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New proximal point algorithms for convex minimization. (English) Zbl 0778.90052
Two modifications of the classical proximal point algorithm for the optimization problem min xR n f(x) are studied, under the only assumption that f:R n R{+} is a proper lower- semicontinuous convex function. In both of them, starting with an initial point x 0 R n , an auxiliary sequence {y k } k=0 of approximately optimal solutions is computed according to x k+1 =J λ k x k :=argmin xR n f (x) + (1/2λ k ) x-y k 2 , {λ k } k=0 being a sequence of positive numbers. In the first variant, a sequence {φ k } k=0 of strictly convex quadratic functions is generated in such a way that, for all k0 and xR n , φ k+1 (x)-f(x)(1-α k )(φ k (x)-f(x)) for some α k [0,1) and one sets y k =(1-α k )x k +α k ν k , with ν k being the minimizer of φ k . For this algorithm, it is proved that f(x k )-inf xR n f(x)=O1 / j=0 k-1 λ j 2 while, for the second variant, f(x k )-inf xR n f(x)=O(1/(k+1) 2 ).
90C25Convex programming
90-08Computational methods (optimization)