Above regularization for constrained generalized complementarity problems

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The convergence rates for the regularized solutions is also shown under ordinary conditions in the theory of nonlinear ill-posed problems. We also give a numerical example as an illustration.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 44-51 This paper is available online at http://stdb.hnue.edu.vn ABOVE REGULARIZATION FOR CONSTRAINED GENERALIZED COMPLEMENTARITY PROBLEMS Nguyen Thi Thuy Hoa1 and Nguyen Buong2 1 Informatics Center, Hanoi University of Home Affairs 2 Institute of Information Technology, Vietnamese Academy of Science and Technology Abstract. The purpose of this paper is to use the Tikhonov regularization method to solve a constrained generalized complementarity problems, that is to find an element x ˜ ∈ C : g(˜x) ≤ 0, h(˜x) ≤ 0, ⟨g(˜ x), h(˜ x)⟩Em = 0, where C is a convex n closed subset in E , g(x) and h(x) are two continuous functions from an Euclidian space En to Em and ⟨., .⟩En denotes the scalar product of En . The convergence rates for the regularized solutions is also shown under ordinary conditions in the theory of nonlinear ill-posed problems. We also give a numerical example as an illustration. Keywords: Convex set, Tikhonov regularization, continuous function. 1. Introduction Let g(x) and h(x) be two continuous functions from an Euclidian space En to Em where the scalar product and norm of En are denoted by ⟨., .⟩En and ∥.∥En , respectively. The problem of finding an element x˜ ∈ En such that x) ≤ 0, h(˜ g(˜ x) ≤ 0, ⟨g(˜ x), h(˜ x)⟩Em = 0, (1.1) where the symbol y = (y1 , ..., ym ) ≤ 0 is meant that yi ≤ 0, i = 1, ..., m, is called a generalized complementarity problem (GCP). In the case that m = n, g(x) = −x, and h(x) = −F (x), a continuous function in En , (1.1) is the classical nonlinear complementarity problem (NCP), we find an element x ∈ En satisfying x ≥ 0, F (x) ≥ 0, ⟨x, F (x)⟩En = 0, Received September 12, 2014. Accepted October 23, 2014. Contact Nguyen Thi Thuy Hoa, e-mail address: nguyenhoanvhn@gmail.com 44
Above regularization for constrained generalized complementarity problems which has attracted much attention due to its various applications (see, [6, 8, 10]). There exist several methods for the solution of the NCP [3, 4, 7]. All of these methods are proposed for solving an equivalent minimization problem or a system of equations. The particular class of methods to be considered are the so-called regularization methods, which are designed to handle ill-posed problems. An ill-posed problem may be difficult to solve since small errors in the computations can lead to a totally wrong solution. The Tikhonov-regularization scheme in [5, 9] for NCP consists in solving a sequence of complementarity problems: xε ≥ 0, Fε (xε ) ≥ 0, ⟨xε , Fε (xε )⟩En = 0, where Fε (x) = F (x) + εx, and ε is a small parameter of regularization. In [12] the regularization xε is defined on the basis of H(ε, z) = 0 ⇐⇒ ε = 0, x ∈ S 0 , where S 0 denotes the solution set of NCP, z := (ε, x), H(ε, z) := (ε, G(ε, z))T , Gi (ε, x) := ϕ(xi , Fε,i (x)), i = 1, ..., n, where Fε,i is the ith component of Fε . The convegence of the method is established only for the P0 -function F . Moreover, estimating the convergence rate of the regularized solutions is still an opened question. Recently, for GCP (1.1), without P0 -property for g