Let H be a real Hilbert space and let T: H Ä 2 H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit v # T &1 0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.
Academic Press
In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim. 14 (1976), 877-898] and Kamimura and Takahashi [J. Approx. Theory 106 (2000), 226-240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem. (2000): 47H05, 47J25.
Mathematics Subject Classifications
In this paper, we study a dynamic portfolio-consumption optimization problem when the market price of risk is driven by linear Gaussian processes. We show sufficient conditions to verify that an explicit solution derived from the Hamilton-Jacobi-Bellman equation is in fact an optimal solution to the portfolio selection problem.
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