scholarly journals Optimality conditions for set-valued optimization problems via scalarization function

2020 ◽  
Vol 3 (SI3) ◽  
pp. first
Author(s):  
Manh Linh Ha ◽  
Dinh Huy Nguyen ◽  
Thi Thanh Truc Nguyen
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Pareto Optimal ◽  
Optimal Solutions ◽  
Science And Engineering ◽  
Primal Problem ◽  
Multi Objective ◽  
Scalarization Method ◽  
Close Relationship ◽  
Scalarization Function

One of the most important and popular topics in optimization problems is to find its optimal solutions, especially Pareto optimal points, a well-known solution introduced in multi-objective optimization. This topic is one of the oldest challenges in many issues related to science, engineering and other fields. Many important practical-problems in science and engineering can be expressed in terms of multi-objective/ set-valued optimization problems in order to achieve the proper results/ properties. To find the Pareto solutions, a corresponding scalarization problem has been established and studied. The relationships between the primal problem and its scalarization one should be investigated for finding optimal solutions. It can be shown that, under some suitable conditions, the solutions of the corresponding scalarization problem have uniform spread and have a close relationship to Pareto optimal solutions for the primal one. Scalarization has played an essential role in studying not only numerical methods but also duality theory. It can be usefully applied to get relationships/ important results between other fields, for example optimization, convex analysis and functional analysis. In scalarization, we ussually use a kind of scalarized-functions. One of the first and the most popular scalarized-functions used in scalarization method is the Gerstewitz function. In the paper, we mention some problems in set-valued optimization. Then, we propose an application of the Gerstewitz function to these problems. In detail, we establish some optimality conditions for Pareto/ weak solutions of unconstrained/ constrained set-valued optimization problems by using the Gerstewitz function. The study includes the consideration of problems in theoretical approach. Some examples are given to illustrate the obtained results.

Evaluating the ε-Domination Based Multi-Objective Evolutionary Algorithm for a Quick Computation of Pareto-Optimal Solutions

2005 ◽  
Vol 13 (4) ◽  
pp. 501-525 ◽  
Author(s):  
Kalyanmoy Deb ◽  
Manikanth Mohan ◽  
Shikhar Mishra
Keyword(s):  
Steady State ◽  
Evolutionary Algorithm ◽  
Optimization Problems ◽  
Computational Time ◽  
Test Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Nsga Ii ◽  
Multi Objective

Since the suggestion of a computing procedure of multiple Pareto-optimal solutions in multi-objective optimization problems in the early Nineties, researchers have been on the look out for a procedure which is computationally fast and simultaneously capable of finding a well-converged and well-distributed set of solutions. Most multi-objective evolutionary algorithms (MOEAs) developed in the past decade are either good for achieving a well-distributed solutions at the expense of a large computational effort or computationally fast at the expense of achieving a not-so-good distribution of solutions. For example, although the Strength Pareto Evolutionary Algorithm or SPEA (Zitzler and Thiele, 1999) produces a much better distribution compared to the elitist non-dominated sorting GA or NSGA-II (Deb et al., 2002a), the computational time needed to run SPEA is much greater. In this paper, we evaluate a recently-proposed steady-state MOEA (Deb et al., 2003) which was developed based on the ε-dominance concept introduced earlier (Laumanns et al., 2002) and using efficient parent and archive update strategies for achieving a well-distributed and well-converged set of solutions quickly. Based on an extensive comparative study with four other state-of-the-art MOEAs on a number of two, three, and four objective test problems, it is observed that the steady-state MOEA is a good compromise in terms of convergence near to the Pareto-optimal front, diversity of solutions, and computational time. Moreover, the ε-MOEA is a step closer towards making MOEAs pragmatic, particularly allowing a decision-maker to control the achievable accuracy in the obtained Pareto-optimal solutions.

