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Register a new userOn some rescaled shape optimization problems
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We consider Cheeger-like shape optimization problems of the form $$minbig{|Omega|^alpha J(Omega) : Omegasubset Dbig}$$ where $D$ is a given bounded domain and $alpha$ is above the natural scaling. We show the existence of a solution and analyze as $J(Omega)$ the particular cases of the compliance functional $C(Omega)$ and of the first eigenvalue $lambda_1(Omega)$ of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.
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We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $R^d$, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case $p>d$, where the existence occurs under very mild conditions, and the case $ple d$, where additional assumptions have to be made on the data.
We suggest a new greedy strategy for convex optimization in Banach spaces and prove its convergent rates under a suitable behavior of the modulus of uniform smoothness of the objective function.
In this paper, optimal actuator shape for nonlinear parabolic systems is discussed. The system under study is an abstract differential equation with a locally Lipschitz nonlinear part. A quadratic cost on the state and input of the system is considered. The existence of an optimal actuator shape has been established in the literature. This paper focuses on driving the optimality conditions for actuator shapes belonging to a Banach space. The application of the theory to the optimal actuator shape design for railway track model is considered.
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin condition on the free boundary. We show the existence of an optimal domain under rather general assumptions and we study the cases when the optimal domains are open sets and have a finite perimeter.
In this paper, we propose some new proximal quasi-Newton methods with line search or without line search for a special class of nonsmooth multiobjective optimization problems, where each objective function is the sum of a twice continuously differentiable strongly convex function and a proper convex but not necessarily differentiable function. In these new proximal quasi-Newton methods, we approximate the Hessian matrices by using the well known BFGS, self-scaling BFGS, and the Huang BFGS method. We show that each accumulation point of the sequence generated by these new algorithms is a Pareto stationary point of the multiobjective optimization problem. In addition, we give their applications in robust multiobjective optimization, and we show that the subproblems of proximal quasi-Newton algorithms can be regarded as quadratic programming problems. Numerical experiments are carried out to verify the effectiveness of the proposed method.
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