scholarly journals A duality theorem for a multiple objective fractional optimization problem

1986 ◽  
Vol 34 (3) ◽  
pp. 415-425 ◽  
Author(s):  
T. Weir
Keyword(s):  
Duality Theorem ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Proper Efficiency ◽  
Dual Solution ◽  
Multiple Objective ◽  
Fractional Optimization ◽  
Linear Problems

Characterizations of efficiency and proper efficiency are given for classes of multiple objective fractional optimization problems. These results are then applied to the case of multiple objective fractional linear problems. A dual problem is given for the multiple objective fractional problem and it is shown that for a properly efficient primal solution the dual solution is also properly efficient.

scholarly journals Mixed E-duality for E-differentiable Vector Optimization Problems Under (Generalized) V-E-invexity

2021 ◽  
Vol 2 (3) ◽  
Author(s):  
Najeeb Abdulaleem
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Equality Constraints ◽  
Vector Optimization Problems ◽  
Duality Theorems ◽  
Differentiable Vector

AbstractIn this paper, a class of E-differentiable vector optimization problems with both inequality and equality constraints is considered. The so-called vector mixed E-dual problem is defined for the considered E-differentiable vector optimization problem with both inequality and equality constraints. Then, several mixed E-duality theorems are established under (generalized) V-E-invexity hypotheses.

scholarly journals A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 372
Author(s):  
Liu He ◽  
Qi-Lin Wang ◽  
Ching-Feng Wen ◽  
Xiao-Yan Zhang ◽  
Xiao-Bing Li
Keyword(s):  
Existence Theorem ◽  
Lower Order ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Strong Duality ◽  
Higher Order ◽  
Weak Duality ◽  
Duality Theorems ◽  
Benson Proper Efficiency

In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the higher-order Mond-Weir type dual problem for a set-valued optimization problem and obtain the corresponding weak duality, strong duality and converse duality theorems, respectively.

scholarly journals The Use of the Duality Principle to Solve Optimization Problems

2018 ◽  
Vol 6 (1) ◽  
pp. 33
Author(s):  
Rowland Jerry Okechukwu Ekeocha ◽  
Chukwunedum Uzor ◽  
Clement Anetor
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Linear Program ◽  
Duality Gap ◽  
Duality Principle ◽  
Primal Problem ◽  
Optimal Values ◽  
Primal And Dual Problems ◽  
Primal And Dual

<p><span>The duality principle provides that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.<span>  </span>In other words given any linear program, there is another related linear program called the dual. In this paper, an understanding of the dual linear program will be developed. This understanding will give important insights into the algorithm and solution of optimization problem in linear programming. <span> </span>Thus the main concepts of duality will be explored by the solution of simple optimization problem.</span></p>

scholarly journals On G-invexity-type nonlinear programming problems

2015 ◽  
Vol 5 (1) ◽  
pp. 13-20
Author(s):  
Tadeusz Antczak ◽  
Manuel Arana Jiménez
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Extremum Problem ◽  
Sufficient Optimality Conditions ◽  
New Class ◽  
Necessary And Sufficient ◽  
Wolfe Dual

In this paper, we introduce the concepts of KT-G-invexity and WD$-G-invexity for the considered differentiable optimization problem with inequality constraints. Using KT-G-invexity notion, we prove new necessary and sufficient optimality conditions for a new class of such nonconvex differentiable optimization problems. Further, the so-called G-Wolfe dual problem is defined for the considered extremum problem with inequality constraints. Under WD-G-invexity assumption, the necessary and sufficient conditions for weak duality between the primal optimization problem and its G-Wolfe dual problem are also established.

scholarly journals Approximate proper efficiency for multiobjective optimization problems

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 6091-6101
Author(s):  
Ying Gao ◽  
Zhihui Xu
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problem ◽  
Normal Cone ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Proper Efficiency ◽  
Proximal Normal Cone ◽  
Radiant Set ◽  
Benson Proper Efficiency ◽  
Multiobjective Optimization Problems

This paper is devoted to the study of a new kind of approximate proper efficiency in terms of proximal normal cone and co-radiant set for multiobjective optimization problem. We derive some properties of the new approximate proper efficiency and discuss the relations with the existing approximate concepts, such as approximate efficiency and approximate Benson proper efficiency. At last, we study the linear scalarizations for the new approximate proper efficiency under the generalized convexity assumption and give some examples to illustrate the main results.

scholarly journals Pareto Optimality Conditions and Duality for Vector Quadratic Fractional Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
W. A. Oliveira ◽  
A. Beato-Moreno ◽  
A. C. Moretti ◽  
L. L. Salles Neto
Keyword(s):  
Optimality Conditions ◽  
Pareto Optimality ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Sufficient Optimality Conditions ◽  
Objective Functions ◽  
Convex Constraints ◽  
Fractional Optimization ◽  
Convexity Assumptions

One of the most important optimality conditions to aid in solving a vector optimization problem is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker condition. However, to obtain the sufficient optimality conditions, it is necessary to impose additional assumptions on the objective functions and on the constraint set. The present work is concerned with the constrained vector quadratic fractional optimization problem. It shows that sufficient Pareto optimality conditions and the main duality theorems can be established without the assumption of generalized convexity in the objective functions, by considering some assumptions on a linear combination of Hessian matrices instead. The main aspect of this contribution is the development of Pareto optimality conditions based on a similar second-order sufficient condition for problems with convex constraints, without convexity assumptions on the objective functions. These conditions might be useful to determine termination criteria in the development of algorithms.

