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A self-adaptive projection-type method for nonlinear multiple-sets split feasibility problem. (English) Zbl 1272.65053
This very useful paper presents the split problem: Find a vector \(x^* \) such that \[ (1)\quad x^* \in C\text{ and }Ax^* \in Q, \] where \(C \subseteq \mathbb R^n\) and \(Q \subseteq \mathbb R^m\) are nonempty closed convex sets, \(A \in \mathbb R^{m \times n}\) is a matrix. If \(C\) and \(Q\) are intersections of a family of closed convex sets, i.e. \[ (2)\quad C: = \bigcap^t_{i=1}C_i \text{ and } Q: = \bigcap^r_{j=1}Q_j , \] where \(C_i\) \((i= 1,2,\dots, t)\) and \(Q_j\) \((j= 1,2,\dots, r)\) are nonempty closed convex sets in the \(n\)-and \(m\)-dimensional Euclidean spaces, respectively, the problem (1) – (2) is the so-called multiple-sets split feasibility problem (MSFP). The NSFP (nonlinear split feasibility problem) is to find a vector \(x^*\) such that
(3) \(x^* \in C\) and \(F(x^*) \in Q\),
where the mapping \(F: \mathbb R^n \rightarrow \mathbb R^m\) is nonlinear and continuously differentiable. The purpose of this work is to emphasize the necessity of using the nonlinear models, and equally, the authors propose numerical algorithms for solving the nonlinear models. The proximity function for NSFP and MNSFP (which is exploited to reformulate the NSFP or NMSFP as an optimization problem) is introduced. The authors test the proposed method on an NSFP and report the numerical results diagrammatically.
The proposed method is globally convergent. To illustrate the effect of the proposed algorithm, an example is used.

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
Full Text: DOI
[1] DOI: 10.1007/BF02142692 · Zbl 0828.65065
[2] DOI: 10.1137/S0036144593251710 · Zbl 0865.47039
[3] DOI: 10.1016/S1076-5670(08)70157-5
[4] DOI: 10.1088/0266-5611/18/2/310 · Zbl 0996.65048
[5] DOI: 10.1088/0266-5611/20/1/006 · Zbl 1051.65067
[6] DOI: 10.1088/0266-5611/21/6/017 · Zbl 1089.65046
[7] DOI: 10.1088/0031-9155/51/10/001
[8] DOI: 10.1016/j.jmaa.2006.05.010 · Zbl 1253.90211
[9] DOI: 10.1088/0266-5611/21/5/009 · Zbl 1080.65033
[10] DOI: 10.1016/j.laa.2007.03.002 · Zbl 1135.65022
[11] DOI: 10.1088/0266-5611/22/6/007 · Zbl 1126.47057
[12] DOI: 10.1088/0266-5611/20/4/014 · Zbl 1066.65047
[13] DOI: 10.1088/0266-5611/21/5/017 · Zbl 1080.65035
[14] DOI: 10.1109/TNS.2004.834824
[15] Phuoc , KT , Rousse , A , Pittman , M , Rousseau , JP , Malka , V , Fritzler , S , Umstadter , D and Hulin , D .X-ray radiation from nonlinear Thomson Scattering of an intense femtosecond laser on relativistic electrons in a Helium plasma, Phys Rev Lett. 91 (2003), pp. 195001:1–4
[16] Shao LZ, A survey of beam intensity optimization in IMRT (2005)
[17] DOI: 10.1088/0266-5611/25/11/115001 · Zbl 1185.65102
[18] DOI: 10.1007/BF01584073 · Zbl 0227.90044
[19] DOI: 10.7153/mia-07-45 · Zbl 1086.49007
[20] Auslender A, Optimisation: Méthodes Numéiques (1976)
[21] DOI: 10.1007/BF03007664 · Zbl 0352.47023
[22] Golshtein EG, Modified Lagrangians and Monotone Maps in Optimization (1996)
[23] Censor Y, The split feasibility model leading to a unified approach for inversion problems in intensity-modulated radiation therapy (2005)
[24] DOI: 10.1023/A:1013048729944 · Zbl 0998.65066
[25] DOI: 10.1016/j.ejor.2008.03.004 · Zbl 1163.58305
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