and h, we [2] presented an other approach to regularizing (1.1) and gave an estimate of convergence for our method. In this paper, using this idea, we consider the following constrained GCP (CGCP): find an element x˜ ∈ C : g(˜ x) ≤ 0, h(˜ x) ≤ 0, ⟨g(˜ x), h(˜x)⟩Em = 0, (1.2) where C is a convex closed subset in En . Put φi (x) = max{0, gi (x)}, i = 1, m, φm+i (x) = max{0, hi (x)}, i = 1, m. Clearly, Si := {x ∈ En : gi (x) ≤ 0} = {x ∈ En : φi (x) = 0}, i = 1, m, Si+m := {x ∈ En : hi (x) ≤ 0} = {x ∈ En : φm+i (x) = 0}, i = 1, m, the functions φj are also continuous, nonegative on En and φj (y) = 0 ∀y ∈ ∩N j=1 Sj , j = 1, ..., 2m := N. Moreover, ∩N j=1 Sj = {x ∈ E : g(x) ≤ 0, h(x) ≤ 0}. n 45
Nguyen Thi Thuy Hoa and Nguyen Buong Evidently, ∑ m ⟨g(˜ x), h(˜ x)⟩Em = gi (˜ x) = 0 ⇐⇒ gi (˜ x)hi (˜ x)hi (˜ x) = 0, i = 1, ..., m. i=1 Therefore, we consider the functions fi (x) = gi (x)hi (x) and set S0 = {x ∈ En : fi (x) = 0, i = 1, ..., m}. (1.3) Then, CGCP (1.2) is equivalent to find an element x˜ belonging to C ∩ (∩N i=0 Si ) such that φj (˜ x) = min φj (x), j = 1, ..., N, (1.4) x∈C∩(∩N i=0 Si ) In order to solve this vector optimization problem, we consider the following unconstrained scalar optimization problem: Find an element x˜ ∈ En such that Fα (xα ) = minn Fα (x), (1.5) x∈E where Fα (x) is defined by ∑ N Fα (x) = ∥x − PC (x)∥ + 2 ∥F (x)∥2Em + αµj φj (x) + α∥x − x+ ∥2En , j=1 ≤ µ1 < µj < µj+1 < 1, j = 2, ..., N − 1, F (x) = (f1 (x), ..., fm (x))T , (1.6) where PC (x) is the metric projection of x ∈ En on C and x+ is some element in En . It is well-known [14] that problem (1.5) has a solution xα for each α > 0. 2. Main results We have the following results. Theorem 2.1. Let αk → 0, as k → ∞. Then every sequence {xk }, where xk := xαk is a solution of (1.5) with α replaced by αk , has a convergent subsequence. The limit of every convergent subsequence is an x+ - Cminimal norm solution (x+ - CMNS). If, in addition, the x+ - CMNS x˜ is unique, then lim xk = x˜. k→∞ 46
Above regularization for constrained generalized complementarity problems Proof. From (1.5) it follows ∑ N µ ∥xk − PC (xk )∥2 + ∥F (xk )∥2Em + αk j φj (xk ) + αk ∥xk − x+ ∥2En j=1 ∑ N µ ≤ ∥y − PC (y)∥ + 2 ∥F (y)∥2Em + αk j φj (y) + αk ∥y − x+ ∥2En , (2.1) j=1 for each fixed element y ∈ E . Taking y ∈ C ∩ (∩N n j=0 Sj ), we have that y − PC (y) = 0, F (y) = 0 and φj (xk ) ≥ φj (y) = 0, j = 1, ..., N. Then, from (2.1) it implies ∥xk − x+ ∥En ≤ ∥y − x+ ∥En . (2.2) Consequently, {xk } is bounded. Let {xl } ⊂ {xk } be such that xl → x as l → ∞. We shall prove that x is a solution of (1.2). Indeed, from (2.1) and φj (x) ≥ 0 ∀x ∈ En , we obtain that 0 ≤ ∥xl − PC (xl )∥2 , ∥F (xl )∥2Em ≤ αl ∥y − x+ ∥2En . Tending towards l → ∞ in the last inequality, the continuous property of PC and F gives x ∈ C and F (x) = 0, i.e., x ∈ C ∩ S0 . Now, we prove that x ∈ S1 . For any element y ∈ C ∩ S0 , from (2.1) and φj (x) ≥ 0∀x ∈ En , j = 1, ..., N , we can write ∑ N µ −µ1 φ1 (xl ) + αl1−µ1 ∥xl − x+ ∥2En ≤ φ1 (y) + αl j φj (y) + αl1−µ1 ∥y − x+ ∥2En . j=2 After passing l → ∞ in the last inequality we obtain φ1 (x) ≤ φ1 (y) ∀y ∈ C ∩ S0 . Note that x is a local minimizer of φ1 on C ∩ S0 . But, because of C ∩ S0 ∩ S1 ̸= ∅, x also is a global minimizer of φ1 on En . It means that x ∈ S1 . Further, we prove that x ∈ S2 . For any y ∈ C ∩ S0 ∩ S1 , from (2.1) and φj (x) ≥ 0 ∀x ∈ En , j = 1, ..., N we have ∑ N µ −µ2 l φ2 (x ) + αl1−µ2 ∥xl − x+ ∥2En ≤ φ2 (y) + αl j φj (y) + αl1−µ2 ∥y − x+ ∥2En . j=3 By a similar argument, we obtain x ∈ S2 , and x ∈ Sj , j = 3, ..., N − 1. Consequently, N −1 x ∈ C ∩ (∩j=0 Sj ). Finally, we have to prove that x is a solution of (1.2). As above, for −1 N −1 any y ∈ C ∩ (∩N j=0 Sj ), from (2.1) it deduces that φN (x) ≤ φN (y) ∀y ∈ C ∩ (∩j=0 Sj ). Hence, φN (x) = miny∈C∩(∩N −1 Sj ) φN (y), i.e., x ∈ SN . The x+ - CMNS property of x is j=0 followed from (2.2). The theorem is now proven. Theorem 2.2. Assume that the following conditions hold: (i) F is differentiable, 47
Nguyen Thi Thuy Hoa and Nguyen Buong (ii) there exists L > 0 such that ∥F ′ (˜ x) − F ′ (z)∥Em ≤ L∥˜ x − z∥En for z in some neighbouhood of x˜ (iii) there exists ω ∈ Em such that x˜ − x+ = F ′ (˜ x)∗ ω (iv) L∥ω∥Em < 1. Then, we have √ ∥xk − x˜∥En = O( αk ). Proof. Using (1.5) and (1.6) with x = x˜ we obtain ∑ N µ ∥x − PC (x )∥ + ∥F (x k k 2 k )∥2Em + αk j φj (xk ) + αk ∥xk − x˜∥2En j=1 ≤ αk [∥˜ x − x+ ∥2En − ∥xk − x+ ∥2En + ∥xk − x˜∥2En ]. (2.3) Since x˜ ∈ ∩N j=0 Sj , then φj (x ) ≥ φj (˜ k x) = 0, j = 1, ..., N. From this fact and (2.3) we have ∥F (xk )∥2Em + αk ∥xk − x˜∥2En ≤ 2αk ⟨ω, F ′ (˜ x − xk )⟩Em . x)(˜ (2.4) Note that condition (ii) implies F (xk ) = F ′ (˜ x)(xk − x˜) + rk (2.5) with 1 ∥rk ∥Em ≤ L∥xk − x˜∥2En . (2.6) 2 Combining (2.4)-(2.6) leads to ∥F (xk )∥2Em + αk ∥xk − x˜∥2En ≤ 2αk ∥ω∥Em ∥F (xk )∥Em + αk ∥ω∥Em L∥xk − x˜∥2En . Thus, ∥F (xk )∥2Em + αk (1 − ∥ω∥Em L)∥xk − x˜∥2Rn ≤ 2αk ∥ω∥Em ∥F (xk )∥Em . (2.7) Therefore, ∥F (xk )∥Em = O(αk ). Then, from (2.7) it follows √ ∥xk − x˜∥En = O( αk ). The theorem is proven. 48
Above regularization for constrained generalized complementarity problems Remarks: (i) In the case that the given functions fi and φj are smooth, then the functional Fα , will have the same property, if instead of φj in (1.5)-(1.6) we use φ2j (see, [14]). (ii) In practice, the set C is chosen such that C ∩ (∩N j=0 Sj ) contains only one solution of (1.2). Then, all of the sequence {xk } converges to the solution, as in k → ∞. (iii) The regularized solution xk of (1.5) can be approximated by the conjugate gradient method in [14], the method of trust region in [15] or [13]. 3. Nummerical results In this section, in order to show the performence of the above method, we present some numerical results for the case: g(x1 , x2 ) = (x21 + x22 − 25, −x1 + 3), h(x1 , x2 ) = ((x1 − 3)2 + x22 − 4, x1 − 4), (3.1) and the set C = C1 ∩ C2 , defined as follows: C1 = {(x1 , x2 ) ∈ E2 : x1 − 4x2 + 1 ≤ 0}, C2 = {(x1 , x2 ) ∈ E2 : 2x1 − x2 − 2 ≥ 0}. (3.2) √ Clearly, problem √ (1.1) has four solutions x1 = √ (3, −2), x 2 = (3, 2), x3 = (4, 3), x = (4, − 3) and x = (3, 2), x = (4, 3) are solutions of (1.1) and (1.2), but 4 2 3 1 x2 is a minimal norm solution. Here, we choose: µ1 = 16 , µ2 = 81 , µ3 = 14 and µ4 = 12 with x+ = (0, 0). The parameter αk = (1 + k)−1 and the starting point for the case k = 1 is x0 = (20; 45) ∈ / C. We minimize the following Tikhonov functional: ∑ 4 µ Fαk (x) = ∥x − PC (x)∥2E2 + ∥F (x)∥2E2 + ej (x) + αk ∥x − x+ ∥2E2 , αk j φ j=1 where PC (x) = 21 [PC1 (x) + PC2 (x)] (see, [1]), PCi (x) = x if x ∈ Ci and if x ∈ / Ci then PCi (x) = Pli (x) with the line li = {(x1 , x2 ) : ai x1 + bi x2 = ci and ai , bi , ci are the coefficients, defined in (3.2). { x21 + x22 − 25 if x21 + x22 − 25, e1 (x) = φ1 (x) = φ (3.1) 0 if x21 + x22 − 25 ≤ 0 { (x1 − 3)2 for x1 > 3 and any x2 , e2 (x) = φ22 (x) = φ (3.2) 0 for x1 ≤ 3 and any x2 , 49