scholarly journals Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators

2022 ◽  
Vol 12 (1) ◽  
pp. 93
Author(s):  
Jutamas Kerdkaew ◽  
Rabian Wangkeeree ◽  
Rattanaporn Wangkeeree
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Data Uncertainty ◽  
Global Optimality ◽  
Optimal Solutions ◽  
Duality Theorems ◽  
Uncertain Optimization ◽  
Kkt Point ◽  
Robust Duality

<p style='text-indent:20px;'>In this paper, a robust optimization problem, which features a maximum function of continuously differentiable functions as its objective function, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisfies the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were first introduced by Jayakumar and Srisatkunarajah [<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.</p>

scholarly journals A Method for Integration of Preferences to a Multi-Objective Evolutionary Algorithm Using Ordinal Multi-Criteria Classification

2021 ◽  
Vol 26 (2) ◽  
pp. 27
Author(s):  
Alejandro Castellanos-Alvarez ◽  
Laura Cruz-Reyes ◽  
Eduardo Fernandez ◽  
Nelson Rangel-Valdez ◽  
Claudia Gómez-Santillán ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Selective Pressure ◽  
Optimization Problems ◽  
Region Of Interest ◽  
Decision Makers ◽  
Multiple Objective ◽  
Objective Functions ◽  
Multi Objective ◽  
Imperfect Knowledge ◽  
Real World Problems

Most real-world problems require the optimization of multiple objective functions simultaneously, which can conflict with each other. The environment of these problems usually involves imprecise information derived from inaccurate measurements or the variability in decision-makers’ (DMs’) judgments and beliefs, which can lead to unsatisfactory solutions. The imperfect knowledge can be present either in objective functions, restrictions, or decision-maker’s preferences. These optimization problems have been solved using various techniques such as multi-objective evolutionary algorithms (MOEAs). This paper proposes a new MOEA called NSGA-III-P (non-nominated sorting genetic algorithm III with preferences). The main characteristic of NSGA-III-P is an ordinal multi-criteria classification method for preference integration to guide the algorithm to the region of interest given by the decision-maker’s preferences. Besides, the use of interval analysis allows the expression of preferences with imprecision. The experiments contrasted several versions of the proposed method with the original NSGA-III to analyze different selective pressure induced by the DM’s preferences. In these experiments, the algorithms solved three-objectives instances of the DTLZ problem. The obtained results showed a better approximation to the region of interest for a DM when its preferences are considered.

ONE IMPROVED AGENT GENETIC ALGORITHM — RING-LIKE AGENT GENETIC ALGORITHM FOR GLOBAL NUMERICAL OPTIMIZATION

2009 ◽  
Vol 26 (04) ◽  
pp. 479-502 ◽  
Author(s):  
BIN LIU ◽  
TEQI DUAN ◽  
YONGMING LI
Keyword(s):  
Genetic Algorithm ◽  
Numerical Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Genetic Operators ◽  
Global Numerical Optimization ◽  
Candidate Solution ◽  
Low Dimension ◽  
Multi Agent ◽  
Agent Structure

In this paper, a novel genetic algorithm — dynamic ring-like agent genetic algorithm (RAGA) is proposed for solving global numerical optimization problem. The RAGA combines the ring-like agent structure and dynamic neighboring genetic operators together to get better optimization capability. An agent in ring-like agent structure represents a candidate solution to the optimization problem. Any agent interacts with neighboring agents to evolve. With dynamic neighboring genetic operators, they compete and cooperate with their neighbors, and they can also use knowledge to increase energies. Global numerical optimization problems are the most important ones to verify the performance of evolutionary algorithm, especially of genetic algorithm and are mostly of interest to the corresponding researchers. In the corresponding experiments, several complex benchmark functions were used for optimization, several popular GAs were used for comparison. In order to better compare two agents GAs (MAGA: multi-agent genetic algorithm and RAGA), the several dimensional experiments (from low dimension to high dimension) were done. These experimental results show that RAGA not only is suitable for optimization problems, but also has more precise and more stable optimization results.

A Dynamics-Based Optimal Trajectory Generation for Controlling an Automated Excavator

2010 ◽  
Vol 224 (10) ◽  
pp. 2109-2119 ◽  
Author(s):  
S Yoo ◽  
C-G Park ◽  
S-H You ◽  
B Lim
Keyword(s):  
Optimal Trajectory ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Linear Constraints ◽  
Control Points ◽  
Weighting Functions ◽  
B Splines ◽  
Finite Dimensional ◽  
Motion Optimization ◽  
Save Energy

This article presents a new methodology to generate optimal trajectories in controlling an automated excavator. By parameterizing all the actuator displacements with B-splines of the same order and with the same number of control points, the coupled actuator limits, associated with the maximum pump flowrate, are described as the finite-dimensional set of linear constraints to the motion optimization problem. Several weighting functions are introduced on the generalized actuator torque so that the solution to each optimization problems contains the physical meaning. Numerical results showing that the generated motions of the excavator are fairly smooth and effectively save energy, which can prevent mechanical wearing and possibly save fuel consumption, are presented. A typical operator's manoeuvre from experiments is referred to bring out the standing features of the optimized motion.

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