On continuous analog of four parameter regularized projection minimization method

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© V.G. Malinov (Orenburg)
1. Consider the minimization problem
/(x) -¦" inf, x & lt-E Q С H,
where Q is a given convex set from the Hilbert space H normed by the scalar product ||x|| = (x. x)½- /(x) is defined, and convex, continuously Frechet differentiable function on Я, and its gradients satisfy the Lipschitz condition. Suppozed that m/f (x) = /* & gt- -oo, x € Q- Q* = {x € Q: /x) = /*} ф 0. A normal solution of problem (1) is defined to be a point x*? Q* with minimal norm. The problem of its determining is ill-posed, and to solve it we must use various regularization methods. 2. Let us construct and study a regularization method for problem (1) in the case when Q is specified exactly and the gradient of /(x) is given approximately, based on the continuous projection second order method
& lt-r (t)x"-(t) + x'-(f) + x (f) = PQ[y (t) +?(f)(7i (t)x'-(f) -72(f)Tj (y (*), t))], i & gt- 0, x (0) =x°, x'-(0) = x1,
where Pq[v] is the projection of v onto the set Q- x^x1 6 Q are the initial points- T'-5(y (t), t) = = f (y (t), t) + T (t)y (t) у (t) € Q, t ^ 0 is an approximation at the point y (?) of the exact gradient Т'- (x (t), t) — f (x (f)) + r (?)x (?), xCff,() 0, of the Tikhonov function T (x (t)) = /x (?)) + r (t)||x||2/2, x € Я, t ^ 0- x = x (f) G C2[0,+oo) — c (i), a (i), /?(?), 71 (?), 72(f), t (?), 6(t) are parameters of the method (2) — the vector 7i (i)x (t) — 72(f)Тй (у (?), t) allows to regulate the direction of motion to minimum by change values of the parameters 71(f) and 72(f) — The report suggests sufficient conditions for the convergence of the method (2) to a normal solution of problem (1). It is necessary to note, that: 1) in [1] method (2) were studied for nonlinear constrained problems- 2) method (2) is a continuous analog of the iterative four-parameter minimization method from [2].
1. Малинов В. Г. О регуляризованном непрерывном проекционном методе минимизации второго порядка // Алгоритмический анализ неустойчивых задач: Тез. докл. Всерос. науч. конф. Екатеринбург: Изд-во Уральского ун-та, 2001. С. 230−231.
2. Malinov V.G. A Four-Parameter Two-Step Regularized Projection Method for Minimization // Computational Mathematics and Mathematical Phisics. V. 39. № 4. P. 539−544.

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