Directed Multiobjective Optimization Based on the Weighted Hypervolume Indicator

TLDR: The algorithm W-HypE, which implements the previous concepts, is introduced, and the effectiveness of the search, directed towards the user's preferred solutions, is shown using an extensive set of experiments including the necessary statistical performance assessment.

Abstract: Recently, there has been a large interest in set-based evolutionary algorithms for multi objective optimization. They are based on the definition of indicators that characterize the quality of the current population while being compliant with the concept of Pareto-optimality. It has been shown that the hypervolume indicator, which measures the dominated volume in the objective space, enables the design of efficient search algorithms and, at the same time, opens up opportunities to express user preferences in the search by means of weight functions. The present paper contains the necessary theoretical foundations and corresponding algorithms to (i) select appropriate weight functions, to (ii) transform user preferences into weight functions and to (iii) efficiently evaluate the weighted hypervolume indicator through Monte Carlo sampling. The algorithm W-HypE, which implements the previous concepts, is introduced, and the effectiveness of the search, directed towards the user's preferred solutions, is shown using an extensive set of experiments including the necessary statistical performance assessment. Copyright © 2013 John Wiley & Sons, Ltd.

Figures

Figure 10: Illustration of Tchebycheff approach.

Figure 5: Simple weight functions: objective stressing (exponential).

Figure 6: Simple weight functions: preference point (normal distribution).

Table 3: Used parameters of the biobjective 0/1 knapsack problem in PISA.

Figure 1: Illustration of the standard hypervolume indicator for a set of 3-objective objective vectors (left) and the weighted hypervolume indicator IwH(A,{r}) (volume of the gray shape in the right-hand plot) for a set A of nine solutions (black dots) of a bi-objective problem. The plot shows an example of a weight function w(z), where for all objective vectors z that are not dominated by A or not enclosed by r the function w is not plotted. The plot is taken from Fig. 1 in (Auger et al., 2012) and updated.

Figure 14: Pareto front approximations on bi-objective ZDT1 (front shown as solid line) for the following weight functions: (a) w1 (exponential), (b) w2 (preference point), (c) w3 (Tchebycheff), (d) w4 (desirability function), (e) w5 (uniform region), (f) w6 (ε-constraint), (g) IBEA, (h) NSGA-II. The employed weight functions are shown as contour lines and as gray shading (larger weight in darker regions).

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Directed multiobjective optimization based on the weighted hypervolume indicator" ?

The algorithm W-HypE which implements the above concepts is introduced and the effectiveness of the search, directed towards the user ’ s preferred solutions, is shown using an extensive set of experiments including the necessary statistical performance assessment. 

Q2. What is the way to determine the quality of a set of solutions?

Unary set indicators, such as the hypervolume indicator, can now be used to represent the quality of a whole set of solutions by a single scalar value. 

Q3. What is the main step when formalizing user preferences in terms of the weighted hypervolume?

The main step when formalizing user preferences in terms of the weighted hypervolume is to choose the underlying weight function. 

Q4. What is the weighted hypervolume indicator of A with respect to R?

The weighted hypervolume indicator IwH(A,R) for the set A of nine points equals the integral of the weight function over the objective space that is weakly dominated by the set A and which weakly dominates the reference point r = (r1, r2). 

Q5. What is the way to combine q weight density functions?

The authors here present only one possibility, namely to combine q weight density functions w1(z), . . . ,wq(z) by a linear combinationwlc(z) = p1w1(z) + . . . + pqwq(z) (9)where the pi are positive real numbers that sum up to one, i.e., p1 + . . . + pq = 1.In order to sample the weight density function wlc(z) constructed according to (9), random samples can be generated using the following steps: