On the strong convergence of a general Iterative algorithm
aa r X i v : . [ m a t h . D S ] F e b ON THE STRONG CONVERGENCE OF A GENERAL ITERATIVEALGORITHM
RAMZI MAY AND ZAHRA BIN ALI
Abstract.
Let Q be a nonempty closed and convex subset of a real Hilbert space H , T : Q → Q a nonexpansive mapping which has a least one fixed point f : Q → H aLipschitzian function, and F : Q → H a Lipschitzian and strongly monotone mapping.We prove, under appropriate conditions on the functions f and F , the control realsequences { α n } and { β n } , and the error term { e n } , that for any starting point x in Q, the sequence { x n } generated by the iterative process x n +1 = β n x n + (1 − β n ) P Q ( α n f ( x n ) + ( I − α n F ) T x n + e n )converges strongly to the unique solution of the variational inequality problemFind q ∈ C such that h F ( q ) − f ( q ) , x − q i ≥ x ∈ C where C = F ix ( T ) is the set of fixed points of T. Our main result unifies and extendsmany well-known previous results. Introduction
Let H be a real Hilbert space with inner product h ., . i and associated norm k . k , Q a nonempty closed and convex subset of H , T : Q → Q a nonexpansive mapping (i.e., k T x − T y k ≤ k x − y k for all x, y ∈ Q ) such that C := F ix ( T ) = { x ∈ Q : T x = x } isnonempty, f : Q → H a Lipschitzian mapping with coefficient α ≥
0, and F : Q → H a Lipschitizian mapping with coefficient κ > . We assume that F is strongly monotoneoperator with coefficient η > , which means that h F ( x ) − F ( y ) , x − y i ≥ η k x − y k for all x, y ∈ Q. We assume in addition that α < η.
Then it is easily seen that the operator g := F − f is strongly monotone with coefficient η − α ; hence the variational inequality problemFind q ∈ C such that h F ( q ) − f ( q ) , x − q i ≥ x ∈ C, (VIP)has a unique solution which we denote by q ∗ . In the present work, we are concerned with the construction of a general iterativealgorithm that generates sequences converging strongly to q ∗ . Let us first recall someprevious results related to this purpose. In the particular case Q = H , f ≡ u a Date : February 11, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Hilbert space; Variational inequality problem; Fixed points of nonexpansivemappings; Projection operator; Iterative algorithm. constant, and F = I the identity mapping from H into itself, Halpern [6] introducedthe iterative process (cid:26) x ∈ H ( a starting point) x n +1 = α n u + (1 − α n ) T x n , (1.1)with { α n } ∈ [0 , . He established that if α n = n θ for all n ≥ θ ∈ ]0 ,
1[ then thegenerated sequence { x n } converges strongly to q ∗ which is in this case equal to P C ( u )where P C : H → C is the metric projection from H onto the closed and convex subset C = F ix ( T ) . He also proved that the conditions(C1) lim n → + ∞ α n = 0 , (C2) P + ∞ n =0 α n = + ∞ , are necessary for the strong convergence of the algorithm (1.1).In 1977, Lions [7] extended the result of Halpern. In fact, he proved the strongconvergence of sequences { x n } generated by process (1.1) to q ∗ provided the sequence { α n } satisfies the necessary conditions (C1)-(C2) and the supplementary condition(C3) lim n → + ∞ α n +1 − α n α n = 0 . In 2000, Moudafi [10] considered the case when Q = H , f : H → H is a contractionwith coefficient α ∈ [0 , , F = I the identity mapping from H into itself. He introducedthe algorithm (cid:26) x ∈ H (a starting point) x n +1 = α n f ( x n ) + (1 − α n ) T x n , (1.2)where { α n } ∈ ]0 , . He established, under the conditions (C1), (C2) and(C4) lim n → + ∞ α n +1 − α n α n +1 α n = 0 , the strong convergence of any sequence generated by this algorithm to q ∗ which, inthis case, is equal to the unique fixed point of the contraction mapping P C ◦ f. In 2004, Xu [17] improved Moudafi result; in fact, he followed a new approach toprove the strong convergence property of the algorithm (1.2) provided the sequence { α n } satisfies the conditions (C1), (C2) and(C5) lim n → + ∞ α n +1 − α n α n = 0 or P + ∞ n =0 | α n +1 − α n | < + ∞ . Xu [16] has also considered the case when Q = H , f = u a constant and F = A a η − strongly positive self