ψ(z)〉 for ψ = (ψ₁, ψ₂, …, ψ_n). Then, for each multiplier x ≥ 0, we need only to find z_x ∈ Z such that

, see [19].

Under the above assumptions (2.4), we introduce an approximate problem AVI_P(F, X) associated with VI _P(F, X): finding x^* ∈ X such that

()

where

satisfies

for arbitrary

. Its dual problem AVI_D(F, X) is to find x^* ∈ X such that

()

where

satisfies

for arbitrary

Definition 2.5. The gap function of AVI_P(F, X) is defined by .

Definition 2.6. A point is called an ε-solution to AVI_P(F, X) if for given ε > 0.

The optimal solution sets of AVI_P(F, X) and AVI_D(F, X) are denoted by and , respectively. The following proposition ensures that is nonempty.

Proposition 2.7. If there exists a point such that

()

and

is not normal to X at x^*, then

is nonempty.

Proof. Since is not normal to X at x^*, there exists a point x₀ ∈ X such that . ∀ x ∈ X, set x_t = tx₀ + (1 − t)x for t ∈ (0,1], then we have and . Note the condition (2.9), we obtain . Letting t → 0, it follows from the condition (b) in (2.4) that , that is, .

In the following part, we focus our attention on solving AVI_P(F, X). Let Γ(y, x) : Rⁿ × Rⁿ → Rⁿ denote an auxiliary mapping, continuous in x and y, strongly monotone in y, that is,

()

for some β > 0. We consider the auxiliary variational inequality associated with Γ, whose solution w(x) satisfies

()

In view of the strong monotonicity of Γ(y, x) with respect to y, this auxiliary variational inequality has a unique solution w(x).

Proposition 2.8. The mapping w : X → X is continuous on X. Furthermore, x is a solution to AVI_P(F, X) if and only if x = w(x).

Proof. The first part of the proposition follows from Theorem 5.4 in [1]. To prove the second part, we first suppose that x = w(x). This yields , that is, x solves AVI_P(F, X). Conversely, suppose that x solves AVI_P(F, X), then

()

and from (2.11), we have

()

Adding the two preceding inequalities, one obtains

()

and we conclude, from the strong monotonicity of Γ with respect to y, that x = w(x).

Let ρ < 1, α < β be two positive numbers. Let l be the smallest nonnegative integer satisfying

()

where

satisfies

for arbitrary

. The existence of a finite l will be proved in Proposition 2.9. The composite mapping G is defined, for every x ∈ X, by

()

If x^* is a solution to AVI_P(F, X), then we have w(x^*) = x^*, l = 0 and

Proposition 2.9. The operator G is well defined for every x ∈ X. Moreover, we have

()

where L is the number given in (2.4)-(c).

Proof. From the definition of w(x), we have

()

Suppose (2.15) does not hold for any finite integer l, that is,

()

Note the assumption (2.4)-(b), we obtain

()

therefore,

()

Since α < β, (2.21) is in contradiction with (2.18). To prove the second part, we notice that

()

if α ≤ β − Lρ^l, which means the second conclusion of Proposition 2.9 holds.

Proposition 2.10. If , then , we have

()

Proof. Let y(x) = x + ρ^l(w(x) − x), then and

()

Since

, so w(x) ≠ x. Therefore,

()

For all

, there holds

()

By combining (2.25) with (2.26), we obtain

, that is,

3. A Proximal Analytic Center Cutting Plane Algorithm

Algorithm 3.1 offered in this section is a modification of the algorithm in [5]. Algorithm 3.1 is described as follows.

Algorithm 3.1. Let β be the strong monotonicity constant of Γ(x, y) with respect to y, and let α ∈ (0, β), ε ∈ (0,1) be two constants. Set k = 0, A^k = A ∈ R^m×n, b^k = b, and ε^k = ε.

Step 1 (computation of the center). Find an approximate analytic center x^k of X^k = {x ∈ Rⁿ∣A^kx ≤ b^k}.

Step 2 (stopping criterion). If g_P(x^k) ≤ ε, stop.

Step 3 (solving the approximate auxiliary variational inequality problem). Find w(x^k), such that

()

where

satisfies

Step 4 (construction of the approximate cutting plane). Let and , where l_k is the smallest integer that satisfies

()

where

satisfies

Let ,

()

Increase k by one and go to Step 1.

