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.
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 Informatics Center, Hanoi University of Home Affairs 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 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 g(˜ x) ≤ 0, h(˜ x) ≤ 0, ⟨g(˜ x), h(˜ x)⟩Em = 0, (1.1) where the symbol y = (y1 , , ym ) ≤ 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) = ⇐⇒ ε = 0, x ∈ S , where S 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) = ∀y ∈ ∩N j=1 Sj , j = 1, , 2m := N Moreover, n ∩N j=1 Sj = {x ∈ E : g(x) ≤ 0, h(x) ≤ 0} 45 Nguyen Thi Thuy Hoa and Nguyen Buong Evidently, ⟨g(˜ x), h(˜ x)⟩Em = m ∑ gi (˜ x)hi (˜ x) = ⇐⇒ gi (˜ 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) = x∈C∩(∩N i=0 Si ) φj (x), j = 1, , N, (1.4) 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), x∈E (1.5) where Fα (x) is defined by Fα (x) = ∥x − PC (x)∥ + ∥F (x)∥2Em + N ∑ 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 > 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 ∥xk − PC (xk )∥2 + ∥F (xk )∥2Em + N ∑ µ αk j φj (xk ) + αk ∥xk − x+ ∥2En j=1 ≤ ∥y − PC (y)∥ + ∥F (y)∥2Em + N ∑ µ αk j φj (y) + αk ∥y − x+ ∥2En , (2.1) j=1 for each fixed element y ∈ E Taking y ∈ C ∩ (∩N j=0 Sj ), we have that y − PC (y) = 0, F (y) = and φj (xk ) ≥ φj (y) = 0, j = 1, , N Then, from (2.1) it implies n ∥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) ≥ ∀x ∈ En , we obtain that ≤ ∥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 φ1 (xl ) + αl1−µ1 ∥xl − x+ ∥2En ≤ φ1 (y) + N ∑ µ −µ1 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) ≥ ∀x ∈ En , j = 1, , N we have l φ2 (x ) + αl1−µ2 ∥xl − x+ ∥2En ≤ φ2 (y) + N ∑ µ −µ2 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 − Consequently, N −1 Sj ) Finally, we have to prove that x is a solution of (1.2) As above, for x ∈ C ∩ (∩j=0 N −1 −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 > 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 < Then, we have √ ∥xk − x˜∥En = O( αk ) Proof Using (1.5) and (1.6) with x = x˜ we obtain ∥x − PC (x )∥ + ∥F (x k k k )∥2Em + N ∑ µ αk j φj (xk ) + αk ∥xk − x˜∥2En j=1 ≤ αk [∥˜ x − x+ ∥2En − ∥xk − x+ ∥2En + ∥xk − x˜∥2En ] (2.3) k Since x˜ ∈ ∩N x) = 0, j = 1, , N From this fact and (2.3) we j=0 Sj , then φj (x ) ≥ φj (˜ have ∥F (xk )∥2Em + αk ∥xk − x˜∥2En ≤ 2αk ⟨ω, F ′ (˜ x)(˜ x − xk )⟩Em (2.4) Note that condition (ii) implies F (xk ) = F ′ (˜ x)(xk − x˜) + rk (2.5) ∥rk ∥Em ≤ L∥xk − x˜∥2En (2.6) with 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 Therefore, ∥F (xk )∥Em = O(αk ) Then, from (2.7) it follows √ ∥xk − x˜∥En = O( αk ) The theorem is proven 48 (2.7) 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] 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 + ≤ 0}, C2 = {(x1 , x2 ) ∈ E2 : 2x1 − x2 − ≥ 0} (3.2) √ Clearly, problem (1.1) has four solutions x = (3, −2), x = (3, 2), x = (4, 3), √ √ x = (4, − 3) and x = (3, 2), x = (4, 3) are solutions of (1.1) and (1.2), but 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 = is x0 = (20; 45) ∈ / C We minimize the following Tikhonov functional: Fαk (x) = ∥x − PC (x)∥2E2 + ∥F (x)∥2E2 + ∑ µ αk j φj (x) + αk ∥x − x+ ∥2E2 , 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 ) : x1 + bi x2 = ci and , bi , ci are the coefficients, defined in (3.2) { x21 + x22 − 25 if x21 + x22 − 25, (3.1) φ1 (x) = φ1 (x) = if x21 + x22 − 25 ≤ φ2 (x) = φ22 (x) = { (x1 − 3)2 for x1 > and any x2 , for x1 ≤ and any x2 , (3.2) 49 Nguyen Thi Thuy Hoa and Nguyen Buong φ3 (x) = φ3 (x) = { (x1 − 3)2 + x22 − if (x1 − 3)2 + x22 − ≤ 0 φ4 (x) = φ24 (x) = if (x1 − 3)2 + x22 − > 0, { (x1 − 4)2 for x1 > and any x2 , for x1 ≤ and any x2 (3.3) (3.4) 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 10 50 100 200 400 800 1000 x1 1.0199206 1.7190803 2.7953346 2.9615843 2.2984413 2.9908316 2.9940854 2.9966234 x2 0.2686316 1.5334557 1.9894142 1.9996203 1.9999334 1.9999745 1.9999906 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, 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convergent conjugate gradient methods Science in China, 46, pp 253-261 51 ... monotonicity in regularization methods for nonlinear complementarity problems Hamburger Beitrage zur Angewandten Mathematik, Rein A, preprint 125 50 Above regularization for constrained generalized complementarity. .. )∥Em Therefore, ∥F (xk )∥Em = O(αk ) Then, from (2.7) it follows √ ∥xk − x˜∥En = O( αk ) The theorem is proven 48 (2.7) Above regularization for constrained generalized complementarity problems. . .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