adjoint bounded linear operator from H to H . He proved thestrong convergence of the algorithm (cid:26) x ∈ H (a starting point) x n +1 = α n u + ( I − α n A ) T x n to the unique solution q ∗ of (VIP) provided the real sequence { α n } satisfies the conditions(C1), (C2) and (C5). Let us notice here that, in this case, q ∗ is the unique minimizerof the strongly quadratic convex function h Ax, x i − h u, x i over the closed and convexsubset C = F ix ( T ) . Later in 2006, Mariano and Xu [8] established that the previous strong convergenceresult remains true in the more general case when f : H → H is a Lipschitzian mappingwith constant α strictly less than η . HE STRONG CONVERGENCE OF A GENERAL ITERATIVE ALGORITHM 3
On the other hand, Yamada [18] studied the particular case when Q = H , f ≡ . Heproved that if the sequence { α n } satisfies the conditions (C1), (C2) and (C3) then forevery starting point x ∈ H the sequence { x n } generated by the iterative process x n +1 = ( I − α n F ) T x n converges strongly to q ∗ . In 2010, Tiang [14], by combining the iterative method of Yamada and the method ofMariano and Xu, has introduced the general algorithm (cid:26) x ∈ H (a starting point) x n +1 = α n f ( x n ) + ( I − α n F ) T x n . He established the strong convergence of this algorithm to q ∗ provided that the realsequence { α n } satisfies the conditions (C1), (C2) and (C5).Later, in 2011, Ceng, Ansari and Yao [5] extended Tiang’s result to the case where Q is not necessary equal to the whole space H . Precisely, they proved that if the sequence { α n } satisfies the conditions (C1), (C2) and (C5), then for any starting point x in Q the sequence { x n } defined by the scheme x n +1 = P Q ( α n f ( x n ) + ( I − α n F ) T x n )converges strongly to q ∗ . In this paper, inspired by the previous works and the papers [19] and [3], we introducethe following general hybrid and perturbed algorithm: (cid:26) x ∈ Q (a starting point) x n +1 = β n x n + (1 − β n ) P Q ( α n f ( x n ) + ( I − α n F ) T x n + e n ) , (HPA)where { α n } and { β n } are two real sequences in [0 ,
1] and { e n } is a sequence in H repre-senting the perturbation term. Roughly speaking, we will prove that any sequence { x n } generated by the algorithm (HPA) converges strongly to q ∗ provided that the sequence { α n } satisfies only the necessary conditions (C1) and (C2), the sequence { β n } is not tooclose to 0 or 1 , and the perturbation term { e n } is relatively small with respect to { α n } .The paper is organized as follows. In the next section, we recall some tools lemmasthat will be useful frequently in the proof of the results of the paper. The section 3is devoted to the study of the convergence of an imcite algorithm associated to thealgorithm (HPA). The strong convergence of (HPA) will investigate in Section4. TheLast section will be devoted to the study of the limit case where the strong monotonicitycoefficient of F is equal to the Lipschitzian coefficient of f .2. Preliminaries
In this section, we recall some classical results that will be useful in the proof of themain theorems of the paper.The first result is a simple but powerful lemma proved by Xu in [15]. This lemma isa generalization of a result due to Bertsekas (see [Lemma 1.5.1, [2]]).
RAMZI MAY AND ZAHRA BIN ALI
Lemma 2.1. let { a n } be a sequence of nonnegative real numbers such that: a n +1 ≤ (1 − γ n ) a n + γ n r n + δ n , n ≥ , where { γ n } ∈ [0 , and { r n } and { δ n } are two real sequences such that (1) P + ∞ n =0 γ n = + ∞ ;(2) P + ∞ n =0 | δ n | < + ∞ ;(3) lim sup n → + ∞ r n ≤ . Then the sequence { a n } converges to . The second result is the following lemma due to Suzuki [13]
Lemma 2.2.
Let { z n } and { w n } be two bounded sequence in a Banach space E and let { β n } be a sequence in [0 , with < lim inf n → + ∞ β n ≤ lim sup n → + ∞ β n < . Suppose that z n +1 = β n z n + (1 − β n ) w n , n ≥ and lim sup n → + ∞ ( k w n +1 − w n k − k z n +1 − z n k ) ≤ . Then lim n → + ∞ k z n − w n k = 0 . The last result of this section is a particular case of the well-known demiclosednessprinciple (see [[1], Corollary 4.18]).