Solving Two Multi-Objective Optimization Problems Using Evolutionary Algorithm

2011 ◽  
pp. 218-233 ◽  
Author(s):  
Ruhul A. Sarker ◽  
Hussein A. Abbass ◽  
Charles S. Newton
Keyword(s):  
Evolutionary Algorithms ◽  
Optimization Problems ◽  
Benchmark Problems ◽  
Pareto Optimal ◽  
Multi Criteria Decision Making ◽  
Pareto Optimal Solutions ◽  
Assessment Methodology ◽  
Optimal Solutions ◽  
Multi Objective Optimization ◽  
Multi Objective

Being capable of finding a set of pareto-optimal solutions in a single run is a necessary feature for multi-criteria decision making, Evolutionary algorithms (EAs) have attracted many researchers and practitioners to address the solution of Multi-objective Optimization Problems (MOPs). In a previous work, we developed a Pareto Differential Evolution (PDE) algorithm to handle multi-objective optimization problems. Despite the overwhelming number of Multi-objective Evolutionary Algorithms (MEAs) in the literature, little work has been done to identify the best MEA using an appropriate assessment methodology. In this chapter, we compare our algorithm with twelve other well-known MEAs, using a popular assessment methodology, by solving two benchmark problems. The comparison shows the superiority of our algorithm over others.

Solving Expensive Multi-Objective Optimization Problems by Kriging Model with Multi-Point Updating Strategy

2014 ◽  
Vol 685 ◽  
pp. 667-670 ◽  
Author(s):  
Ding Han ◽  
Jian Rong Zheng
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Average Distance ◽  
Kriging Model ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Multi Objective Optimization ◽  
Good Convergence ◽  
Multi Objective

A method which utilizes Kriging model and a multi-point updating strategy is put forward for solving expensive multi-objective optimization problems. Assisted by a defined cheaper multi-objective optimization problem and a maximum average distance criterion, multiple updating points can be found. The proposed method is tested on two numerical functions and a ten-bar truss problem, the results show that the proposed method is efficient in obtaining Pareto optimal solutions with good convergence and diversity when the same computation resource is used comparing with two other methods.

Tracing the Envelope of the Objective-Space in Multi-Objective Topology Optimization

2011 ◽  
Author(s):  
Inna Turevsky ◽  
Krishnan Suresh
Keyword(s):  
Topology Optimization ◽  
Eigenvalue Problems ◽  
Optimization Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Topological Sensitivity ◽  
Numerical Examples ◽  
Multi Objective ◽  
Objective Space

In multi-objective problems, one is often interested in generating the envelope of the objective-space, where the envelope is, in general, a superset of pareto-optimal solutions. In this paper, we propose a method for tracing the envelope of multi-objective topology optimization problems, and generating the corresponding topologies. The proposed method exploits the concept of topological sensitivity, and is applied to bi-objective optimization, namely eigenvalue-volume, eigenvalue-eigenvalue and compliance-eigenvalue problems. The robustness and efficiency of the method is illustrated through numerical examples.

Multi Objective Design Optimization of Two Bar Truss Using NSGA II and TOPSIS

2014 ◽  
Vol 984-985 ◽  
pp. 419-424
Author(s):  
P. Sabarinath ◽  
M.R. Thansekhar ◽  
R. Saravanan
Keyword(s):  
Design Optimization ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Nsga Ii ◽  
Multi Objective ◽  
Total Displacement ◽  
Design Variables ◽  
Conflicting Objectives

Arriving optimal solutions is one of the important tasks in engineering design. Many real-world design optimization problems involve multiple conflicting objectives. The design variables are of continuous or discrete in nature. In general, for solving Multi Objective Optimization methods weight method is preferred. In this method, all the objective functions are converted into a single objective function by assigning suitable weights to each objective functions. The main drawback lies in the selection of proper weights. Recently, evolutionary algorithms are used to find the nondominated optimal solutions called as Pareto optimal front in a single run. In recent years, Non-dominated Sorting Genetic Algorithm II (NSGA-II) finds increasing applications in solving multi objective problems comprising of conflicting objectives because of low computational requirements, elitism and parameter-less sharing approach. In this work, we propose a methodology which integrates NSGA-II and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) for solving a two bar truss problem. NSGA-II searches for the Pareto set where two bar truss is evaluated in terms of minimizing the weight of the truss and minimizing the total displacement of the joint under the given load. Subsequently, TOPSIS selects the best compromise solution.

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