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Application of our new fixed point iterative procedure to constrained optimization problemsand split feasibility problems

Dalam dokumen A Study on Fixed Point Iterative Procedures (Halaman 86-91)

APPLICATION AND CONCLUSION

7.1 Application of our new fixed point iterative procedure to constrained optimization problemsand split feasibility problems

This section is allocated to some applications of our new fixed point iterative procedure (2.18). Let 𝐻𝐻be a real Hilbert spacewith inner product βŒ©βˆ™,βˆ™βŒͺand normβ€–βˆ™β€–, respectively. Let 𝐢𝐢be a nonempty closedconvex subset of 𝐻𝐻and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻a nonlinear operator. 𝑇𝑇is said to be:

(1) 𝐿𝐿-Lipschitzian if there exists a constant𝐿𝐿 > 0 such that

‖𝑇𝑇𝑇𝑇 βˆ’ 𝑇𝑇𝑇𝑇‖ ≀ 𝐿𝐿‖𝑇𝑇 βˆ’ 𝑇𝑇‖,βˆ€ 𝑇𝑇,𝑇𝑇 ∈ 𝐢𝐢.

An 𝐿𝐿-Lipschitzian will be contraction if 𝐿𝐿 ∈(0, 1), and non-expansive if 𝐿𝐿 = 1. (2) Monotone if βŒ©π‘‡π‘‡π‘‡π‘‡ βˆ’ 𝑇𝑇𝑇𝑇,𝑇𝑇 βˆ’ 𝑇𝑇βŒͺ β‰₯ 0,βˆ€ 𝑇𝑇,𝑇𝑇 ∈ 𝐢𝐢.

(3) πœ†πœ†-strongly monotone if there exists a constant πœ†πœ† > 0 such that

βŒ©π‘‡π‘‡π‘‡π‘‡ βˆ’ 𝑇𝑇𝑇𝑇,𝑇𝑇 βˆ’ 𝑇𝑇βŒͺ β‰₯ πœ†πœ†β€–π‘‡π‘‡ βˆ’ 𝑇𝑇‖2,βˆ€ 𝑇𝑇,𝑇𝑇 ∈ 𝐢𝐢.

(4) 𝜈𝜈-inverse strongly monotone (𝜈𝜈-ism) if there exists a constant 𝜈𝜈 > 0 such that

βŒ©π‘‡π‘‡π‘‡π‘‡ βˆ’ 𝑇𝑇𝑇𝑇,𝑇𝑇 βˆ’ 𝑇𝑇βŒͺ β‰₯ πœˆπœˆβ€–π‘‡π‘‡π‘‡π‘‡ βˆ’ 𝑇𝑇𝑇𝑇‖2,βˆ€ 𝑇𝑇,𝑇𝑇 ∈ 𝐢𝐢.

The variational inequality problem defined by 𝐢𝐢and 𝑇𝑇will be denoted by𝑉𝑉𝑉𝑉(𝐢𝐢,𝑇𝑇). These were initially studied by Kinderlehrer and Stampachhia [10].

Thevariational inequality problem 𝑉𝑉𝑉𝑉(𝐢𝐢,𝑇𝑇) is the problem of finding a vector 𝒛𝒛in 𝐢𝐢such that βŒ©π‘‡π‘‡π’›π’›,𝒛𝒛 βˆ’ π‘˜π‘˜βŒͺ β‰₯ 0 for all π‘˜π‘˜ ∈ 𝐢𝐢. The set of all such vectors which solvevariational inequality 𝑉𝑉𝑉𝑉(𝐢𝐢,𝑇𝑇)problem is denoted byΞ©(𝐢𝐢,𝑇𝑇). The variationalinequality problem is connected with various kinds of problems such as the convexminimization problem, the complementarity problem, the problem of

finding a point𝑒𝑒 ∈ 𝐻𝐻satisfying 0 = 𝑇𝑇𝑒𝑒 and so on. The existence and approximation of solutionsare important aspects in the study of variational inequalities. The variationalinequality problem 𝑉𝑉𝑉𝑉(𝐢𝐢,𝑇𝑇) is equivalent to the fixed point problem, that isto find π‘‡π‘‡βˆ— ∈ 𝐢𝐢such that