Lemma 2.3.
Let H be an Hilbert space, Q a closed convex and nonempty subset of H ,and T : Q → Q a nonexpansive mapping. If { x n } is a sequence in Q weakly convergingto some x such that { x n − T x n } converges strongly to , then x ∈ F ix ( T ) . The convergence of an implicit algorithm
In this section, we prove the strong convergence of the perturbed and implicit algorithm x t = P Q ( tf ( x t ) + ( I − tF ) T x t + e ( t ))as t → + to the unique solution q ∗ of the variational inequality problem (VIP) providedthat the perturbation term e ( t ) is sufficiently small. More precisely, we will prove thefollowing theorem. Theorem 3.1. let δ ∗ := 2 η − ακ and e :]0 , δ ∗ [ → H such that lim t → + k e ( t ) k t = 0 . Then for every t ∈ ]0 , δ ∗ [ there exists a unique x t ∈ Q such that x t = P Q ( tf ( x t ) + ( I − tF ) T x t + e ( t )) . HE STRONG CONVERGENCE OF A GENERAL ITERATIVE ALGORITHM 5
Moreover, x t converges strongly in H as t → + toward q ∗ the unique solution of thevariational inequality problem (VIP). The proof relies essentially on the following lemma which will be also used in the nextsection devoted to the study of the strong convergence of the algorithm (HPA).
Lemma 3.2.
Let δ ∈ ]0 , δ ∗ [ . For every t ∈ ]0 , δ ] , the mapping S t : Q → H defined by S t ( x ) = tf ( x ) + ( I − tF ) T x is Lipschitzian with coefficient − tσ where σ := η − α − κ δ . Proof.
Let t ∈ ]0 , δ ] and x, y ∈ Q. We have k ( I − tF ) T x − ( I − tF ) T y k = k T x − T y k − t h F ( T x ) − F ( T y ) , T x − T y i + t k F ( T x ) − F ( T y ) k ≤ (cid:0) − tη + t κ (cid:1) k T x − T y k ≤ (cid:18) − t ( η − tκ (cid:19) k x − y k . Hence by using the elementary inequality √ − x ≤ − x , for all x ∈ [0 , , we deduce that k ( I − tF ) T x − ( I − tF ) T y k ≤ (cid:18) − t ( η − tκ (cid:19) k x − y k . Therefore, k S t ( x ) − S t ( y ) k ≤ t k f ( x ) − f ( y ) k + k ( I − tF ) T x − ( I − tF ) T y k≤ (cid:18) tα + 1 − t ( η − tκ (cid:19) k x − y k = (cid:18) − t ( η − α − tκ (cid:19) k x − y k≤ (cid:18) − t ( η − α − κ δ (cid:19) k x − y k = (1 − σ t ) k x − y k . This completes the proof. (cid:3)
Now we are in position to prove the main result of this section.
Proof.
Let δ ∈ ]0 , δ ∗ [ be a fixed real. Let t ∈ ]0 , δ ] . Since the operator P Q is nonexpansive,it follows from the previous lemma that the two mapping ϕ t and φ t defined from Q to RAMZI MAY AND ZAHRA BIN ALI Q by ϕ t ( x ) = P Q ( S t ( x )) ,φ t ( x ) = P Q ( S t ( x ) + e ( t ))are contractions with the same coefficient 1 − tσ ∈ [0 , . Hence the classical Banach fixedpoint theorem ensures the existence of a unique x t and y t in Q such that x t = P Q ( S t ( x t ))and y t = P Q ( S t ( y t ) + e ( t )) . Using again Lemma 3.2 and the fact that P Q is nonexpansive,we obtain k x t − y t k ≤ (1 − tσ ) k x t − y t k + k e ( t ) k , which implies that k x t − y t k ≤ k e ( t ) k tσ . Hence, from the assumption on e ( t ) , we get k x t − y t k → t → + . Therefore in order to prove that x t → q ∗ as t → + it suffices to prove that y t → q ∗ as t → + . To do this let us first show that the family ( y t )
Hence, y t − T y t → H as t → + . (3.1) HE STRONG CONVERGENCE OF A GENERAL ITERATIVE ALGORITHM 7
On the other hand, since ( y t )
In this section we study the strong convergence of the averaged and perturbed algo-rithm x n +1 = β n x n + (1 − β n ) P Q ( α n f ( x n ) + ( I − α n F ) T x n + e n ) (HPA)which can be seen as a discreet version of the implicit algorithm studied in previoussection. Theorem 4.1.