End of Algorithm 3.1

4. Convergence Analysis

In [20], the authors proposed a column generation scheme to generate the polytope X^k, and they proved if k satisfies the following inequality

()

where ε < 1/2 is a constant, the scheme will stop with a feasible solution, that is, they can find a vector a_k+1 such that

with | | a_k+1|| = 1, Γ contains a full-dimensional closed ball with ε < 1/2 radius. In other words, there exists the smallest k(ε) such that X^k(ε) generated by the column generation scheme does not contain the ball with ε < 1/2 radius, and it is known as the finite cut property. It is easy to know that the result of Theorem 6.6 in [20] also holds without much change for our Algorithm 3.1 using the approximate centers. That is, by using the row generation scheme, there exists the smallest k(ρ) such that X^k(ρ) generated in Step 4 in Algorithm 3.1 does not contain the ball with ρ radius lying inside the polytope X. This result plays an important role in proving the convergence of the described Algorithm 3.1 in Section 3.

Theorem 4.1. Let the polyhedron X have nonempty interior, and let be nonempty. Assumption (2.4) holds. Then either Algorithm 3.1 stops with a solution to AVI_P(F, X) after a finite number of iterations, or there exists a subsequence of the infinite sequence {x^k} that converges to a point in .

Proof. Assume that for every iteration k, and let . From Proposition 2.10, we know that y^* ∈ X^k never lies on H^k for any k. Let be an arbitrary sequence of point in the interior of X converging to y^* and δ_i a sequence of positive numbers such that lim _i→+∞δ_i = 0 and that the sequence of closed balls lies in the interior of X. Note that . From finite cut property, we know that there exists the smallest index k(i) and a point such that satisfies

()

, there exists a point

on the segment

such that

. Since X is compact, we can extract from {x^k(i)} _i∈N a convergent subsequence {x^k(i)} _i∈S. Denote by

its limit point, we have

()

From Proposition 2.9, we know that l_k(i) is bounded. Consequently, we can extract form {l_k(i)} _i∈S a constant subsequence

. Now from the continuity of w(x) for fixed k and the relations (2.15) and (4.3), it follows by taking the limit in (4.3) that

()

By Proposition 2.10, we conclude that

Theorem 4.2. Under the conditions of Theorem 4.1, either Algorithm 3.1 stops with a solution to AVI_P(F, X) after a finite number of iterations, or there exists a subsequence of the infinite sequence {x^k} that converges to a point in .

Proof. Since ε ∈ (0,1), ε^k ∈ (0,1). At the end of Step 4 in Algorithm 3.1 we increase k by one, so we have ε^k+1 < ε^k, ε^k → 0 as k → ∞. Moreover, as k → ∞ in Algorithm 3.1, where denotes the zero vector. This means as k → ∞. Therefore, from the second result of Theorem 4.1, we know is the solution to the problem VI _P(F, X).

5. Conclusions

In [5], the authors proposed a cutting plane method for solving the quasimonotone variational inequalities, but throughout the paper they employed the exact information of the mapping F from X to Rⁿ. Just like the discussion in the first part of our paper, sometimes, it is not so easy or even impossible to compute the exact values of the mapping F. Motivated by this fact, we consider constructing an approximate problem AVI_P(F, X) of VI _P(F, X), and try out a proximal analytic center cutting plane algorithm for solving AVI_P(F, X). In contrast to [5], our algorithm can be viewed as an approximation algorithm, and it is easier to implement than [5] since it only requires the inexact information of the corresponding mapping. At the same time, the conditions we impose on the corresponding mappings are more relaxed, for example, [5] requires the mapping F satisfies the Lipschitz condition, but we only require that the so-called approximate Lipschitz condition (2.4)-(c) holds.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant nos. 11171138, 11171049) and Higher School Research Project of Educational Department of Liaoning Province (Grant no. L2010235).