Nguyen Thi Thuy Hoa and Nguyen Buong { (x1 − 3)2 + x22 − 4 if (x1 − 3)2 + x22 − 4 > 0, e3 (x) = φ3 (x) = φ (3.3) 0 if (x1 − 3)2 + x22 − 4 ≤ 0 { (x1 − 4)2 for x1 > 4 and any x2 , e4 (x) = φ24 (x) = φ (3.4) 0 for x1 ≤ 4 and any x2 We use the standard program, presented in [11] as a procedure, and that the final result for each step is the starting point for the next. The following table of numerical results indicates that the method works quite well in practice. k x1 x2 1 1.0199206 0.2686316 10 1.7190803 1.5334557 50 2.7953346 1.9894142 100 2.9615843 1.9996203 200 2.2984413 1.9999334 400 2.9908316 1.9999745 800 2.9940854 1.9999906 1000 2.9966234 1.9999960 Acknowledgements. This work was supported by the National Foundation for Science and Technology Development, Vietnam. REFERENCES [1] R. E. Brack Jr, 1973. Properties of fixed poit sets of nonexpansive mappings in Banach spaces. Trans. Amer. Math. Soc. 179, pp. 251-262. [2] Ng. Buong and Ng. Th. Th. Hoa, 2009. Tikhonov regularization for nonlinear complemetarity problems. Intern. J. of Math. Analysis, 3(34), pp. 1683-1691. [3] Sh.-q Du, 2012. A nonsmooth Levenberg-Marquardt method for generalized complementarity problems. Journal of Information and Computing Science, 7(4), pp. 267-271. [4] F. Facchinei and J. Soares, 1997. A new merit function for nonlinear complementarity problems and related algorithm. SIAM Journal on Optimization, 7, pp. 225-247. [5] F. Facchinei and C. Kanzow, 1997. Beond monotonicity in regularization methods for nonlinear complementarity problems. Hamburger Beitrage zur Angewandten Mathematik, Rein A, preprint 125. 50
Above regularization for constrained generalized complementarity problems [6] M. C. Ferris and J.-S. Pang, 1997. Engineering and economic applications of complementarity problems. SIAM Review, 39, pp. 669-713. [7] A. Fischer, 1997. Solution of monotone complementarity problems with locally Lipschitzian function. Mathematical programming, pp. 513-532. [8] M. Fukushima, 1996. Merit functions for variational inequality and complementarity problems, in Nonlinear Optimization and Applications. G. Di Pillo and F. Giannessi eds., Plenum Publishing Corporation, New York, pp. 155-170. [9] M. S. Gowda and R. Sznajder, 1998. On the limiting behavior of the trajectory of regularized solutions of a p0 complementarity problem, in M. Fukushima and L. Qi, eds. Reformulation - Nonsmooth, Pieewise Smmoth, Semismooth and Smoothing Methods, pp. 371-379, Kluwer Academic Publishers. [10] P. T. Harker, 1995. Complementarity problem, in Handbook of Global Optimization. R. Horst and P. Pardalos, eds, Kluwer Academic Publishers, Boston, pp. 271-338. [11] J. H. Mathews, 1987. Numerical Methods for Computer Science. Enginneering and Mathematics, Prentice-Hall, INC. Englewood Clift, New Jesey. [12] D. Sun, 1997. A regularization Newton method for solving nonlinear complementarity problems. Applied Mathematics Report AMR 97/15, School of Mathematics, the university of New South Wales, Sydney. [13] A. Vardi, 1985. A trust region algorithm for equality constrained minimization: convergence properties and implementation. SIAM J. Numer. Anal. 4, pp. 575-591. [14] F. P. Vasilev, 1980. Numerical methods for solving extreme problems. M: Nauka, (in Russian). [15] Y. Yuan, 2003. A class of globally convergent conjugate gradient methods. Science in China, 46, pp. 253-261. 51