Stability Of Optimal Solutions Research Articles

Analysis of a Two-Dimensional Thermal Cloaking Problem on the Basis of Optimization

For a two-dimensional model of thermal scattering, inverse problems arising in the development of tools for cloaking material bodies on the basis of a mixed thermal cloaking strategy are considered. By applying the optimization approach, these problems are reduced to optimization ones in which the role of controls is played by variable parameters of the medium occupying the cloaking shell and by the heat flux through a boundary segment of the basic domain. The solvability of the direct and optimization problems is proved, and an optimality system is derived. Based on its analysis, sufficient conditions on the input data are established that ensure the uniqueness and stability of optimal solutions.

Optimization method in problems of acoustic cloaking of material bodies

Optimization problems for a three-dimensional model of acoustic scattering are formulated and studied. These problems arise in designing tools for cloaking material bodies by applying the wave flow method. The cloaking effect is achieved due to an optimal choice of variable parameters of the inhomogeneous isotropic medium occupying the sought shell. The solvability of direct and optimization problems for the acoustic scattering model is proved, and sufficient conditions ensuring the uniqueness and stability of optimal solutions are established.

Analysis of the 3D acoustic cloaking problems using optimization method

Control problems for the 3D model of acoustic scattering which describes scattering acoustic waves by a permeable obstacle with the form of a spherical layer are considered. These problems arise while developing the design technologies of acoustic cloaking devices using the wave flow method. The solvability of direct and control problems for the acoustic scattering model under study is proved. The sufficient conditions which provide local uniqueness and stability of optimal solutions are established.

Analysis and optimization in problems of cloaking of material bodies for the Maxwell equations

We consider control problems for the 3D Maxwell equations describing electromagnetic wave scattering in an unbounded inhomogeneous medium that contains a permeable isotropic obstacle with cloaking boundary. Such problems arise when studying cloaking problems by the optimization method. The boundary coefficient occurring in the impedance boundary condition plays the role of a control. We study the solvability of the control problem and derive optimality systems that describe necessary conditions for the extremum. By analyzing the constructed optimality systems, we justify sufficient conditions imposed on the input data providing the uniqueness and stability of optimal solutions.

Optimal Replenishment Policy for Weibull-Distributed Deteriorating Items with Trapezoidal Demand Rate and Partial Backlogging

An inventory model for Weibull-distributed deteriorating items is considered so as to minimize the total cost per unit time in this paper. The model starts with shortage, allowed partial backlogging, and trapezoidal demand rate. By analyzing the model, an efficient solution procedure is proposed to determine the optimal replenishment and the optimal order quantity and the average total costs are also obtained. Finally, numerical examples are provided to illustrate the theoretical results and a sensitivity analysis of the major parameters with respect to the stability of optimal solution is also carried out.

Stability estimates in the problem of cloaking material bodies for Maxwell’s equations

A control problem is considered for the three-dimensional Maxwell equations in the exterior of an impenetrable body with a boundary partly covered for cloaking. The role of the control is played by the surface impedance of the covered part of the boundary, which enters into the impedance boundary condition. The solvability of the control problem is proved, and an optimality system describing necessary conditions for an extremum is derived. An analysis of the optimality system yields sufficient conditions on the initial data that ensure the uniqueness and stability of optimal solutions for a particular cost functional.

2d Shape Optimization Via Genetic Algorithm

In this study, among different algorithms that have been introduced for obtaining optimal shapes and structures, an advanced optimization method for distinct shapes and also non-significant ones is illustrated. In order to investigate the efficiency of the method, a specific structural member (safety belt) is analyzed. The optimization process is to optimize the member via genetic algorithm, in order to have minimum weight; meanwhile having the ability to support the loading and also sustaining the generated tension stresses under the range of the allowable limit. The main goal of the present work is to focus on the existence of an optimal shape of the member optimized by genetic algorithm, having necessary conditions of optimality for a safety belt, and stability of optimal solutions under some prescribed perturbations.

