Bi-objective hypervolume-based Pareto optimization

Abstract

The search for the best trade-off solutions with respect to several criteria (also called the Pareto set) is the main approach pursued in multi-objective optimization when no additional preferences are associated to the objectives. This problem is known to be compliant with the maximization of the hypervolume (or S-metric), consisting in the Lebesgue measure of the dominated region covered by a set of solutions in the objective space, and bounded by a reference point. While several variants of population-based metaheuristics like evolutionary algorithms address formulations maximizing the hypervolume, the use of gradient-based algorithms for this task has been largely neglected in the literature. Therefore, this paper proposes to solve bi-objective problems by hypervolume maximization through a sequential quadratic programming algorithm. After theoretical developments including the analytical expression of the sensitivities of the hypervolume expressed as functions of the gradient of the objectives, the method is applied to six benchmark test cases, demonstrating the efficiency of the proposed method in comparison with a scalarization of the objectives, and with a state-of-the-art multi-objective genetic algorithm. This method is believed to provide an interesting alternative to metaheuristics when the gradients of the objective functions are available at a limited additional cost, a situation which is encountered in versatile applications, for instance with adjoint methods implemented in computational solid mechanics or fluid dynamics.

Appendix: definition of the six test cases

$$\begin{aligned}&\mathbf{TC1} \hbox { (concave version of Rendon [8])} \left\{ \begin{array}{rl} \mathop {\min }\limits _{\mathbf{X}} &{} \left\{ \begin{array}{lll} f_1(\mathbf{x}) &{}=&{} - (1/(x_1^2 + x_2^2 + 1)) \\ f_2(\mathbf{x}) &{}=&{} - (x_1^2 + 3 x_2^2 + 1) \\ \end{array}\right. \\ \hbox {s.t.:} &{} x_i \in [-3,3], i=1,2 \\ \end{array}\right. \end{aligned}$$

(14)

$$\begin{aligned}&\mathbf{TC2} \hbox { (Murata [8])} \left\{ \begin{array}{rl} \mathop {\min }\limits _{\mathbf{X}} &{} \left\{ \begin{array}{lll} f_1(\mathbf{x}) &{}=&{} 2\sqrt{x_1} \\ f_2(\mathbf{x}) &{}=&{} x_1(1-x_2) + 5 \\ \end{array}\right. \\ \hbox {s.t.:} &{} x_1 \in [1,4], x_2 \in [1,2] \\ \end{array}\right. \end{aligned}$$

(15)

$$\begin{aligned}&\mathbf{TC3} \hbox { (Srinivas [8])} \left\{ \begin{array}{rl} \mathop {\min }\limits _{\mathbf{X}} &{} \left\{ \begin{array}{lll} f_1(\mathbf{x}) &{}=&{} (x_1 - 2)^2 + (x_2 - 1)^2 + 2 \\ f_2(\mathbf{x}) &{}=&{} 9 x_1 - (x_2-1)^2 \\ \end{array}\right. \\ \hbox {s.t.:} &{} x_1^2 + x_2^2 - 225 \le 0 \\ &{} x_1 - 3 x_2 + 10 \le 0 \\ &{} x_i \in [-20,20], i=1,2 \\ \end{array}\right. \end{aligned}$$

(16)

$$\begin{aligned}&\mathbf{TC4} \hbox { (Tanaka [8])} \left\{ \begin{array}{rl} \mathop {\min }\limits _{\mathbf{X}} &{} \left\{ \begin{array}{lll} f_1(\mathbf{x}) &{}=&{} x_1 \\ f_2(\mathbf{x}) &{}=&{} x_2 \\ \end{array}\right. \\ \hbox {s.t.:} &{} -(x_1^2) - (x_2^2) + 1 + 0.1 \cos (16\arctan (x_1/x_2)) \le 0 \\ &{} -0.5 + (x_1 - 0.5)^2 + (x_2 - 0.5)^2 \le 0 \\ &{} x_i \in [0,\pi ], i=1,2 \\ \end{array}\right. \end{aligned}$$

(17)

$$\begin{aligned}&\mathbf{TC5} \hbox { (Osyczka [8])} \left\{ \begin{array}{rl} \mathop {\min }\limits _{\mathbf{X}} &{} \left\{ \begin{array}{lll} f_1(\mathbf{x}) &{}=&{} -(25(x_1-2)^2 + (x_2-2)^2 \\ &{}&{} + (x(3)-1)^2 + (x(4)-4)^2 \\ &{}&{} + (x(5)-1)^2) \\ f_2(\mathbf{x}) &{}=&{} x_1^2 + x_2^2 + x(3)^2 + x(4)^2 + x(5)^2 + x(6)^2\\ \end{array}\right. \\ \hbox {s.t.:} &{} -(x_1 + x_2 - 2) \le 0 \\ &{} -(6 - x_1 - x_2) \le 0 \\ &{} -(2 - x_2 + x_1) \le 0 \\ &{} -(2 - x_1 + 3 x_2) \le 0 \\ &{} -(4 - (x_3 - 3)^2 - x_4) \le 0 \\ &{} -((x_5-3)^2 + x_6 - 4) \le 0 \\ &{} x_1 \in [0,10], x_2 \in [0,10], x_3 \in [1,5] \\ &{} x_4 \in [0,6], x_5 \in 1,5], x_6 \in [0,10] \\ \end{array}\right. \end{aligned}$$

(18)

$$\begin{aligned}&\begin{array}{ll} \mathbf{TC6} &{} \hbox { (Deb's M-LFS with additional}\\ &{} \hbox { linear constraints [9] )} \end{array} \left\{ \begin{array}{rl} \mathop {\min }\limits _{\mathbf{X}} &{} \left\{ \begin{array}{lll} f_1(\mathbf{x}) &{}=&{} x_1 \\ f_2(\mathbf{x}) &{}=&{} g(\mathbf{x}) (1 - f_1(\mathbf{x})/g(\mathbf{x})) \\ \end{array}\right. \\ \hbox {s.t.:} &{} a \left| \sin (b\pi (\sin (\theta )(f_2(\mathbf{x})-e) \right. \\ &{} \quad \quad +\left. \cos (\theta )f_1(\mathbf{x}))^c)\right| ^d \\ &{} \quad \quad - \cos (\theta )(f_2(\mathbf{x})-e) \\ &{} \quad \quad - \sin (\theta )f_1(\mathbf{x}) \le 0\\ &{} 0.75 - x_1 - x_2 \le 0 \\ &{} 0.75 - x_2 - x_3 \le 0 \\ &{} 0.75 - x_3 - x_4 \le 0 \\ &{} 0.75 - x_4 - x_5 \le 0 \\ &{} a = 40, b = 4, c = 1, d = 2, e = -1 \\ &{} x_i \in [0,1], i=1,\dots ,5 \\ \end{array}\right. \end{aligned}$$

(19)