Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization

2020 ◽  
Author(s):  
Giovanni Paolo Crespi ◽  
Andreas H. Hamel ◽  
Matteo Rocca ◽  
Carola Schrage
Keyword(s):  
Complete Lattice ◽  
Optimization Problems ◽  
Multicriteria Decision Making ◽  
Approximate Solutions ◽  
Set Optimization ◽  
Mathematical Economics ◽  
New Approach ◽  
Power Set ◽  
Preordered Set ◽  
Set Relations

Via a family of monotone scalar functions, a preorder on a set is extended to its power set and then used to construct a hull operator and a corresponding complete lattice of sets. Functions mapping into the preordered set are extended to complete lattice-valued ones, and concepts for exact and approximate solutions for corresponding set optimization problems are introduced and existence results are given. Well-posedness for complete lattice-valued problems is introduced and characterized. The new approach is compared with existing ones in vector and set optimization. Its relevance is shown by means of many examples from multicriteria decision making, statistics, and mathematical economics and finance.

scholarly journals A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

2021 ◽  
Author(s):  
Gemayqzel Bouza ◽  
Ernest Quintana ◽  
Christiane Tammer
Keyword(s):  
Optimality Conditions ◽  
Solution Method ◽  
Optimization Problems ◽  
Solution Concept ◽  
Convergence Result ◽  
Descent Method ◽  
Set Optimization ◽  
Steepest Descent Method ◽  
Uncertainty Set ◽  
Set Relations

AbstractIn this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020).

scholarly journals An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 143
Author(s):  
Elisabeth Köbis ◽  
Markus A. Köbis ◽  
Xiaolong Qin
Keyword(s):  
Numerical Methods ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Set Optimization ◽  
Linear Spaces ◽  
Minimal Elements ◽  
Real Linear

This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality approach. As particular cases, we investigate the use of the prominent Tammer–Weidner nonlinear scalarizing functionals, without assuming any topology, in our context. We also derive numerical methods to obtain approximate minimal elements of families of finitely many sets by means of our obtained results.

Qualitative properties of solutions to set optimization problems

2021 ◽  
Vol 40 (2) ◽  
Author(s):  
Lam Quoc Anh ◽  
Nguyen Huu Danh ◽  
Pham Thanh Duoc ◽  
Tran Ngoc Tam
Keyword(s):  
Optimization Problems ◽  
Set Optimization ◽  
Qualitative Properties ◽  
Qualitative Properties Of Solutions

scholarly journals Construction of Consistent Comparison Matrix by Macharis’ Method Revisit

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Yi-Fong Lin
Keyword(s):  
Recent Article ◽  
Multicriteria Decision Making ◽  
Preference Ranking ◽  
Original Design ◽  
Analytic Hierarchy ◽  
New Approach ◽  
Comparison Matrix ◽  
Famous Paper ◽  
Hierarchy Process ◽  
Original Construction

A famous paper that has been cited more than four hundred times tried to combine (a) the preference ranking organization method for enrichment evaluations (PROMETHEE) and (b) the analytic hierarchy process (AHP) to construct a new method for multicriteria decision-making problems. The paper developed a consistent comparison matrix for their AHP by the defined first row and then they allowed the expert to change several entries in the comparison matrix. Hence, how to construct a new comparison matrix that is (i) consistent and (ii) satisfying the assigned values by the expert becomes a challenging problem. A recent article provided a reply to the above problem by the construction of all entries for the comparison matrix. However, they did not follow the original design proposed by the famous paper. In this paper, we present a new approach with a proposition that satisfies the original design of the famous paper and also achieves two goals (i) and (ii). The research gap of proof is fulfilled by this paper. Our findings explain that the original construction of the famous paper to develop a consistent comparison matrix only by the first row with several assigned values by an expert is indeed workable under two additional restrictions proposed by the recent article. We believe that after our proposition, researchers have the confidence to execute the original design of the paper that has been cited more than four hundred times.

scholarly journals A Dai-Liao Hybrid Hestenes-Stiefel and Fletcher-Revees Methods for Unconstrained Optimization

2021 ◽  
Vol 2 (1) ◽  
pp. 33
Author(s):  
Nasiru Salihu ◽  
Mathew Remilekun Odekunle ◽  
Also Mohammed Saleh ◽  
Suraj Salihu
Keyword(s):  
Unconstrained Optimization ◽  
Line Search ◽  
Optimization Problems ◽  
Convex Combination ◽  
Gradient Methods ◽  
Approximate Solutions ◽  
Step Size ◽  
Conjugacy Condition ◽  
Inexact Line Search ◽  
Wolfe Line Search

Some problems have no analytical solution or too difficult to solve by scientists, engineers, and mathematicians, so the development of numerical methods to obtain approximate solutions became necessary. Gradient methods are more efficient when the function to be minimized continuously in its first derivative. Therefore, this article presents a new hybrid Conjugate Gradient (CG) method to solve unconstrained optimization problems. The method requires the first-order derivatives but overcomes the steepest descent method’s shortcoming of slow convergence and needs not to save or compute the second-order derivatives needed by the Newton method. The CG update parameter is suggested from the Dai-Liao conjugacy condition as a convex combination of Hestenes-Stiefel and Fletcher-Revees algorithms by employing an optimal modulating choice parameterto avoid matrix storage. Numerical computation adopts an inexact line search to obtain the step-size that generates a decent property, showing that the algorithm is robust and efficient. The scheme converges globally under Wolfe line search, and it’s like is suitable in compressive sensing problems and M-tensor systems.

On quasi approximate solutions for nonsmooth robust semi-infinite optimization problems

2019 ◽  
Vol 35 (3) ◽  
pp. 417-426 ◽  
Author(s):  
CHANOKSUDA KHANTREE ◽  
RABIAN WANGKEEREE ◽  
◽  
Keyword(s):  
Constraint Qualification ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Data Uncertainty ◽  
Necessary Condition ◽  
Duality Theorems ◽  
Approximate Form ◽  
Locally Lipschitz ◽  
Approximate Duality

This paper devotes to the quasi ε-solution for robust semi-infinite optimization problems (RSIP) involving a locally Lipschitz objective function and infinitely many locally Lipschitz constraint functions with data uncertainty. Under the fulfillment of robust type Guignard constraint qualification and robust type Kuhn-Tucker constraint qualification, a necessary condition for a quasi ε-solution to problem (RSIP). After introducing the generalized convexity, we give a sufficient optimality for such a quasi ε-solution to problem (RSIP). Finally, we also establish approximate duality theorems in term of Wolfe type which is formulated in approximate form.

Sign in / Sign up

Export Citation Format

Share Document