References

1 Harker P. T. and Pang J.-S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Mathematical Programming B. (1990) 48, no. 2, 161–220, https://doi.org/10.1007/BF01582255, MR1073707, ZBL0734.90098.
10.1007/BF01582255
Web of Science® Google Scholar
2 Goffin J.-L. and Vial J.-P., Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method, Optimization Methods & Software. (2002) 17, no. 5, 805–867, https://doi.org/10.1080/1055678021000060829, MR1953822, ZBL1065.90060.
10.1080/1055678021000060829a
Web of Science® Google Scholar
3 Luo Z. Q. and Sun J., An analytic center based column generation algorithm for convex quadratic feasibility problems, SIAM Journal on Optimization. (1999) 9, no. 1, 217–235, https://doi.org/10.1137/S1052623495294943, MR1662587.
10.1137/S1052623495294943
Web of Science® Google Scholar
4 Goffin J. L., Marcotte P., and Zhu D., An analytic center cutting plane method for pseudomonotone variational inequalities, Operations Research Letters. (1997) 20, no. 1, 1–6, https://doi.org/10.1016/S0167%2D6377(96)00029%2D6, MR1428382, ZBL0899.90157.
10.1016/S0167-6377(96)00029-6
Web of Science® Google Scholar
5 Marcotte P. and Zhu D. L., A cutting plane method for solving quasimonotone variational inequalities, Computational Optimization and Applications. (2001) 20, no. 3, 317–324, https://doi.org/10.1023/A:1011219303531, MR1857061, ZBL1054.90073.
10.1023/A:1011219303531
Web of Science® Google Scholar
6 Hintermüller M., A proximal bundle method based on approximate subgradients, Computational Optimization and Applications. (2001) 20, no. 3, 245–266, https://doi.org/10.1023/A:1011259017643, MR1857057, ZBL1054.90053.
10.1023/A:1011259017643
Web of Science® Google Scholar
7 Kiwiel K. C., An algorithm for nonsmooth convex minimization with errors, Mathematics of Computation. (1985) 45, no. 171, 173–180, https://doi.org/10.2307/2008055, MR790650, ZBL0584.65034.
10.1090/S0025-5718-1985-0790650-5
Web of Science® Google Scholar
8 Solodov M. V., On approximations with finite precision in bundle methods for nonsmooth optimization, Journal of Optimization Theory and Applications. (2003) 119, no. 1, 151–165, https://doi.org/10.1023/B:JOTA.0000005046.70410.02, MR2028443, ZBL1094.90046.
10.1023/B:JOTA.0000005046.70410.02
Web of Science® Google Scholar
9 Bertsekas D. P., Nonlinear Programming, 1995, Athena Scientific, Belmont, Mass, USA.
Google Scholar
10 Bertsekas D. P. and Tsitsiklis J. N., Parallel and Distributed Computation, 1989, Prentice-Hall, Englewood Cliffs, NJ, USA.
Google Scholar
11 Bonnans J. F., Gilbert J. C., Lemaréchal C., and Sagastizábal C., Optimization Numerique: Aspects Theoriques et Pratiques, 1997, Springer, Berlin, Germany, MR1613833.
Google Scholar
12 Lemaréchal C., Lagrangian decomposition and nonsmooth optimization: bundle algorithm, prox iteration, augmented Lagrangian, Nonsmooth optimization: Methods and Applications, 1992, Gordon and Breach, Philadelphia, Pa, USA, 201–216, MR1263502, ZBL1050.90553.
Google Scholar
13 He Y. R., Minimizing and stationary sequences of convex constrained minimization problems, Journal of Optimization Theory and Applications. (2001) 111, no. 1, 137–153, https://doi.org/10.1023/A:1017575415432, MR1850682, ZBL0987.90067.
10.1023/A:1017575415432
Web of Science® Google Scholar
14 Pinkus A., Density in approximation theory, Surveys in Approximation Theory. (2005) 1, 1–45, MR2221565, ZBL1067.41003.
Google Scholar
15 Auslender A., Optimization, Méthodes, 1976, Numériques, Masson, Paris.
Google Scholar
16 Hammer B., Universal approximation of mappings on structured objects using the folding architecture, Technical Report. Reihd P, Heft, 1996, no. 183, Universitat Osnabruck, Fachbereich Informatic.
Google Scholar
17 Hammer B. and Sperschneider V., P. P. Wang, Neural networks can approximate mappings on structured objects, Proceedings of the International Conference on Computational Inteliigence and Neural Network, 1997, 211–214.
Google Scholar
18 Hornik K., Stinchcombe M., and White H., Multilayer feedforward networks are universal approximators, Neural Networks. (1989) 2, no. 5, 359–366, 2-s2.0-0024880831.
10.1016/0893-6080(89)90020-8
Web of Science® Google Scholar
19 Kiwiel K. C., A proximal bundle method with approximate subgradient linearizations, SIAM Journal on Optimization. (2006) 16, no. 4, 1007–1023, https://doi.org/10.1137/040603929, MR2219129, ZBL1104.65055.
10.1137/040603929
Web of Science® Google Scholar
20 Goffin J. L., Luo Z. Q., and Ye Y., Complexity analysis of an interior cutting plane method for convex feasibility problems, SIAM Journal on Optimization. (1996) 6, no. 3, 638–652, https://doi.org/10.1137/S1052623493258635, MR1402198, ZBL0856.90088.
10.1137/S1052623493258635
Web of Science® Google Scholar

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A Proximal Analytic Center Cutting Plane Algorithm for Solving Variational Inequality Problems

Abstract

1. Introduction

2. Basic Concepts and Results

3. A Proximal Analytic Center Cutting Plane Algorithm

4. Convergence Analysis

5. Conclusions

Acknowledgments

References

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A Proximal Analytic Center Cutting Plane Algorithm for Solving Variational Inequality Problems

Abstract

1. Introduction

2. Basic Concepts and Results

3. A Proximal Analytic Center Cutting Plane Algorithm

4. Convergence Analysis

5. Conclusions

Acknowledgments

References

Citing Literature

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