π‘‡π‘‡βˆ— = πΉπΉπœ‡πœ‡π‘‡π‘‡βˆ— =𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)π‘‡π‘‡βˆ—,

where πœ‡πœ‡ > 0 is a constant and 𝑃𝑃𝐢𝐢is the metric projection from 𝐻𝐻 onto 𝐢𝐢andπΉπΉπœ‡πœ‡ ≔ 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡). If 𝑇𝑇is 𝐿𝐿-Lipschitzian and πœ†πœ†-strongly monotone, then theoperator πΉπΉπœ‡πœ‡is a contraction on 𝐢𝐢provided that 0 < πœ‡πœ‡ < 2πœ†πœ† 𝐿𝐿⁄ 2. In this case, anapplication of Banach contraction principle (Theorem 1.6.2) implies thatΞ©(𝐢𝐢,𝑇𝑇) = {π‘‡π‘‡βˆ—}and thesequence of the Picard iteration process, given by

𝑇𝑇𝑛𝑛+1 =πΉπΉπœ‡πœ‡π‘‡π‘‡π‘›π‘›, 𝑛𝑛 ∈ β„• converges strongly toπ‘‡π‘‡βˆ—.

Construction of fixed points of non-expansive operators is an important subject in the theory of non-expansive operators and has applications in a number of applied areas such as image recovery and signal processing(see[7, 8,13]). For instance,split feasibility problem of 𝐢𝐢and 𝑇𝑇(denoted by𝑆𝑆𝐹𝐹𝑃𝑃(𝐢𝐢,𝑇𝑇)) is

to find a point 𝑇𝑇in 𝐢𝐢such that 𝑇𝑇𝑇𝑇 ∈ 𝑄𝑄 (7.1)

where𝐢𝐢is a closed convex subset of a Hilbert space 𝐻𝐻1,𝑄𝑄is a closed convexsubset of another Hilbert space 𝐻𝐻2 and 𝑇𝑇:𝐻𝐻1 β†’ 𝐻𝐻2is a bounded linear operator.The 𝑆𝑆𝐹𝐹𝑃𝑃(𝐢𝐢,𝑇𝑇)is said to be consistent if (7.1) has a solution. It is easy to seethat 𝑆𝑆𝐹𝐹𝑃𝑃(𝐢𝐢,𝑇𝑇) is consistent if and only if the following fixed point problem has asolution:

find𝑇𝑇 ∈ 𝐢𝐢such that𝑇𝑇 =𝑃𝑃𝐢𝐢�𝑉𝑉 βˆ’ π›Ύπ›Ύπ‘‡π‘‡βˆ—οΏ½π‘‰π‘‰ βˆ’ 𝑃𝑃𝑄𝑄�𝑇𝑇�𝑇𝑇, (7.2) where𝑃𝑃𝐢𝐢and 𝑃𝑃𝑄𝑄are the orthogonal projections onto 𝐢𝐢and 𝑄𝑄, respectively; 𝛾𝛾 > 0, andπ‘‡π‘‡βˆ—is the adjoint of 𝑇𝑇. Note that for sufficient small 𝛾𝛾 > 0, the operator𝑃𝑃𝐢𝐢�𝑉𝑉 βˆ’ π›Ύπ›Ύπ‘‡π‘‡βˆ—οΏ½π‘‰π‘‰ βˆ’ 𝑃𝑃𝑄𝑄�𝑇𝑇� in (7.2) is non-expansive.

In the view of Lemma 4.1.3, we have the following sharper results which containour iterative procedure (2.18) faster than the one of the iterative procedures

defined by (2.17), (2.16), (2.15), (2.14), (2.13), (2.11) and (2.1). These results deal withvariational inequality problems.

Theorem 7.1.1.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Suppose{𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛},{𝛾𝛾𝑛𝑛} ∈ [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈ (0,1). Thenfor πœ‡πœ‡ ∈ (0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑧𝑧𝑛𝑛,

𝑧𝑧𝑛𝑛 = (1βˆ’ 𝛾𝛾𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛾𝛾𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•, οΏ½ converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

Proof.Since 𝑇𝑇 ≔ 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡) is non-expansive, hence the result follows fromTheorem 4.2.1.∎

The Theorem 7.1.1 leads the following corollaries for iterative procedures defined by (2.16), (2.17), (2.14), (2.15), (2.13), (2.11) and (2.1) respectively.