Let { e n } be a sequence in H and { α n } ∈ ]0 , and { β n } ∈ [0 , two realsequences such that: (i) α n → and P + ∞ n =0 α n = + ∞ (ii) One of the two following two conditions is satisfied: (h1) 0 < lim inf n → + ∞ β n ≤ lim sup n → + ∞ β n < . (h2) lim sup n → + ∞ β n < , either β n +1 − β n α n → or P + ∞ n =0 | β n +1 − β n | < ∞ and either α n +1 − α n α n → or P + ∞ n =0 | α n +1 − α n | < ∞ . RAMZI MAY AND ZAHRA BIN ALI (iii) P + ∞ n =0 k e n k < ∞ or k e n k α n → . Then for every initial guess x ∈ Q , the sequence { x n } generated by the algorithm (HPA)converges strongly in H to q ∗ the unique solution of the variational inequality problem(VIP).Proof. Since we are interested only in studying the asymptotic behavior of the sequence { x n } and α n → as n → ∞ , we can assume without loss of generality that for all n ∈ N ,α n ∈ ]0 , δ ] where δ ∈ ]0 , δ ∗ [ is a fixed real. Let { y n } be the sequence defined as follows (cid:26) y = x y n +1 = β n y n + (1 − β n ) P Q ( α n f ( y n ) + ( I − α n F ) T y n ) , n ≥ . Using the fact P Q is nonexpansive and Lemma 3.2, we obtain k y n +1 − x n +1 k ≤ β n k y n − x n k + (1 − β n ) k P Q ( S α n ( y n )) − P Q ( S α n ( x n ) + e n ) k≤ [ β n + (1 − β n ) (1 − σ α n )] k y n − x n k + (1 − β n ) k e n k≤ (1 − γ n ) k y n − x n k + k e n k , where γ n = σ (1 − β n ) α n . Since lim sup n → + ∞ β n < , there exists a > n ∈ N suchthat aα n ≤ γ n ≤ n ≥ n . Hence, by applying Lemma 2.1, we deduce that y n − x n → . (4.1)Therefore it suffices to prove that the sequence { y n } converges strongly to q ∗ to concludethat { x n } also converges strongly to q ∗ . Let us first show that { y n } is bounded in H . Let q ∈ F ix ( T ) . For every n ∈ N , wehave k y n +1 − q k ≤ β n k y n − q k + (1 − β n )[ k P Q ( S α n ( y n )) − P Q ( S α n ( q )) k + k P Q ( S α n ( q )) − P Q ( q ) k ] ≤ β n k y n − q k + (1 − β n )[ k S α n ( y n ) − S α n ( q ) k + k S α n ( q ) − q k ] ≤ β n k y n − q k + (1 − β n ) [(1 − σ α n ) k y n − q k + α n k f ( q ) − F ( q ) k ] . The last inequality immediately implies that that the sequence v n := max {k y n − q k , k f ( q ) − F ( q ) k σ } is decreasing. Therefore the sequence { y n } is bounded in H and so are the sequences { f ( y n ) } and { F ( T y n ) } . Now we are going to prove that y n − T y n → . (4.2)Let us first assume that the condition (h1) is satisfied. For every n ∈ N , set z n = P Q ( α n f ( y n ) + ( I − α n F ) T y n ) . We have, the sequences { y n } and { z n } are bounded in H , y n +1 = β n y n + (1 − β n ) z n for every n, HE STRONG CONVERGENCE OF A GENERAL ITERATIVE ALGORITHM 9 and k z n +1 − z n k ≤ ( α n + α n +1 ) sup m ≥ k f ( y m ) − F ( T y m ) k + k T y n +1 − T y n k≤ ( α n + α n +1 ) sup m ≥ k f ( y m ) − F ( T y m ) k + k y n +1 − y n k which implies lim sup n → + ∞ k z n +1 − z n k − k y n +1 − y n k ≤ . Therefore, from Lemma 2.2, we deduce that z n − y n → , which combined with the fact that k z n − T y n k = k P Q ( α n f ( y n ) + ( I − α n F ) T y n ) − P Q ( T y n ) k≤ α n sup m ≥ k f ( y m ) − F ( T y m ) k → n → + ∞ , implies the required result (4.2).Let us now establish (4.2) under the assumption (h2). A simple computation usingthe fact that the sequence { y n } and { P Q ( α n f ( y n ) + ( I − α n F ) T y n ) } are bounded in H , ensures the existence of two real constants M , M > n ∈ N , k y