Existence and uniqueness of global solution of nonlinear schrödinger equations onR 2

No. 4 A ROBUST A L G O R I T H M F O R O P T I M I Z A T I O N PROBLEMS 373 Hence for sufficientlylarge k E K, from (4.11) we know W~+n0k = 0. On the other hand, from the fact IJ+ n Q*I = IQ~I and (4.11) we have IJ+ M Q~l -- IQ~l, hence, from the definition of ~ , we easily obtain [2k = ~ for sufficiently large k E K. Thus s ~ (k E K ) are all defined by (14), which contradicts the fact that s } is all defined b y / 1 0 ) for any k E K. Hence (4.7) is true and the lemma follows. We conclude this paper by proving the following theorem. T h e o r e m 4.3. If (H2), (H3) and (H4) are satisfied, then the infinite point sequence {x k } generated by our algorithm converges superllnearly to the unique optimal solution a:* of (NP). Proof. From (4.11) and Lemma 4.2, Qk is equal to some fixed Q for sufficiently large k. The algorithm in Section 3 is equivalent to the variable metric algorithm described in [8] on the hyperplane {x l aTx ----bj (j e Q)}. According to the results of [8], we obtain the conclusion of our theorem. References [1] D. Goldfarb. Extension of Davidon's Variable Metric Method to Maximization under Linear Inequality and Equality Constraints. SIAM J. Applied Math., 1969, 17: 739-764. [2] J. Rosen. The Gradient Projection Method for Nonlinear Programming, I. Linear Constraints. J. Soc. Indust. Appl. Math., 1960, 8: 181-217. [3] H.-Y. Kwei, F. Wu and Y.-L. Lal. Extension of a Variable Metric Algorithm to a Linearly Constrained Optimization Problem--A Variation of Goldfarb's Algorithm. In: K.B. Haley, Ed., OR'78 (NorthHolland, Amsterdam, 1979, 955-974. [4] F. Wu. A Super-linear Convergent Gradient Projection Type Algorithm for Linearly Constrained Problems. European Journal os Operational Research, 1984, 16: 334-344. [5] D.-Z. Du and J. Sun. A New Gradient Projection Method. Math. Numer. Sinica (Chinese Series), 1984, 6(4): 378-386. [6] H.-Y. Kwei and D.-Z. Du. A Superlinearly Convergent Method to Linearly Constrained Optimization Problems under Degeneracy. Acta Math. Appl. Sinics, 1984, 1(1): 76-84. [7] Q.-G. Zeng. Extension of Rosen's Gradient Projection Method and the Proof of Its Global Convergence. Acts Math. Appl. Sinica, 1991, 14(3): 312-322 (in Chinese). [8] M.J.D. Powell. Some Global Convergence Properties of a Variable Metric Algorithm for Minimization without Exact Line Searches. SIAM-AMS Proceeding, Vol.9, 1976. Error Correction On Vol.13, No.4, p.414 of our journal, the title should be: EXISTENCE AND UNIQUENESS OF GLOBAL SOLUTION NONLINEAR SCHRODINGER EQUATIONS ON R2 OF  (...truncated)