scholarly journals Policy Iterations Without Selection Property

10.29007/9rn9 ◽  
2018 ◽  
Author(s):  
Assale Adje
Keyword(s):  
Fixed Point ◽  
Minimization Problem ◽  
Optimal Solution ◽  
Good Choice ◽  
Selection Property ◽  
Perturbation Parameters

In this paper, we propose a modified policy iterations algorithm which does not rely on the selection property. The selection property is the key argument to make improvements during policy iterations. Indeed, a new policy is computed as an optimal solution of a minimization problem. However, in some cases, it might be difficult to prove that an optimal solution exists. To overcome this issue, the new policy is computed as a guaranteed sub-optimal solution of the minimization problem. The good choice of the perturbation parameters preserves the advantages of the original policy iterations algorithm such as the computation of a post-fixed point at each step and the convergence to a fixed point.

Projection algorithms for composite minimization

2017 ◽  
Vol 33 (3) ◽  
pp. 389-397
Author(s):  
XIAONAN YANG ◽  
◽  
HONG-KUN XU ◽  
Keyword(s):  
Hilbert Space ◽  
Minimization Problem ◽  
Convex Functions ◽  
Gradient Descent ◽  
Optimal Solution ◽  
Finite Family ◽  
Local Function ◽  
Projection Algorithms ◽  
Local Functions ◽  
Convex Subsets

Parallel and cyclic projection algorithms are proposed for minimizing the sum of a finite family of convex functions over the intersection of a finite family of closed convex subsets of a Hilbert space. These algorithms consist of two steps. Once the kth iterate is constructed, an inner circle of gradient descent process is executed through each local function, and then a parallel or cyclic projection process is applied to produce the (k + 1) iterate. These algorithms are proved to converge to an optimal solution of the composite minimization problem under investigation upon assuming boundedness of the gradients at the iterates of the local functions and the stepsizes being chosen appropriately.

scholarly journals The coincidence problem for compositions of set-valued maps

1990 ◽  
Vol 41 (3) ◽  
pp. 421-434 ◽  
Author(s):  
H. Ben-El-Mechaiekh
Keyword(s):  
Fixed Point ◽  
Convex Domains ◽  
Coincidence Problem ◽  
Selection Property ◽  
Locally Convex

The main purpose of this work is to give a general and elementary treatment of the fixed point and the coincidence problems for compositions of set-valued maps with not necessarily locally convex domains and to display, once more, the central rôle played by the selection property.

scholarly journals A Fixed-Point Subgradient Splitting Method for Solving Constrained Convex Optimization Problems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 377
Author(s):  
Nimit Nimana
Keyword(s):  
Fixed Point ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Splitting Method ◽  
Optimal Solution ◽  
Bilevel Optimization ◽  
Convergence Properties ◽  
Nonlinear Operators ◽  
Convex Optimization Problems ◽  
Feasible Points

In this work, we consider a bilevel optimization problem consisting of the minimizing sum of two convex functions in which one of them is a composition of a convex function and a nonzero linear transformation subject to the set of all feasible points represented in the form of common fixed-point sets of nonlinear operators. To find an optimal solution to the problem, we present a fixed-point subgradient splitting method and analyze convergence properties of the proposed method provided that some additional assumptions are imposed. We investigate the solving of some well known problems by using the proposed method. Finally, we present some numerical experiments for showing the effectiveness of the obtained theoretical result.

scholarly journals Strong Convergence of a Unified General Iteration fork-Strictly Pseudononspreading Mapping in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Dao-Jun Wen ◽  
Yi-An Chen ◽  
Yan Tang
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Minimization Problem ◽  
Iterative Sequence ◽  
Mean Convergence ◽  
Strictly Pseudononspreading Mapping ◽  
General Iterative Method

We introduce a unified general iterative method to approximate a fixed point ofk-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of ak-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.

Fast Connectivity Minimization on Large-Scale Networks

2021 ◽  
Vol 15 (3) ◽  
pp. 1-25
Author(s):  
Chen Chen ◽  
Ruiyue Peng ◽  
Lei Ying ◽  
Hanghang Tong
Keyword(s):  
Minimization Problem ◽  
Computational Efficiency ◽  
Large Scale ◽  
Critical Infrastructure ◽  
Optimal Solution ◽  
Polynomial Algorithms ◽  
Social Network Mining ◽  
The Best Approximation ◽  
Special Cases ◽  
Large Scale Networks

The connectivity of networks has been widely studied in many high-impact applications, ranging from immunization, critical infrastructure analysis, social network mining, to bioinformatic system studies. Regardless of the end application domains, connectivity minimization has always been a fundamental task to effectively control the functioning of the underlying system. The combinatorial nature of the connectivity minimization problem imposes an exponential computational complexity to find the optimal solution, which is intractable in large systems. To tackle the computational barrier, greedy algorithm is extensively used to ensure a near-optimal solution by exploiting the diminishing returns property of the problem. Despite the empirical success, the theoretical and algorithmic challenges of the problems still remain wide open. On the theoretical side, the intrinsic hardness and the approximability of the general connectivity minimization problem are still unknown except for a few special cases. On the algorithmic side, existing algorithms are hard to balance between the optimization quality and computational efficiency. In this article, we address the two challenges by (1) proving that the general connectivity minimization problem is NP-hard and is the best approximation ratio for any polynomial algorithms, and (2) proposing the algorithm CONTAIN and its variant CONTAIN + that can well balance optimization effectiveness and computational efficiency for eigen-function based connectivity minimization problems in large networks.

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