scholarly journals ϵ-Henig Saddle Points and Duality of Set-Valued Optimization Problems in Real Linear Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Zhi-Ang Zhou
Keyword(s):  
Saddle Point ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Saddle Points ◽  
Linear Spaces ◽  
Equivalent Characterization ◽  
Generalized Cone Subconvexlikeness ◽  
Real Linear ◽  
The Relationship

We studyϵ-Henig saddle points and duality of set-valued optimization problems in the setting of real linear spaces. Firstly, an equivalent characterization ofϵ-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between theϵ-Henig saddle point of the Lagrangian set-valued map and theϵ-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.

Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1168-1182
Author(s):  
Hongwei Liang ◽  
Zhongping Wan
Keyword(s):  
Optimization Problem ◽  
Saddle Points ◽  
Sufficient Conditions ◽  
Proper Efficiency ◽  
Solid Cone ◽  
Linear Spaces ◽  
New Class ◽  
Benson Proper Efficiency ◽  
Real Linear ◽  
Necessary And Sufficient

Abstract A new class of generalized convex set-valued maps termed relatively solid generalized cone-subconvexlike maps is introduced in real linear spaces not equipped with any topology. This class is a generalization of generalized cone-subconvexlike maps and relatively solid cone-subconvexlike maps. Necessary and sufficient conditions for Benson proper efficiency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality. The results generalize and improve some corresponding ones in the literature. Some examples are afforded to illustrate our results.

scholarly journals A Generalized Alternative Theorem of Partial and Generalized Cone Subconvexlike Set-Valued Maps and Its Applications in Linear Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Zhi-Ang Zhou ◽  
Jian-Wen Peng
Keyword(s):  
Optimization Problems ◽  
Proper Efficiency ◽  
Linear Spaces ◽  
Alternative Theorem ◽  
Equivalent Characterization

We first introduce a new notion of the partial and generalized cone subconvexlike set-valued map and give an equivalent characterization of the partial and generalized cone subconvexlike set-valued map in linear spaces. Secondly, a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map was presented. Finally, Kuhn-Tucker conditions of set-valued optimization problems were established in the sense of globally proper efficiency.

scholarly journals METHODS OF FORMULATING AND SOLVING OPTIMIZATION PROBLEMS OF CUTTING LOGS OF LARGE SIZE BY BAR-SAWING METHOD WITH CUTTING OUT ONE BAR AND FIVE PAIRS OF SIDE EDGING BOARDS WITH SUBSEQUENT SAWING LUMBER FOR EDGED BOARDS

2017 ◽  
Vol 7 (1) ◽  
pp. 137-150
Author(s):  
Агапов ◽  
Aleksandr Agapov
Keyword(s):  
Mathematical Model ◽  
Objective Function ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Target Function ◽  
Cross Sectional ◽  
Optimization Task ◽  
Cutting Out ◽  
The Mathematical Model ◽  
The Relationship

For the first time the mathematical model of task optimization for this scheme of cutting logs, including the objective function and six equations of connection. The article discusses Pythagorean area of the logs. Therefore, the target function is represented as the sum of the cross-sectional areas of edging boards. Equation of the relationship represents the relationship of the diameter of the logs in the vertex end with the size of the resulting edging boards. This relationship is described through the use of the Pythagorean Theorem. Such a representation of the mathematical model of optimization task is considered a classic one. However, the solution of this mathematical model by the classic method is proved to be problematic. For the solution of the mathematical model we used the method of Lagrange multipliers. Solution algorithm to determine the optimal dimensions of the beams and side edging boards taking into account the width of cut is suggested. Using a numerical method, optimal dimensions of the beams and planks are determined, in which the objective function takes the maximum value. It turned out that with the increase of the width of the cut, thickness of the beam increases and the dimensions of the side edging boards reduce. Dimensions of the extreme side planks to increase the width of cut is reduced to a greater extent than the side boards, which are located closer to the center of the log. The algorithm for solving the optimization problem is recommended to use for calculation and preparation of sawing schedule in the design and operation of sawmill lines for timber production. When using the proposed algorithm for solving the optimization problem the output of lumber can be increased to 3-5 %.

scholarly journals On Well-Founded and Recursive Coalgebras

2020 ◽  
pp. 17-36
Author(s):  
Jiří Adámek ◽  
Stefan Milius ◽  
Lawrence S. Moss
Keyword(s):  
General Proof ◽  
Initial Algebra ◽  
Equivalent Characterization ◽  
Algebra Morphism ◽  
Recursion Theorem ◽  
The Relationship ◽  
Well Foundedness

AbstractThis paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor’s General Recursion Theorem that every well-founded coalgebra is recursive, and we study conditions which imply the converse. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.

A saddle point characterization of efficient solutions for interval optimization problems

2017 ◽  
Vol 58 (1-2) ◽  
pp. 193-217 ◽  
Author(s):  
Debdulal Ghosh ◽  
Debdas Ghosh ◽  
Sushil Kumar Bhuiya ◽  
Lakshmi Kanta Patra
Keyword(s):  
Saddle Point ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Interval Optimization

scholarly journals Optimization problems, first order approximated optimization problems and their connections

2012 ◽  
Vol 28 (1) ◽  
pp. 133-141
Author(s):  
EMILIA-LOREDANA POP ◽  
◽  
DOREL I. DUCA ◽  
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Saddle Points ◽  
Optimal Solutions ◽  
First Order

In this paper, we attach to the optimization problem ... where X is a subset of Rn, f : X → R, g = (g1, ..., gm) : X → Rm and h = (h1, ..., hq) : X → Rq are functions, the (0, 1) − η− approximated optimization problem (AP). We will study the connections between the optimal solutions for Problem (AP), the saddle points for Problem (AP), optimal solutions for Problem (P) and saddle points for Problem (P).

scholarly journals A new approach to optimality in a class of nonconvex smooth optimization problems

2018 ◽  
Vol 34 (1) ◽  
pp. 01-07
Author(s):  
TADEUSZ ANTCZAK ◽  
Keyword(s):  
Approximation Method ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Extremum Problem ◽  
Optimal Solutions ◽  
New Approach ◽  
Smooth Optimization

In this paper, a new approximation method for a characterization of optimal solutions in a class of nonconvex differentiable optimization problems is introduced. In this method, an auxiliary optimization problem is constructed for the considered nonconvex extremum problem. The equivalence between optimal solutions in the considered differentiable extremum problem and its approximated optimization problem is established under (Φ, ρ)-invexity hypotheses.

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