A Calculus of EPI-Derivatives Applicable to Optimization

AbstractWhen an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first- and secondorder conditions for optimality can be stated in terms of the first- and second-order epi-derivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem's solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epi-derivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of "amenable" functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.

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The second-order composed radial derivatives of perturbation mappings of parametric set-valued optimization problems

Science and Technology Development Journal - Natural Sciences ◽

10.32508/stdjns.v4i3.838 ◽

2020 ◽

Vol 4 (3) ◽

pp. First

Author(s):

Phạm Lê Bạch Ngọc ◽

Nguyen Thanh Tung ◽

Nguyen Huynh Nghia

Keyword(s):

Sensitivity Analysis ◽

Optimization Problem ◽

Optimization Problems ◽

Second Order ◽

Pareto Solution ◽

Generalized Differentiability ◽

Radial Derivative ◽

Set Valued Mapping ◽

Solution Mapping ◽

Derivatives Of

In the paper, we study the generalized differentiability in set-valued optimization, namely stydying the second-order composed radial derivative of a given set-valued mapping. Inspired by the adjacent cone and the higher-order radial con in Anh NLH et al. (2011), we introduce the second-order composed radial derivative. Then, its basic properties are investigated and relationships between the second-order compsoed radial derivative of a given set-valued mapping and that of its profile are obtained. Finally, applications of this derivative to sensitivity analysis are studied. In detail, we work on a parametrized set-valued optimization problem concerning Pareto solutions. Based on the above-mentioned results, we find out sensitivity analysis for Pareto solution mapping of the problem. More precisely, we establish the second-order composed radial derivative for the perturbation mapping (here, the perturbation means the Pareto solution mapping concerning some parameter). Some examples are given to illustrate our results. The obtained results are new and improve the existing ones in the literature.

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Shape Optimization of Hydraulic Structures: an Example of an Optimum Design of a Fish Passage

10.29007/2k64 ◽

2018 ◽

Author(s):

Pat Prodanovic ◽

Cedric Goeury ◽

Fabrice Zaoui ◽

Riadh Ata ◽

Jacques Fontaine ◽

...

Keyword(s):

Numerical Model ◽

Shape Optimization ◽

Objective Function ◽

Optimum Design ◽

Optimization Problem ◽

A Priori ◽

Coupled System ◽

Fish Passage ◽

Hydraulic Structures ◽

Design Variables

This paper presents a practical methodology developed for shape optimization studies of hydraulic structures using environmental numerical modelling codes. The methodology starts by defining the optimization problem and identifying relevant problem constraints. Design variables in shape optimization studies are configuration of structures (such as length or spacing of groins, orientation and layout of breakwaters, etc.) whose optimal orientation is not known a priori. The optimization problem is solved numerically by coupling an optimization algorithm to a numerical model. The coupled system is able to define, test and evaluate a multitude of new shapes, which are internally generated and then simulated using a numerical model. The developed methodology is tested using an example of an optimum design of a fish passage, where the design variables are the length and the position of slots. In this paper an objective function is defined where a target is specified and the numerical optimizer is asked to retrieve the target solution. Such a definition of the objective function is used to validate the developed tool chain. This work uses the numerical model TELEMAC- 2Dfrom the TELEMAC-MASCARET suite of numerical solvers for the solution of shallow water equations, coupled with various numerical optimization algorithms available in the literature.

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Derivatives of Hadamard type in vector optimization

Mathematica Slovaca ◽

10.1515/ms-2017-0113 ◽

2018 ◽

Vol 68 (2) ◽

pp. 421-430

Author(s):

Karel Pastor

Keyword(s):

Nonlinear Analysis ◽

Optimality Conditions ◽

Vector Optimization ◽

Optimality Condition ◽

Optimization Problem ◽

Vector Optimization Problem ◽

Dini Derivatives ◽

Scalar Optimality ◽

Derivatives Of

Abstract In our paper we will continue the comparison which was started by Vsevolod I. Ivanov [Nonlinear Analysis 125 (2015), 270–289], where he compared scalar optimality conditions stated in terms of Hadamard derivatives for arbitrary functions and those which was stated for ℓ-stable functions in terms of Dini derivatives. We will study the vector optimization problem and we show that also in this case the optimality condition stated in terms of Hadamard derivatives is more advantageous.

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Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

Journal of Elasticity ◽

10.1007/s10659-021-09836-6 ◽

2021 ◽

Author(s):

V. Calisti ◽

A. Lebée ◽

A. A. Novotny ◽

J. Sokolowski

Keyword(s):

Circular Inclusion ◽

Elasticity Tensor ◽

Second Order ◽

Topological Derivative ◽

Microstructural Changes ◽

Topological Derivatives ◽

Spatial Dimensions ◽

Elasticity Tensors ◽

Derivatives Of ◽

Optimization Of Microstructures

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

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