scholarly journals On the differentiation of the functional in distributed optimization problems with imperfect contact

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 775-783
Author(s):  
Aigul Manapova
Keyword(s):  
Numerical Solutions ◽  
Optimization Problems ◽  
Normal Component ◽  
Imperfect Contact ◽  
Convection Diffusion ◽  
The Cost ◽  
Adjoint Operators ◽  
Second Order Elliptic Equations ◽  
Direct Problems ◽  
Discontinuous Data

We investigate issues of numerical solving of optimal control problems for second order elliptic equations with non-self-adjoint operators - convection-diffusion problems. Control processes are described by semi-linear convection-diffusion equation with discontinuous data and solutions (states) subject to the boundary interface conditions of imperfect type (i.e., problems with a jump of the coefficients and the solution on the interface; the jump of the solution is proportional to the normal component of the flux). Controls are involved in the coefficients of diffusion and convective transfer. We prove differentiability and Lipshitz continuity of the cost functional, depending on a state of the system and a control. The calculation of the gradients uses the numerical solutions of direct problems for the state and adjoint problems.

scholarly journals An approximation of problems of optimal control on the coefficients of elliptic convection-diffusion equations with an imperfect contact matching condition

2019 ◽  
Vol 21 (2) ◽  
pp. 187-214 ◽  
Author(s):  
Fedor F. Lubyshev ◽  
Aigul R. Manapova
Keyword(s):  
Optimal Control ◽  
Optimization Problems ◽  
Normal Component ◽  
Adjoint Operator ◽  
Variable Coefficients ◽  
State Equations ◽  
Imperfect Contact ◽  
Convection Diffusion ◽  
Difference Approximations ◽  
The Difference

We consider nonlinear optimization problems for processes described by non-self-adjoint elliptic equations of convection-diffusion problems with an imperfect contact matching conditions. These are the problems with a jump of the coefficients and of the solution on the interface; the jump of the solution is proportional to the normal component of the flux. Variable coefficients multiplying the highest and the lowest derivatives in the equation and the coefficients by nonlinear terms in the equations of state are used as controls. Finite difference approximations of optimization problems are constructed and investigated. For the approximation of state equations we propose a new ``modified difference scheme'' in which the variable grid coefficients in the principal part of the difference operator are computed using method other than traditionally applied in the theory of difference schemes. The problem's correctness is investigated. The accuracy estimation of difference approximations with respect to the state are obtained. Convergence rate of approximations with respect to cost functional is estimated, too. Weak convergence with respect to control is proved. The presence of a non-self-adjoint operator causes certain difficulties in constructing and studying approximations of differential equations describing discontinuous states of controlled processes, in particular, in proving the difference approximations well-posedness, and in studying the relationship between the original optimal control problem and the approximate mesh problem. The approximations are regularized. The obtained results will be heavily used later in solving problems associated with the development of effective methods for the numerical solution to the constructed finite-dimensional mesh optimal control problems and their computer implementation.

A weak derivative approach to optimization of threshold parameters in a multicomponent maintenance system

2001 ◽  
Vol 38 (02) ◽  
pp. 386-406 ◽  
Author(s):  
Bernd Heidergott
Keyword(s):  
Optimization Problems ◽  
Estimation Algorithm ◽  
Replacement Policy ◽  
Cost Performance ◽  
Weak Derivative ◽  
Preventive Replacement ◽  
Maintenance System ◽  
Threshold Optimization ◽  
Age Replacement ◽  
The Cost

We consider a multicomponent maintenance system controlled by an age replacement policy: when one of the components fails, it is immediately replaced; all components older than a threshold age θ are preventively replaced. Costs are associated with each maintenance action, such as replacement after failure or preventive replacement. We derive a weak derivative estimator for the derivative of the cost performance with respect to θ. The technique is quite general and can be applied to many other threshold optimization problems in maintenance. The estimator is easy to implement and considerably increases the efficiency of a Robbins-Monro type of stochastic approximation algorithm. The paper is self-contained in the sense that it includes a proof of the correctness of the weak derivative estimation algorithm.

Analytical Calculation of Deformations and Kinematic Analysis of a Flat Truss with an Arbitrary Number of Panels

2021 ◽  
pp. 113-122
Author(s):  
М. Н. Кирсанов ◽  
О. В. Воробьев
Keyword(s):  
Kinematic Analysis ◽  
Arbitrary Number ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Analytical Calculation ◽  
Middle Part ◽  
Asymptotic Approximations ◽  
Induction Method ◽  
Particular Solutions

Постановка задачи. Разыскиваются аналитические зависимости прогиба и смещения опоры плоской фермы решетчатого вида от числа панелей. Ферма имеет сдвоенную решетку, прямолинейный нижний и приподнятый в средней части верхний пояс. Результаты. Для двух видов нагружения по формуле Максвелла-Мора получены аналитические зависимости прогибов конструкции от нагрузки, размеров и числа панелей. Для обобщения серии частных решений с различным числом панелей ферм на произвольный случай использован метод индукции и аналитические возможности системы компьютерной математики Maple. Для некоторых решений получены асимптотические приближения. Показано распределение усилий в элементах фермы. Выводы. Полученные формулы могут быть использованы в задачах оптимизации и как тестовые для оценки приближенных численных решений. Выявлены случаи геометрической изменяемости фермы при числе панелей, кратном трем. Приведен алгоритм выявления соответствующего распределения возможных скоростей шарниров. Statement of the problem. Analytical dependences of the deflection and displacement of the support of a flat lattice truss on the number of panels are being sought. The truss has a double lattice, a rectilinear lower belt and an upper belt raised in the middle part. Results. For two types of loading, according to the Maxwell-Mohr formula, analytical dependences of the deflections of the structure on the load, dimensions and number of panels are obtained. To generalize a series of particular solutions for trusses with different numbers of panels for an arbitrary case, the induction method and the analytical capabilities of the Maple computer mathematics system were used. For some solutions, asymptotic approximations are obtained. The distribution of forces in the rods of the structure is shown. Conclusions. The obtained formulas can be used in optimization problems and as test ones for evaluating approximate numerical solutions. Cases of geometric variability of the truss with the number of panels being a multiple of three are revealed. An algorithm for identifying the corresponding distribution of possible velocities of the joints is presented.

