scholarly journals Optimality and Duality with Respect to b-(E,m)-Convex Programming

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 774
Author(s):  
Bo Yu ◽  
Jiagen Liao ◽  
Tingsong Du
Keyword(s):  
Convex Programming ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Optimality Criteria ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Symmetric Duality ◽  
Convex Mappings ◽  
Uniqueness Of The Solution ◽  
Optimality And Duality

Noticing that E -convexity, m-convexity and b-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the ( E , m ) -convex sets and the b- ( E , m ) -convex mappings are introduced. The properties concerning operations that preserve the ( E , m ) -convexity of the proposed mappings are derived. The unconstrained and inequality constrained b- ( E , m ) -convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b- ( E , m ) -convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz–John necessary optimality criteria for nonlinear multi-objective b- ( E , m ) -convex programming are established. The Wolfe-type symmetric duality theorems under the b- ( E , m ) -convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in b- ( E , m ) -convex programming.

scholarly journals Optimality and duality for nonsmooth semi-infinite E-convex multi-objective programming with support functions

2020 ◽  
Vol 11 ◽  
pp. 17
Author(s):  
Tarek Emam
Keyword(s):  
Strong Duality ◽  
Sufficient Optimality Conditions ◽  
Convex Programming Problem ◽  
Duality Theorems ◽  
Support Functions ◽  
Primal Problem ◽  
Multi Objective ◽  
Multi Objective Programming ◽  
Optimality And Duality ◽  
Convexity Assumptions

In this paper, we study a nonsmooth semi-infinite multi-objective E-convex programming problem involving support functions. We derive sufficient optimality conditions for the primal problem. We formulate Mond-Weir type dual for the primal problem and establish weak and strong duality theorems under various generalized E-convexity assumptions.

scholarly journals Optimality and Duality for Nondifferentiable Minimax Fractional Programming with Generalized Convexity

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Anurag Jayswal
Keyword(s):  
Fractional Programming ◽  
Generalized Convexity ◽  
Optimality Criteria ◽  
Sufficient Optimality Conditions ◽  
View Point ◽  
Generalized Convex Functions ◽  
Duality Results ◽  
Minimax Fractional Programming ◽  
Optimality And Duality ◽  
Dual Models

We establish several sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems from a view point of generalized convexity. Subsequently, these optimality criteria are utilized as a basis for constructing dual models, and certain duality results have been derived in the framework of generalized convex functions. Our results extend and unify some known results on minimax fractional programming problems.

scholarly journals Optimality and duality without a constraint qualification for minimax programming

2003 ◽  
Vol 67 (1) ◽  
pp. 121-130 ◽  
Author(s):  
Houchun Zhou ◽  
Wenyu Sun
Keyword(s):  
Optimality Conditions ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dual Model ◽  
Duality Theorems ◽  
Minimax Programming ◽  
Optimality And Duality ◽  
Necessary And Sufficient

Without the need of a constraint qualification, we establish the optimality necessary and sufficient conditions for generalised minimax programming. Using these optimality conditions, we construct a parametric dual model and a parameter-free mixed dual model. Several duality theorems are established.

scholarly journals Sufficient Conditions and Duality Theorems for Nondifferentiable Minimax Fractional Programming

2011 ◽  
Vol 2011 ◽  
pp. 1-22
Author(s):  
Shun-Chin Ho
Keyword(s):  
Optimality Conditions ◽  
Fractional Programming ◽  
Sufficient Conditions ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Invex Functions ◽  
Duality Results ◽  
Wolfe Duality ◽  
Minimax Fractional Programming

We consider nondifferentiable minimax fractional programming problems involving B-(p, r)-invex functions with respect to η and b. Sufficient optimality conditions and duality results for a class of nondifferentiable minimax fractional programming problems are obtained undr B-(p, r)-invexity assumption on objective and constraint functions. Parametric duality, Mond-Weir duality, and Wolfe duality problems may be formulated, and duality results are derived under B-(p, r)-invex functions.

scholarly journals Optimality and duality for nonsmooth semi-infinite multiobjective programming with support functions

2017 ◽  
Vol 27 (2) ◽  
pp. 205-218 ◽  
Author(s):  
Yadvendra Singh ◽  
S.K. Mishra ◽  
K.K. Lai
Keyword(s):  
Generalized Convexity ◽  
Multiobjective Programming ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Support Functions ◽  
Primal Problem ◽  
Special Cases ◽  
Multiobjective Programming Problem ◽  
Optimality And Duality ◽  
Convexity Assumptions

In this paper, we consider a nonsmooth semi-infinite multiobjective programming problem involving support functions. We establish sufficient optimality conditions for the primal problem. We formulate Mond-Weir type dual for the primal problem and establish weak, strong and strict converse duality theorems under various generalized convexity assumptions. Moreover, some special cases of our problem and results are presented.

scholarly journals On semidifferentiable interval-valued programming problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kin Keung Lai ◽  
Avanish Shahi ◽  
Shashi Kant Mishra
Keyword(s):  
Saddle Point ◽  
Optimality Conditions ◽  
Minimization Problem ◽  
Optimal Solution ◽  
Optimality Criteria ◽  
Lagrangian Function ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Preinvex Functions ◽  
Interval Valued

AbstractIn this paper, we consider the semidifferentiable case of an interval-valued minimization problem and establish sufficient optimality conditions and Wolfe type as well as Mond–Weir type duality theorems under semilocal E-preinvex functions. Furthermore, we present saddle-point optimality criteria to relate an optimal solution of the semidifferentiable interval-valued programming problem and a saddle point of the Lagrangian function.

scholarly journals ON OPTIMALITY AND DUALITY IN INTERVAL-VALUED VARIATIONAL PROBLEM WITH B-(p,r)-INVEXITY

2021 ◽  
Author(s):  
Indira Priyadarshini Debnath ◽  
Nisha Pokharna
Keyword(s):  
Variational Problem ◽  
Optimization Problem ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Real World Problem ◽  
Sufficiency Theorem ◽  
Optimality And Duality ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Interval Valued

In this paper, we consider a class of interval-valued variational optimization problem. We extend the definition of B -( p,r )- invexity which was originally defined for scalar optimization problem to the interval-valued variational problem. The necessary and sufficient optimality conditions for the problem have been established under B -( p,r )-invexity assumptions. An application, showing utility of the sufficiency theorem in real-world problem, has also been provided. In addition to this, for an interval- optimization problem Mond-Weir and Wolfe type duals are presented and related duality theorems have been proved. Non-trivial examples verifying the results have also been presented throughout the paper.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 529-539
Author(s):  
Xianghu Liu
Keyword(s):  
Approximate Controllability ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Evolution System ◽  
Controlled Systems ◽  
Uniqueness Of The Solution ◽  
Fractional Evolution Systems ◽  
Evolution Systems

Abstract The aim of this study is to investigate the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. To achieve this, we first transform the mild solution of the Hilfer fractional evolution system into a fixed point problem for a condensing map. Then, by using the topological degree approach, we present sufficient conditions for the existence and uniqueness of the solution of the Hilfer fractional evolution systems. Using the variational approach, we obtain sufficient conditions for the finite approximate controllability of semilinear controlled systems. Finally, an example is provided to illustrate main results.

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