scholarly journals A notion of local proper efficiency in the Borwein sense in vector optimisation

2003 ◽  
Vol 45 (1) ◽  
pp. 75-89 ◽  
Author(s):  
B. Jiménez ◽  
V. Novo
Keyword(s):  
Efficient Solutions ◽  
Proper Efficiency ◽  
Proper Solution ◽  
Sufficient Condition ◽  
Strict Efficiency ◽  
Proper Efficient Solutions

AbstractIn this paper we define two types of proper efficient solutions in the Borwein sense for vector optimisation problems and we compare them with the notions of local Borwein, Ishizuka-Tuan, Kuhn-Tucker and strict efficiency. A sufficient condition for a proper solution is also proved.

scholarly journals Second-Order Optimality Conditions for Set-Valued Optimization Problems Under Benson Proper Efficiency

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Qilin Wang ◽  
Guolin Yu
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Efficient Solutions ◽  
Second Order ◽  
Proper Efficiency ◽  
Second Order Optimality Conditions ◽  
Benson Proper Efficiency ◽  
Necessary And Sufficient ◽  
Proper Efficient Solutions

Some new properties are obtained for generalized second-order contingent (adjacent) epiderivatives of set-valued maps. By employing the generalized second-order adjacent epiderivatives, necessary and sufficient conditions of Benson proper efficient solutions are given for set-valued optimization problems. The results obtained improve the corresponding results in the literature.

scholarly journals Optimality criteria for deterministic discrete-time infinite horizon optimization

2005 ◽  
Vol 2005 (1) ◽  
pp. 57-80 ◽  
Author(s):  
Irwin E. Schochetman ◽  
Robert L. Smith
Keyword(s):  
Discrete Time ◽  
Efficient Solution ◽  
Optimality Criterion ◽  
Infinite Horizon ◽  
Optimality Criteria ◽  
Efficient Solutions ◽  
Regularity Conditions ◽  
Sufficient Condition ◽  
Infinite Horizon Optimization ◽  
Continuous State

We consider the problem of selecting an optimality criterion, when total costs diverge, in deterministic infinite horizon optimization over discrete time. Our formulation allows for both discrete and continuous state and action spaces, as well as time-varying, that is, nonstationary, data. The task is to choose a criterion that is neither too overselective, so thatnopolicy is optimal, nor too underselective, so thatmostpolicies are optimal. We contrast and compare the following optimality criteria: strong, overtaking, weakly overtaking, efficient, and average. However, our focus is on the optimality criterion of efficiency. (A solution isefficientif it is optimal to each of the states through which it passes.) Under mild regularity conditions, we show that efficient solutions always exist and thus are not overselective. As to underselectivity, we provide weak state reachability conditions which assure that every efficient solution is also average optimal, thus providing a sufficient condition for average optima to exist. Our main result concerns the case where the discounted per-period costs converge to zero, while the discounted total costs diverge to infinity. Under the assumption that we can reach from any feasible state any feasible sequence of states in bounded time, we show that every efficient solution is also overtaking, thus providing a sufficient condition for overtaking optima to exist.

scholarly journals Strict Efficiency in Vector Optimization with Nearly Convexlike Set-Valued Maps

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Xiaohong Hu ◽  
Zhimiao Fang ◽  
Yunxuan Xiong
Keyword(s):  
Saddle Point ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Well Posedness ◽  
Scalar Problem ◽  
New Type ◽  
Strict Efficiency ◽  
Multiplier Theorems

The concept of the well posedness for a special scalar problem is linked with strictly efficient solutions of vector optimization problem involving nearly convexlike set-valued maps. Two scalarization theorems and two Lagrange multiplier theorems for strict efficiency in vector optimization involving nearly convexlike set-valued maps are established. A dual is proposed and duality results are obtained in terms of strictly efficient solutions. A new type of saddle point, called strict saddle point, of an appropriate set-valued Lagrange map is introduced and is used to characterize strict efficiency.

scholarly journals Multiobjective fractional programming problems and the sufficient condition involving Hb – (p, r)-η- invex function

2021 ◽  
Vol 336 ◽  
pp. 08015
Author(s):  
Xiaoyan Gao ◽  
Huan Niu
Keyword(s):  
Convex Functions ◽  
Fractional Programming ◽  
Efficient Solutions ◽  
Sufficient Condition ◽  
Sufficient Optimality Condition ◽  
Multiobjective Fractional Programming ◽  
Invex Functions ◽  
Arcwise Connected ◽  
Feasible Solutions ◽  
Invex Function

On the basis of arcwise connected convex functions and (p, r) −η - invex functions, we established Hb –(p, r) –η- invex functions. Based on the generalized invex assumption of new functions, the solutions of a class of multiobjective fractional programming problems are studied, and the sufficient optimality condition for the feasible solutions of multiobjective fractional programming problems to be efficient solutions are established and proved.

Strongly Proper Efficient Solutions: Efficient Solutions with Bounded Trade-Offs

2015 ◽  
Vol 168 (3) ◽  
pp. 864-883 ◽  
Author(s):  
Kazhal Khaledian ◽  
Esmaile Khorram ◽  
Majid Soleimani-damaneh
Keyword(s):  
Efficient Solutions ◽  
Trade Offs ◽  
Proper Efficient Solutions
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