An Efficient Algorithm for Computing Hypervolume Contributions

2010 ◽  
Vol 18 (3) ◽  
pp. 383-402 ◽  
Author(s):  
Karl Bringmann ◽  
Tobias Friedrich
Keyword(s):  
Evolutionary Algorithms ◽  
Efficient Algorithm ◽  
Optimization Algorithms ◽  
Multi Objective ◽  
Multiplicative Factor ◽  
Optimal Set ◽  
Hypervolume Indicator ◽  
Additive Term

The hypervolume indicator serves as a sorting criterion in many recent multi-objective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a population of size n with d objectives, a solution with minimal hypervolume contribution in time [Formula: see text](nd/2 log n) for d > 2. This improves all previously published algorithms by a factor of n for all d > 3 and by a factor of [Formula: see text] for d = 3. We also analyze hypervolume indicator based optimization algorithms which remove λ > 1 solutions from a population of size n = μ + λ. We show that there are populations such that the hypervolume contribution of iteratively chosen λ solutions is much larger than the hypervolume contribution of an optimal set of λ solutions. Selecting the optimal set of λ solutions implies calculating [Formula: see text] conventional hypervolume contributions, which is considered to be computationally too expensive. We present the first hypervolume algorithm which directly calculates the contribution of every set of λ solutions. This gives an additive term of [Formula: see text] in the runtime of the calculation instead of a multiplicative factor of [Formula: see text]. More precisely, for a population of size n with d objectives, our algorithm can calculate a set of λ solutions with minimal hypervolume contribution in time [Formula: see text](nd/2 log n + nλ) for d > 2. This improves all previously published algorithms by a factor of nmin{λ,d/2} for d > 3 and by a factor of n for d = 3.

scholarly journals An Ensemble Framework of Evolutionary Algorithm for Constrained Multi-Objective Optimization

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 116
Author(s):  
Junhua Ku ◽  
Fei Ming ◽  
Wenyin Gong
Keyword(s):  
Evolutionary Algorithms ◽  
Real World ◽  
Optimization Problems ◽  
State Of The Art ◽  
Multi Objective Optimization ◽  
The Real ◽  
Multi Objective ◽  
The Past ◽  
Hypervolume Indicator ◽  
Real World Problems

In the real-world, symmetry or asymmetry widely exists in various problems. Some of them can be formulated as constrained multi-objective optimization problems (CMOPs). During the past few years, handling CMOPs by evolutionary algorithms has become more popular. Lots of constrained multi-objective optimization evolutionary algorithms (CMOEAs) have been proposed. Whereas different CMOEAs may be more suitable for different CMOPs, it is difficult to choose the best one for a CMOP at hand. In this paper, we propose an ensemble framework of CMOEAs that aims to achieve better versatility on handling diverse CMOPs. In the proposed framework, the hypervolume indicator is used to evaluate the performance of CMOEAs, and a decreasing mechanism is devised to delete the poorly performed CMOEAs and to gradually determine the most suitable CMOEA. A new CMOEA, namely ECMOEA, is developed based on the framework and three state-of-the-art CMOEAs. Experimental results on five benchmarks with totally 52 instances demonstrate the effectiveness of our approach. In addition, the superiority of ECMOEA is verified through comparisons to seven state-of-the-art CMOEAs. Moreover, the effectiveness of ECMOEA on the real-world problems is also evaluated for eight instances.

A Scalable D Dominant Strategy to Solve Large-Dimensional Multi-Objective Optimization Problems

2014 ◽  
Vol 1037 ◽  
pp. 383-388
Author(s):  
Qiong Yuan ◽  
Guang Ming Dai
Keyword(s):  
Evolutionary Algorithms ◽  
Optimization Problems ◽  
Optimization Algorithms ◽  
Dominant Strategy ◽  
Numerical Comparison ◽  
Multi Objective Optimization ◽  
Dominant Mechanism ◽  
Multi Objective ◽  
Comparison Test ◽  
Research Areas

Solving large-dimensional multi-objective optimization problems is one of the focus research areas of multi-objective optimization evolutionary . When using traditional multi-objective optimization algorithms to solve large-dimensional multi-objective optimization problems,we found that the unsatisfactory optimizing results often exist. To overcome this flaw, in this paper we studied scalable dominant mechanism and proposed a D dominant strategy. According to the superior theory of D strategy ,we improved the current four kinds of typical multi-objective optimization evolutionary algorithms. The numerical comparison test on DTLZ1-6 (20) questions which were solved by the improved algorithms indicated that D strategy had in varying degrees improved the algorithms for solving large-dimensional multi-objective optimization problems .Thus ,we confirmed that the D strategy for solving large-dimensional multi-objective optimization problems is effective.

scholarly journals Drift Analysis and Evolutionary Algorithms Revisited

2018 ◽  
Vol 27 (4) ◽  
pp. 643-666 ◽  
Author(s):  
J. LENGLER ◽  
A. STEGER
Keyword(s):  
Evolutionary Algorithms ◽  
Evolutionary Algorithm ◽  
Random Search ◽  
Optimization Algorithms ◽  
Simple Algorithm ◽  
Linear Functions ◽  
Search Point ◽  
Drift Analysis ◽  
Fixed Constant ◽  
Greedy Optimization

One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a function f: {0,1}n → ℝ. The algorithm starts with a random search point ξ ∈ {0,1}n, and in each round it flips each bit of ξ with probability c/n independently at random, where c > 0 is a fixed constant. The thus created offspring ξ' replaces ξ if and only if f(ξ') ≥ f(ξ). The analysis of the runtime of this simple algorithm for monotone and for linear functions turned out to be highly non-trivial. In this paper we review known results and provide new and self-contained proofs of partly stronger results.

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