Control of boundary impedance in two-dimensional material-body cloaking by the wave flow method

Control problems are considered for a two-dimensional electromagnetic field model describing electromagnetic wave scattering in a unbounded homogeneous medium containing an anisotropic permeable inclusion with a partially covered (cloaked) boundary. The control is a function involved in the impedance boundary condition on the covered part of the boundary. The solvability of the original mixed transmission problem for the two-dimensional Helmholtz equation and of the control problems is proved. Optimality systems describing necessary extremum conditions are derived. The uniqueness and stability of optimal solutions with respect to certain perturbations of the cost functional and the incident wave are established.

Stability estimates for the solutions to inverse extremal problems for the Helmholtz equation

The inverse problems are under study for the Helmholtz equation describing acoustic scattering at a three-dimensional inclusion. Some optimization method reduces these problems to the inverse extremum problems with variable refraction index and boundary source density as controls. We prove that these problems are solvable and derive the optimality systems that describe necessary optimality conditions. Analysis of the optimality systems leads us to some sufficient conditions on the input data ensuring the uniqueness and stability of optimal solutions.

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

We consider control problems for the 2-D Helmholtz equation in an unbounded domain with partially coated boundary. Dirichlet boundary condition is given on one part of the boundary and the impedance boundary condition is imposed on another its part. The role of control in control problem under study is played by boundary impedance. Quadratic tracking–type functionals for the field play the role of cost functionals. Solvability of control problems is proved. The uniqueness and stability of optimal solutions with respect to certain perturbations of both cost functional and incident field are established.

On stability of solutions of the coefficient inverse extremal problems for the stationary convection-diffusion equation

The coefficient inverse extremal problems are studied for the stationary convectiondiffusion equation in a bounded domain under mixed boundary conditions on the boundary of the domain. The role of control is played by the velocity vector of a medium and the functions that are involved in the boundary conditions for temperature. The solvability of the extremal problems is proven both for an arbitrary weakly lower semicontinuous quality functional and for the particular quality functionals. On the basis of analysis of the optimality system some sufficient conditions are established on the initial data providing the uniqueness and stability of optimal solutions under sufficiently small perturbations of both the quality functional and one of the functions involved in the original boundary value problem.

Stability of Discrete Approximations and Necessary Optimality Conditions for Delay-Differential Inclusions

This paper is devoted to the study of optimization problems for dynamical systems governed by constrained delay-differential inclusions with generally nonsmooth and nonconvex data. We provide a variational analysis of the dynamic optimization problems based on their data perturbations that involve finite-difference approximations of time-derivatives matched with the corresponding perturbations of endpoint constraints. The key issue of such an analysis is the justification of an appropriate strong stability of optimal solutions under finite-dimensional discrete approximations. We establish the required pointwise convergence of optimal solutions and obtain necessary optimality conditions for delay-differential inclusions in intrinsic Euler–Lagrange and Hamiltonian forms involving nonconvex-valued subdifferentials and coderivatives of the initial data.

Remanufacturing planning for the reverse Wagner/Whitin models

In this paper the reverse Wagner/Whitin’s dynamic production planning and inventory control model and some of its extensions are studied. In such reverse (product recovery) models, used products arrive to be stored and to be remanufactured at minimum cost. For the reverse model with given demand the zero-inventory-property of optimal solutions is proved, the corresponding Wagner/Whitin algorithm is presented and the stability of optimal solutions is discussed for the case of a large quantity of low cost used products. Furthermore, the model of the alternate application of remanufacturing and manufacturing processes is analysed. Again, for the case of a large quantity of low cost used products the zero-inventory-property is proved, the Wagner/Whitin algorithm is applied to determine the periods in which used products are remanufactured or new products are produced, and the stability of Silver/Meal solutions for this model is sketched.

Sensitivity analysis in pareto optimal stochastic mine planning

Abstract In most planning situations, a decision maker would frequently require knowledge about the stability of optimal solutions. Similarly, in PARETO OPTIMAL STOCHASTIC MINE PLANNING, a decision maker would require knowledge about the stability of the pareto optimal solution set and the preferred solution. This knowledge is gained through sensitivity analysis, and gives the decision maker an indication of the level of risk taken when a particular solution is preferred.