n +1 − y n k ≤ β n k y n − y n − k + (1 − β n ) (cid:13)(cid:13) P Q ( S α n ( y n )) − P Q ( S α n − ( y n − )) (cid:13)(cid:13) + M | β n − β n − |≤ β n k y n − y n − k + (1 − β n ) (cid:13)(cid:13) S α n ( y n ) − S α n − ( y n − ) (cid:13)(cid:13) + M | β n − β n − |≤ β n k y n − y n − k + (1 − β n ) k S α n ( y n ) − S α n ( y n − ) k +(1 − β n ) (cid:13)(cid:13) S α n ( y n − ) − S α n − ( y n − ) (cid:13)(cid:13) + M | β n − β n − |≤ β n k y n − y n − k + (1 − β n )(1 − σ α n ) k y n − y n − k + M | β n − β n − | + M | α n − α n − | = (1 − σ (1 − β n ) α n ) k y n − y n − k + M | β n − β n − | + M | α n − α n − | . Hence, by proceeding as in the proof of (4.1), we infer that k y n +1 − y n k → . (4.3)On the other, for every n ∈ N , we have k y n +1 − T y n k ≤ β n k y n − T y n k + (1 − β n ) k P Q ( S α n ( y n )) − P Q ( T y n ) k≤ β n k y n +1 − T y n k + β n k y n +1 − y n k + k S α n ( y n ) − T y n k≤ β n k y n +1 − T y n k + k y n +1 − y n k + α n k f ( y n ) − F ( T y n ) k . Hence, we obtain the inequality k y n +1 − T y n k ≤ − β n (cid:18) k y n +1 − y n k + α n sup m ≥ k f ( y m ) − F ( T y m ) k (cid:19) , which, combined with (4.3) and the fact that lim sup n → + ∞ β n <
1, implies that k y n +1 − T y n k → . (4.4) The required estimate (4.2), follows from (4.3) and (4.4).Now we are going to apply the fundamental Lemma 2.1, to conclude that { y n } con-verges strongly to q ∗ . But first let us notice that by proceeding as in the proof of Theorem3.1 and by using (4.2) and the fact that { y n } is bounded, we deduce thatlim sup n → + ∞ h y n − q ∗ , f ( q ∗ ) − F ( q ∗ ) i ≤ . Combined with the estimate (4.2), the last inequality yieldslim sup n → + ∞ h T y n − q ∗ , f ( q ∗ ) − F ( q ∗ ) i ≤ . (4.5)Finally, for every n ∈ N , k y n +1 − q ∗ k ≤ β n k y n − q ∗ k + (1 − β n ) k P Q ( S α n ( y n )) − P Q ( q ∗ ) k ≤ β n k y n − q ∗ k + (1 − β n ) k S α n ( y n )) − q ∗ k = β n k y n − q ∗ k + (1 − β n )[ k S α n ( y n )) − S α n ( q ∗ ) k + 2 h S α n ( y n )) − S α n ( q ∗ ) , S α n ( q ∗ ) − q ∗ i + k S α n ( q ∗ ) − q ∗ k ]= β n k y n − q ∗ k + (1 − β n )[ k S α n ( y n )) − S α n ( q ∗ ) k + 2 h S α n ( y n )) − q ∗ , S α n ( q ∗ ) − q ∗ i − k S α n ( q ∗ ) − q ∗ k ] ≤ β n k y n − q ∗ k + (1 − β n )[(1 − α n σ ) k y n − q ∗ k + 2 α n h f ( y n ) − F ( T y n ) , f ( q ∗ ) − F ( q ∗ ) i + 2 α n h T y n − q ∗ , f ( q ∗ ) − F ( q ∗ ) i ] ≤ (1 − γ n ) k y n − q ∗ k + 2 α n (1 − β n ) [ h T y n − q ∗ , f ( q ∗ ) − F ( q ∗ ) i + Cα n ]= (1 − γ n ) k y n − q ∗ k + γ n r n where C > n, γ n = 2 σ (1 − β n ) α n and r n = 1 σ ( h T y n − q ∗ , f ( q ∗ ) − F ( q ∗ ) i + Cα n ) . Using the estimate (4.5), we obtain lim sup r n ≤
0; hence, by applying Lemma 2.1 weconclude, as previously, that the sequence { y n } converges strongly to q ∗ . This completesthe proof of Theorem 4.1. (cid:3) The study of the limit case µ = α Throughout this section, we assume that µ = α. Hence the operator f − F is monotoneand not necessary strongly monotone; so the uniqueness of the solution of the variationalinequality problem (VIP) is no long assured. We assume moreover that (VIP) has atleast one solution. We denote by S V IP the set of the solutions of (VIP). The followingtheorem provides a method to approximate a particular element of the set S V IP . HE STRONG CONVERGENCE OF A GENERAL ITERATIVE ALGORITHM 11
Theorem 5.1.
