scholarly journals Proximal Gradient Method for Solving Bilevel Optimization Problems

2020 ◽  
Vol 25 (4) ◽  
pp. 66
Author(s):  
Seifu Endris Yimer ◽  
Poom Kumam ◽  
Anteneh Getachew Gebrie
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Split Feasibility Problem ◽  
Bilevel Optimization ◽  
Feasibility Problem ◽  
Proximal Gradient Method ◽  
Level Problem ◽  
Upper Level ◽  
The Split Feasibility Problem

In this paper, we consider a bilevel optimization problem as a task of finding the optimum of the upper-level problem subject to the solution set of the split feasibility problem of fixed point problems and optimization problems. Based on proximal and gradient methods, we propose a strongly convergent iterative algorithm with an inertia effect solving the bilevel optimization problem under our consideration. Furthermore, we present a numerical example of our algorithm to illustrate its applicability.

Multi-Objective Robust Optimization Under Interval Uncertainty Using Online Approximation and Constraint Cuts

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
W. Hu ◽  
M. Li ◽  
S. Azarm ◽  
A. Almansoori
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Computational Effort ◽  
Interval Uncertainty ◽  
Test Results ◽  
Robust Solution ◽  
Multi Objective ◽  
Function Calls ◽  
Level Problem ◽  
Upper Level

Many engineering optimization problems are multi-objective, constrained and have uncertainty in their inputs. For such problems it is desirable to obtain solutions that are multi-objectively optimum and robust. A robust solution is one that as a result of input uncertainty has variations in its objective and constraint functions which are within an acceptable range. This paper presents a new approximation-assisted MORO (AA-MORO) technique with interval uncertainty. The technique is a significant improvement, in terms of computational effort, over previously reported MORO techniques. AA-MORO includes an upper-level problem that solves a multi-objective optimization problem whose feasible domain is iteratively restricted by constraint cuts determined by a lower-level optimization problem. AA-MORO also includes an online approximation wherein optimal solutions from the upper- and lower-level optimization problems are used to iteratively improve an approximation to the objective and constraint functions. Several examples are used to test the proposed technique. The test results show that the proposed AA-MORO reasonably approximates solutions obtained from previous MORO approaches while its computational effort, in terms of the number of function calls, is significantly reduced compared to the previous approaches.

Subgradient algorithm for split hierarchical optimization problems

2018 ◽  
Vol 34 (3) ◽  
pp. 391-399
Author(s):  
NIMIT NIMANA ◽  
◽  
NARIN PETROT ◽  
◽  
Keyword(s):  
Hilbert Spaces ◽  
Optimization Problems ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Hierarchical Optimization ◽  
Convergence Results ◽  
Hierarchical Optimization Problems ◽  
Subgradient Algorithm ◽  
The Split Feasibility Problem ◽  
Split Feasibility

In this paper we emphasize a split type problem of some integrating ideas of the split feasibility problem and the hierarchical optimization problem. Working on real Hilbert spaces, we propose a subgradient algorithm for approximating a solution of the introduced problem. We discuss its convergence results and present a numerical example.

On Bilevel Optimization with Inexact Follower

2020 ◽  
Vol 17 (1) ◽  
pp. 74-95 ◽  
Author(s):  
M. Hosein Zare ◽  
Oleg A. Prokopyev ◽  
Denis Sauré
Keyword(s):  
Optimization Problems ◽  
Broad Class ◽  
Bilevel Optimization ◽  
Modeling Framework ◽  
Optimization Framework ◽  
Lower Level Problem ◽  
Level Problem ◽  
Potential Impact ◽  
Upper Level ◽  
Optimal Reaction

Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader’s actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower’s use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems.

An iterative algorithm with inertial technique for solving the split common null point problem in Banach spaces

2021 ◽  
pp. 2250120
Author(s):  
Yan Tang ◽  
Pongsakorn Sunthrayuth
Keyword(s):  
Banach Spaces ◽  
Split Feasibility Problem ◽  
Null Point ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Inertial Algorithm ◽  
The Split Feasibility Problem ◽  
Inertial Technique ◽  
Split Feasibility

In this work, we introduce a modified inertial algorithm for solving the split common null point problem without the prior knowledge of the operator norms in Banach spaces. The strong convergence theorem of our method is proved under suitable assumptions. We apply our result to the split feasibility problem, split equilibrium problem and split minimization problem. Finally, we provide some numerical experiments including compressed sensing to illustrate the performances of the proposed method. The result presented in this paper improves and generalizes many recent important results in the literature.

scholarly journals New Inertial Relaxed CQ Algorithms for Solving Split Feasibility Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Haiying Li ◽  
Yulian Wu ◽  
Fenghui Wang
Keyword(s):  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Linear Operators ◽  
Feasibility Problem ◽  
Variable Step Size ◽  
Step Size ◽  
Bounded Linear ◽  
Variable Step ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The split feasibility problem SFP has received much attention due to its various applications in signal processing and image reconstruction. In this paper, we propose two inertial relaxed C Q algorithms for solving the split feasibility problem in real Hilbert spaces according to the previous experience of applying inertial technology to the algorithm. These algorithms involve metric projections onto half-spaces, and we construct new variable step size, which has an exact form and does not need to know a prior information norm of bounded linear operators. Furthermore, we also establish weak and strong convergence of the proposed algorithms under certain mild conditions and present a numerical experiment to illustrate the performance of the proposed algorithms.

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