An improved proximal method with quasi-distance for nonconvex multiobjective optimization problem

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Fouzia Amir ◽  
Ali Farajzadeh ◽  
Jehad Alzabut
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Convergence Result ◽  
Proximal Method ◽  
Pareto Solution ◽  
Multiobjective Optimization Problems ◽  
Locally Lipschitz ◽  
The Cost ◽  
Constrained Multiobjective Optimization

Abstract Multiobjective optimization is the optimization with several conflicting objective functions. However, it is generally tough to find an optimal solution that satisfies all objectives from a mathematical frame of reference. The main objective of this article is to present an improved proximal method involving quasi-distance for constrained multiobjective optimization problems under the locally Lipschitz condition of the cost function. An instigation to study the proximal method with quasi distances is due to its widespread applications of the quasi distances in computer theory. To study the convergence result, Fritz John’s necessary optimality condition for weak Pareto solution is used. The suitable conditions to guarantee that the cluster points of the generated sequences are Pareto–Clarke critical points are provided.

scholarly journals Integrated Offline and Online Decision Making under Uncertainty

2021 ◽  
Vol 70 ◽  
pp. 77-117
Author(s):  
Allegra De Filippo ◽  
Michele Lombardi ◽  
Michela Milano
Keyword(s):  
Optimization Problems ◽  
Greedy Heuristic ◽  
Management Problem ◽  
Quality Improvements ◽  
Solution Quality ◽  
Convex Optimization Problem ◽  
Routing Problem ◽  
Multi Stage ◽  
The Cost ◽  
Parametric Convex Optimization

This paper considers multi-stage optimization problems under uncertainty that involve distinct offline and online phases. In particular it addresses the issue of integrating these phases to show how the two are often interrelated in real-world applications. Our methods are applicable under two (fairly general) conditions: 1) the uncertainty is exogenous; 2) it is possible to define a greedy heuristic for the online phase that can be modeled as a parametric convex optimization problem. We start with a baseline composed by a two-stage offline approach paired with the online greedy heuristic. We then propose multiple methods to tighten the offline/online integration, leading to significant quality improvements, at the cost of an increased computation effort either in the offline or the online phase. Overall, our methods provide multiple options to balance the solution quality/time trade-off, suiting a variety of practical application scenarios. To test our methods, we ground our approaches on two real cases studies with both offline and online decisions: an energy management problem with uncertain renewable generation and demand, and a vehicle routing problem with uncertain travel times. The application domains feature respectively continuous and discrete decisions. An extensive analysis of the experimental results shows that indeed offline/online integration may lead to substantial benefits.

Shape Sensitivity Analysis Applied to Exterior Acoustics

2001 ◽  
Author(s):  
Gonzalo R. Feijóo ◽  
Assad A. Oberai ◽  
Peter M. Pinsky
Keyword(s):  
Optimization Problems ◽  
Auxiliary Problem ◽  
Adjoint Problem ◽  
The Body ◽  
Structural Acoustics ◽  
Shape Sensitivity ◽  
Shape Sensitivity Analysis ◽  
Primal Problem ◽  
Acoustic Media ◽  
The Cost

Abstract We present a method to calculate the derivative of a functional that depends on the shape of a body immersed in an acoustic media. The functional depends implicitly on the shape through the solution of an exterior acoustic problem. The derivative is calculated in terms of the solution of the primal problem and an auxiliary problem, the adjoint problem. An important aspect of this method is that the cost of calculating the derivative is independent of the number of parameters used to represent the shape of the body. This allows for efficient solution of optimization problems in structural acoustics.

Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

2021 ◽  
Author(s):  
Sundar Namala ◽  
Rizwan Uddin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Solutions ◽  
Second Order ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Integral Methods ◽  
Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

Multiobjective optimization design of hybrid composite laminates for vibration using sequential permutation table method

2021 ◽  
pp. 095440622110504
Author(s):  
Zhao Jing ◽  
Qin Sun ◽  
Yongjie Zhang ◽  
Ke Liang
Keyword(s):  
Multiobjective Optimization ◽  
Fundamental Frequency ◽  
Hybrid Composite ◽  
Composite Laminates ◽  
Optimization Design ◽  
Heuristic Algorithms ◽  
Optimization Problems ◽  
Design Flexibility ◽  
The Cost ◽  
Ply Orientation

Due to the large variable design space in optimization problems of composite laminates, it remains one of the challenging tasks to develop efficient optimization design methods to improve the design flexibility and efficiency. This work presents a sequential permutation table (SPT) method for the multiobjective optimization design of two-material hybrid composite laminates with simply supported boundary conditions, which maximizes the fundamental frequency and minimizes the cost/weight. Based on the vibration analysis of hybrid composite laminates, the approximate linear regularity of the square of fundamental frequency is derived, and two best ply orientations for the two materials are identified, respectively. By assigning one best ply orientation with maximum fundamental frequency at respective stacking positions, and using another best ply orientation to replace plies in the stacking sequence from the mid-plane to the outermost can lead to the optimum. Two multiobjective optimization problems are employed to verify the SPT method, results are compared with those obtained by heuristic algorithms. The obtained better solutions demonstrate the effectiveness and efficiency of the SPT method and its potentials for optimal design of hybrid composite laminates.

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