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

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.

scholarly journals Structure of Pareto Solutions of Generalized Polyhedral-Valued Vector Optimization Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Qinghai He ◽  
Weili Kong
Keyword(s):  
Banach Spaces ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Solution Set ◽  
Pareto Optimal ◽  
Pareto Solution ◽  
Value Set ◽  
Set Valued Mapping ◽  
Optimal Value

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

Second-order sensitivity analysis for parametric equilibrium problems in set-valued optimization

2019 ◽  
Vol 53 (4) ◽  
pp. 1245-1260
Author(s):  
Nguyen Le Hoang Anh
Keyword(s):  
Sensitivity Analysis ◽  
Equilibrium Problems ◽  
Second Order ◽  
Weak Perturbation ◽  
Parametric Equilibrium Problems ◽  
Perturbation Map ◽  
Derivatives Of

In the paper, we first establish relationships between second-order contingent derivatives of a given set-valued map and that of the weak perturbation map. Then, these results are applied to sensitivity analysis for parametric equilibrium problems in set-valued optimization.

A Calculus of EPI-Derivatives Applicable to Optimization

1993 ◽  
Vol 45 (4) ◽  
pp. 879-896 ◽  
Author(s):  
R. A. Poliquin ◽  
R. T. Rockafellar
Keyword(s):  
Convex Function ◽  
Objective Function ◽  
Nonsmooth Analysis ◽  
Optimization Problem ◽  
Second Order ◽  
Smooth Mapping ◽  
Composition Classes ◽  
Central Case ◽  
Derivatives Of ◽  
Amenable Functions

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.

scholarly journals Sensitivity Analysis of Optimization Problems Under Second Order Regular Constraints

1998 ◽  
Vol 23 (4) ◽  
pp. 806-831 ◽  
Author(s):  
J. Frédéric Bonnans ◽  
Roberto Cominetti ◽  
Alexander Shapiro
Keyword(s):  
Sensitivity Analysis ◽  
Optimization Problems ◽  
Second Order

scholarly journals Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives

2021 ◽  
Author(s):  
Tung Nguyen
Keyword(s):  
Optimization Problems ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Second Order ◽  
Set Valued Mapping ◽  
Contingent Epiderivative ◽  
Wolfe Duality ◽  
Order Constraint ◽  
Sufficient Conditions For Optimality ◽  
Second Order Necessary Conditions

We propose a generalized second-order asymptotic contingent epiderivative of a set-valued mapping, study its properties, as well as relations to some second-order contingent epiderivatives, and sufficient conditions for its existence. Then, using these epiderivatives, we investigate set-valued optimization problems with generalized inequality constraints. Both second-order necessary conditions and sufficient  conditions for optimality of the Karush-Kuhn-Tucker type are established under the second-order constraint qualification. An application to Mond-Weir and Wolfe duality schemes is also presented. Some remarks and examples are provided to illustrate our results.

Second-Order Composed Radial Derivatives of the Benson Proper Perturbation Map for Parametric Multi-Objective Optimization Problems

2020 ◽  
Vol 37 (04) ◽  
pp. 2040011
Author(s):  
Qilin Wang ◽  
Xiaoyan Zhang
Keyword(s):  
Lipschitz Continuity ◽  
Optimization Problems ◽  
Second Order ◽  
Proper Efficiency ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Benson Proper Efficiency ◽  
Perturbation Map ◽  
Derivatives Of ◽  
Cone Convexity

In this paper, we introduce second-order composed radial derivatives of set-valued maps and establish some of its properties. By applying this second-order derivative, we obtain second-order sensitivity results for parametric multi-objective optimization problems under the Benson proper efficiency without assumptions of cone-convexity and Lipschitz continuity. Some of our results improve and derive the recent corresponding ones in the literature.

scholarly journals Reliability-Based Design Optimization of Structures Using the Second-Order Reliability Method and Complex-Step Derivative Approximation

2021 ◽  
Vol 11 (11) ◽  
pp. 5312
Author(s):  
Junho Chun
Keyword(s):  
Sensitivity Analysis ◽  
Design Optimization ◽  
Optimization Problems ◽  
Second Order ◽  
Probabilistic Constraints ◽  
Reliability Based Design Optimization ◽  
Step Size ◽  
Reliability Based Design ◽  
Reliability Method ◽  
Derivative Approximation

This paper proposes a reliability-based design optimization (RBDO) approach that adopts the second-order reliability method (SORM) and complex-step (CS) derivative approximation. The failure probabilities are estimated using the SORM, with Breitung’s formula and the technique established by Hohenbichler and Rackwitz, and their sensitivities are analytically derived. The CS derivative approximation is used to perform the sensitivity analysis based on derivations. Given that an imaginary number is used as a step size to compute the first derivative in the CS derivative method, the calculation stability and accuracy are enhanced with elimination of the subtractive cancellation error, which is commonly encountered when using the traditional finite difference method. The proposed approach unifies the CS approximation and SORM to enhance the estimation of the probability and its sensitivity. The sensitivity analysis facilitates the use of gradient-based optimization algorithms in the RBDO framework. The proposed RBDO/CS–SORM method is tested on structural optimization problems with a range of statistical variations. The results demonstrate that the performance can be enhanced while satisfying precisely probabilistic constraints, thereby increasing the efficiency and efficacy of the optimal design identification. The numerical optimization results obtained using different optimization approaches are compared to validate this enhancement.

Second-Order Design Sensitivity Analysis of Mechanical System Dynamics by a Mixed Direct Differentiation Adjoint Variable Method

1998 ◽  
Author(s):  
Javier Urruzola ◽  
José Manuel Jiménez
Keyword(s):  
Sensitivity Analysis ◽  
Design Sensitivity ◽  
Second Order ◽  
Variable Method ◽  
Adjoint Variable Method ◽  
Adjoint Variable ◽  
Direct Differentiation ◽  
Design Variables ◽  
Adjoint Variables ◽  
Derivatives Of

Abstract This paper presents a new approach to second order sensitivity analysis of multibody dynamics. Adjoint variables together with direct differentiation are used to derive first- and second-order derivatives of measures of dynamic response with respect to design variables. It is shown that the proposed method can be compared advantageously to the fully adjoint variable method proposed by Haug in terms of simplicity and numerical cost. In order to validate the algorithm, a simple oscillator example proposed by Haug is solved analytically and by the mixed method, with identical results.

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