Assume that the sequences { α n } , { β n } and { e n } satisfy the same as-sumptions as in Theorem. Then, for every ε > and x ∈ Q , the sequence { x εn } definedby the recursive formula x εn +1 = β n x εn + (1 − β n ) P Q ( α n f ( x εn ) + ((1 − α n ε ) I − α n F ) T x εn + e n ) , n ≥ , converges strongly to q ε the unique solution of the variational inequality problemFind q ∈ C such that h F ( q ) + εq − f ( q ) , x − q i ≥ for all x ∈ C. (VIP ε ) Moreover, the set S V IP is closed and convex and q ε converges strongly as ε → to thenearest element of S V IP to the origin.Proof.
For every ε > , the operator F ε := F + εI is µ + ε strongly monotone and κ + ε Lipschitizian. Since µ + ε > α, the first part of this theorem follows immediately fromTheorem 4.1.Let δ C : C → H be the indicator function associated to the closed, convex andnonempty subset C. We recall that δ C is given by δ C ( x ) = (cid:26) , x ∈ C + ∞ , x / ∈ C It is well-known that δ C is a proper, lower semi continuous, and convex function. Hence,its sub-gradient ∂δ C is a maximal and monotone operator with domain equal to C . Werecall that, for every x ∈ C , ∂δ C ( x ) = { u ∈ H : h u, y − x i ≤ } . It is then easily seen that the set S V IP is equal to A − (0) the set of zeros of the maximaland monotone operator A := F − f + δ C . Therefore, S V IP is a closed and convex subsetof H . On the other hand, the unique solution q ε of (VIP ε ) satisfies the relation − ( F ( q ε ) + εq ε − f ( q ε )) ∈ δ C ( q ε ) , which is equivalent to 0 ∈ q ε + 1 ε A ( q ε ) , since ε δ C ( q ε ) = δ C ( q ε ) . Therefore, q ε = J ε (0) , where for every λ > J λ = ( I + λA ) − is the resolvant of A (for more details, see the pioneer paper [9] of Minty). Hence, fromthe following lemma due to Bruck [4] and Morosanu [11] Lemma 5.2.
Let A : D ( A ) ⊂ H → H be a maximal monotone operator with A − (0) = ∅ . Then for any u ∈ H , ( I + tA ) − u → P A − (0) ( u ) as t → + ∞ . we deduce that q ε converges strongly as ε → P A − (0) = P S V IP (0) which is theelement of S V IP with minimal norm. (cid:3)
Remark 5.3.
From the previous theorem, we expect, but we don’t yet have the justifi-cation, that under some appropriate assumptions on the real sequences { α n } , { β n } and { ε n } , the sequences { x n } generated by the iterative process x n +1 = β n x n + (1 − β n ) P Q ( α n f ( x n ) + ((1 − α n ε n ) I − α n F ) T x n ) , n ≥ , where x is an arbitrary element of Q , converge strongly in H to u ∗ = P S V IP (0) . Letus notice that Reich and Xu in [12] had raised a similar open question related to theconstrained least squares problem
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