Hypervolume Subset Selection with Small Subsets

2019 ◽  
Vol 27 (4) ◽  
pp. 611-637
Author(s):  
Benoît Groz ◽  
Silviu Maniu
Keyword(s):  
Evolutionary Algorithms ◽  
Subset Selection ◽  
Higher Dimensions ◽  
Efficient Algorithms ◽  
Selection Problem ◽  
Multiobjective Evolutionary Algorithms ◽  
Optimal Subset ◽  
Arbitrary Dimensions ◽  
Hypervolume Indicator

The hypervolume subset selection problem (HSSP) aims at approximating a set of [Formula: see text] multidimensional points in [Formula: see text] with an optimal subset of a given size. The size [Formula: see text] of the subset is a parameter of the problem, and an approximation is considered best when it maximizes the hypervolume indicator. This problem has proved popular in recent years as a procedure for multiobjective evolutionary algorithms. Efficient algorithms are known for planar points ([Formula: see text]), but there are hardly any results on HSSP in larger dimensions ([Formula: see text]). So far, most algorithms in higher dimensions essentially enumerate all possible subsets to determine the optimal one, and most of the effort has been directed toward improving the efficiency of hypervolume computation. We propose efficient algorithms for the selection problem in dimension 3 when either [Formula: see text] or [Formula: see text] is small, and extend our techniques to arbitrary dimensions for [Formula: see text].

scholarly journals UNIVERSAL CALABI–YAU ALGEBRA: TOWARDS AN UNIFICATION OF COMPLEX GEOMETRY

2003 ◽  
Vol 18 (30) ◽  
pp. 5541-5612 ◽  
Author(s):  
F. ANSELMO ◽  
J. ELLIS ◽  
D. V. NANOPOULOS ◽  
G. VOLKOV
Keyword(s):  
Lie Algebras ◽  
Complex Geometry ◽  
Algebraic Approach ◽  
Higher Dimensions ◽  
Normal Algebra ◽  
Natural Extensions ◽  
Recurrence Formulae ◽  
Different Dimensions ◽  
Arbitrary Dimensions

We present a universal normal algebra suitable for constructing and classifying Calabi–Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a "dual" construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi–Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi–Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan–Lie algebras. This Universal Calabi–Yau algebra is a powerful tool for deciphering the Calabi–Yau genome in all dimensions.

scholarly journals An Improved MOEA/D Based on Reference Distance for Software Project Portfolio Optimization

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Jing Xiao ◽  
Jing-Jing Li ◽  
Xi-Xi Hong ◽  
Min-Mei Huang ◽  
Xiao-Min Hu ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Evolutionary Algorithms ◽  
Portfolio Optimization ◽  
Optimization Problem ◽  
Experimental Comparison ◽  
Software Project ◽  
Project Portfolio ◽  
Multiobjective Evolutionary Algorithms ◽  
Reference Distance ◽  
Nondominated Sorting

As it is becoming extremely competitive in software industry, large software companies have to select their project portfolio to gain maximum return with limited resources under many constraints. Project portfolio optimization using multiobjective evolutionary algorithms is promising because they can provide solutions on the Pareto-optimal front that are difficult to be obtained by manual approaches. In this paper, we propose an improved MOEA/D (multiobjective evolutionary algorithm based on decomposition) based on reference distance (MOEA/D_RD) to solve the software project portfolio optimization problems with optimizing 2, 3, and 4 objectives. MOEA/D_RD replaces solutions based on reference distance during evolution process. Experimental comparison and analysis are performed among MOEA/D_RD and several state-of-the-art multiobjective evolutionary algorithms, that is, MOEA/D, nondominated sorting genetic algorithm II (NSGA2), and nondominated sorting genetic algorithm III (NSGA3). The results show that MOEA/D_RD and NSGA2 can solve the software project portfolio optimization problem more effectively. For 4-objective optimization problem, MOEA/D_RD is the most efficient algorithm compared with MOEA/D, NSGA2, and NSGA3 in terms of coverage, distribution, and stability of solutions.

Analysis of Solution Quality of Multiobjective Evolutionary Algorithms for Service Restoration in Distribution Systems

2017 ◽  
Vol 28 (6) ◽  
pp. 796-805
Author(s):  
Danilo Sipoli Sanches ◽  
Marcelo Favoretto Castoldi ◽  
João Bosco Augusto London ◽  
Alexandre Cláudio Botazzo Delbem
Keyword(s):  
Evolutionary Algorithms ◽  
Distribution Systems ◽  
Multiobjective Evolutionary Algorithms ◽  
Solution Quality ◽  
Service Restoration
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