Corollary 7.1.2.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Suppose{𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛},{𝛾𝛾𝑛𝑛} ∈ [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈ (0,1). Thenfor πœ‡πœ‡ ∈ (0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑧𝑧𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑧𝑧𝑛𝑛,

𝑧𝑧𝑛𝑛 = (1βˆ’ 𝛾𝛾𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛾𝛾𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•, οΏ½ converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

Corollary 7.1.3.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Suppose{𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛},{𝛾𝛾𝑛𝑛} ∈ [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈ (0,1). Thenfor πœ‡πœ‡ ∈ (0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑧𝑧𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑧𝑧𝑛𝑛,

𝑧𝑧𝑛𝑛 = (1βˆ’ 𝛾𝛾𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛾𝛾𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•, οΏ½ converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

Corollary 7.1.4.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Suppose{𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛},{𝛾𝛾𝑛𝑛} ∈ [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈ (0,1). Thenfor πœ‡πœ‡ ∈ (0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑧𝑧𝑛𝑛,

𝑧𝑧𝑛𝑛 = (1βˆ’ 𝛾𝛾𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛾𝛾𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•,οΏ½ converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

Corollary 7.1.5.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Suppose{𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛} ∈ [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈(0,1). Thenfor πœ‡πœ‡ ∈ (0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; 𝑛𝑛 ∈ β„•, οΏ½ converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

Corollary 7.1.6.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Suppose{𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛} ∈ [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈(0,1). Thenfor πœ‡πœ‡ ∈ (0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•,οΏ½ converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

Corollary 7.1.7.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Suppose{𝛼𝛼𝑛𝑛} ∈ [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈ (0,1). Thenfor πœ‡πœ‡ ∈(0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•, converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

Corollary 7.1.8.Let 𝐢𝐢 be a nonempty closed convex subset of a Hilbert space𝐻𝐻 and 𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻 a 𝐿𝐿-Lipschitzian andπœ†πœ†-strongly monotone operator. Thenfor πœ‡πœ‡ ∈ (0, 2πœ†πœ† 𝐿𝐿⁄ 2), the iterative sequence{𝑇𝑇𝑛𝑛} generated from𝑇𝑇1 ∈ 𝐢𝐢 and definedby

𝑇𝑇𝑛𝑛+1 = 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•, converges weakly toπ‘‡π‘‡βˆ— ∈ Ξ©(𝐢𝐢, 𝑇𝑇).

7.1.9Application to constrained optimization problems

Let 𝐢𝐢be a closedconvex subset of a Hilbert space𝐻𝐻, 𝑃𝑃𝐢𝐢the metric projection of𝐻𝐻onto 𝐢𝐢and𝑇𝑇:𝐢𝐢 β†’ 𝐻𝐻a 𝜈𝜈-ism where 𝜈𝜈 > 0 is a constant. It is well known that 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡π‘‡π‘‡)is non-expansive operator provided that πœ‡πœ‡ ∈(0, 2𝜈𝜈).

The algorithms for signal and image processing are often iterative constrainedoptimization processes designed to minimize a convex differentiable function 𝑇𝑇overa closed convex set 𝐢𝐢 in𝐻𝐻. It is well known that every𝐿𝐿- Lipschitzian operator is2⁄𝐿𝐿-ism. Therefore, we have the following result which generates the sequence ofvectors in the constrained or feasible set 𝐢𝐢which converges weakly to the optimalsolution which minimizes𝑇𝑇.

Theorem 7.1.10.Let 𝐢𝐢 be a closed convex subset of a Hilbert space 𝐻𝐻 and 𝑇𝑇 aconvex and differentiable function on an open set 𝐷𝐷 containing the set 𝐢𝐢. Assumethat βˆ‡π‘‡π‘‡ is an 𝐿𝐿-Lipschitz operator on𝐷𝐷, πœ‡πœ‡ ∈ (0, 2⁄𝐿𝐿)and minimizers of 𝑇𝑇 relativeto the set 𝐢𝐢 exist. For a given𝑇𝑇1 ∈ 𝐢𝐢,let {𝑇𝑇𝑛𝑛} be a sequence in 𝐢𝐢 generated by

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ 𝑇𝑇)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ 𝑇𝑇)𝑇𝑇𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ 𝑇𝑇)𝑇𝑇𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ 𝑇𝑇)𝑧𝑧𝑛𝑛,

𝑧𝑧𝑛𝑛 = (1βˆ’ 𝛾𝛾𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛾𝛾𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ 𝑇𝑇)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•, οΏ½

where {𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛} π‘Žπ‘Žπ‘›π‘›π‘Žπ‘Ž {𝛾𝛾𝑛𝑛}are sequences in [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈ (0,1). Then the sequence {𝑇𝑇𝑛𝑛} converges weakly to a minimizer of𝑇𝑇.

