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Regularization Parameter Selection Methods in Parallel MR Image Reconstruction
Published in Joseph Suresh Paul, Raji Susan Mathew, Regularized Image Reconstruction in Parallel MRI with MATLAB®, 2019
Joseph Suresh Paul, Raji Susan Mathew
Similar to the L-curve, the GCV function also cannot be derived explicitly if the regularization term is non-quadratic. As a result, the parameter selection problem becomes a bilevel optimization model which is very difficult to solve compared to the quadratic case [6]. This difficulty is overcome by introduction of a non-linear GCV function [28,46,47] derived by computing the Jacobian matrix consisting of the partial derivatives of Equation (2.6) with respect to the observed data b. In addition, several modifications of statistical methods such as SURE and Bayesian methods have also been proposed for application to non-quadratic settings.
Surrogate-based bilevel shape optimization for blended-wing–body underwater gliders
Published in Engineering Optimization, 2023
Weixi Chen, Peng Wang, Huachao Dong
Bilevel optimization mainly solves an upper level problem, which is subject to the optimality of the corresponding lower level problem. The upper level problem and the lower level problem are also called the leader and follower problems, respectively. As a form of design, each level contains its own objectives, variables and constraints. The optimization information interacts iteratively between the two levels, thereby remarkably improving the problem-solving performance. Bilevel optimization problems are ubiquitous in daily life. The presence of latent coupling effects between distinct levels of bilevel optimization problems is commonplace, especially in real-world applications, such as the economy (Sinha, Malo, and Deb 2013), logistics (Sun, Gao, and Wu 2008) and transportation (Migdalas 1995). With this in mind, coupling information can be shared between different levels.
Optimality conditions for a multiobjective bilevel optimization problem involving set valued constraints
Published in Optimization, 2021
N. Gadhi, K. Hamdaoui, M. El idrissi
Nowadays, bilevel programming problems are being studied and discussed by several authors at various levels of generality [1–11]. In bilevel optimization, one considers two optimization problems which are connected with each other. We regard the upper level problem which depends on the solution of the lower level problem. To be more exact, the set of solutions of the lower level problem restricts the feasible set of the upper level problem and influences its objective function. On the other hand, the lower level problem is a parametric optimization problem since the objective function of this problem is minimized only with respect to one of the variables.
Some new optimality conditions for semivector bilevel optimization program
Published in Optimization, 2022
Roughly speaking, bilevel optimization problem is an optimization problem dates from decision process involving two players with a hierarchical structure. The leader in upper level announces his variables first, then the followers in lower level react to his variables to optimize their own objective functions. Now it has been well recognized as a theoretically very challenging and practically important area in applied mathematics and other areas. The interested readers are referred to [1–3] for more detailed history and monograph of bilevel optimization problem.