Proof.Since 𝑇𝑇 ≔ 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ 𝑇𝑇) is non-expansive, hence the result follows fromTheorem 4.2.1.∎

7.1.11Application to split feasibility problems

Recall that a mapping 𝑇𝑇in aHilbert space 𝐻𝐻is said to be averaged if 𝑇𝑇can be written as (1 βˆ’ 𝛼𝛼)𝑉𝑉+𝛼𝛼𝑆𝑆, where𝛼𝛼 ∈ (0, 1) and 𝑆𝑆is a non-expansive map on 𝐻𝐻. Setπ‘žπ‘ž(𝑇𝑇): = 12��𝑇𝑇 βˆ’ 𝑃𝑃𝑄𝑄𝑇𝑇�𝑇𝑇�, 𝑇𝑇 ∈ 𝐢𝐢.

Consider the minimization problem findπ‘‡π‘‡βˆˆπΆπΆminπ‘žπ‘ž(𝑇𝑇).

By [17], the gradient of π‘žπ‘žis βˆ‡π‘žπ‘ž =π‘‡π‘‡βˆ—οΏ½π‘‰π‘‰ βˆ’ 𝑃𝑃𝑄𝑄�𝑇𝑇, where π‘‡π‘‡βˆ—is the adjoint of 𝑇𝑇.Since 𝑉𝑉 βˆ’ 𝑃𝑃𝑄𝑄is non-expansive, it follows that βˆ‡π‘žπ‘žis 𝐿𝐿-Lipschitzian with𝐿𝐿 =

‖𝑇𝑇‖2.Therefore, βˆ‡π‘žπ‘žis 1⁄𝐿𝐿-ism and for any 0 < πœ‡πœ‡ < 2⁄𝐿𝐿, 𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘žis averaged.

Therefore,the composition 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘ž) is also averaged. Set𝑇𝑇 ≔ 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡π‘žπ‘ž). Note thatthe solution set of 𝑆𝑆𝐹𝐹𝑃𝑃(𝐢𝐢,𝑇𝑇)is 𝐹𝐹(𝑇𝑇).

We now present an iterative procedure that can be used to find solutions of 𝑆𝑆𝐹𝐹𝑃𝑃(𝐢𝐢,𝑇𝑇).

Theorem 7.1.12.Assume that 𝑆𝑆𝐹𝐹𝑃𝑃(𝐢𝐢,𝑇𝑇)is consistent.

Suppose{𝛼𝛼𝑛𝑛}, {𝛽𝛽𝑛𝑛} π‘Žπ‘Žπ‘›π‘›π‘Žπ‘Ž {𝛾𝛾𝑛𝑛}are sequences in [πœ‰πœ‰, 1βˆ’ πœ‰πœ‰] for all 𝑛𝑛 ∈ β„• and for some πœ‰πœ‰ ∈ (0,1). Let{𝑇𝑇𝑛𝑛} be a sequence in 𝐢𝐢 generated by

𝑇𝑇𝑛𝑛+1 = (1βˆ’ 𝛼𝛼𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘ž)𝑇𝑇𝑛𝑛 +𝛼𝛼𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘ž)𝑇𝑇𝑛𝑛, 𝑇𝑇𝑛𝑛 = (1βˆ’ 𝛽𝛽𝑛𝑛)𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘ž)𝑇𝑇𝑛𝑛 +𝛽𝛽𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘ž)𝑧𝑧𝑛𝑛,

𝑧𝑧𝑛𝑛 = (1βˆ’ 𝛾𝛾𝑛𝑛)𝑇𝑇𝑛𝑛 +𝛾𝛾𝑛𝑛𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘ž)𝑇𝑇𝑛𝑛; βˆ€ 𝑛𝑛 ∈ β„•, οΏ½

where 0 < πœ‡πœ‡ < 2⁄‖𝑇𝑇‖2. Then the sequence{𝑇𝑇𝑛𝑛}converges weakly to a solution of 𝑆𝑆𝐹𝐹𝑃𝑃(𝐢𝐢,𝑇𝑇).

Proof.Since 𝑇𝑇 ≔ 𝑃𝑃𝐢𝐢(𝑉𝑉 βˆ’ πœ‡πœ‡ βˆ‡ π‘žπ‘ž) is non-expansive, hence the result follows fromTheorem 4.2.1.∎

Dalam dokumen A Study on Fixed Point Iterative Procedures (Halaman 86-91)
Unduh sekarang "A Study on Fixed Point..."

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