Amaral–Assunção–Souza (AAS)

This family comprises the AAS1 and AAS2 constructors. Both problems are drawn from “A Partially Derivative-Free Proximal Method for Composite Multiobjective Optimization in the Hölder Setting” [1].

Overview

Both constructors have nvar = 2 and nobj = 2. Their bounds differ as shown below.

The bounds in each row apply componentwise to both variables. Analytical Jacobians and Hessians are not registered for either constructor. Strict-convexity information is not available (nothing) for either problem.

Mathematical formulations

The formulas below describe the objective functions implemented by the constructors. In both cases, let $F:\mathbb{R}^2 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$, where $x = (x_1,x_2) \in \mathbb{R}^2$.

AAS1

The objectives are

\[\begin{aligned} f_1(x) &= \frac{1}{2}\lVert Ax-b\rVert_2^2, & f_2(x) &= \frac{\mu}{p}\lVert Dx\rVert_p^p. \end{aligned}\]

The implemented parameters are

\[A = \begin{bmatrix}2.0 & 0.5\\0.5 & 1.5\end{bmatrix}, \qquad b = \begin{bmatrix}1.0\\-0.5\end{bmatrix}, \qquad p = 1.003, \qquad \mu = 0.9, \qquad D = \begin{bmatrix}1.0 & 0.8\\0.3 & 1.2\end{bmatrix}.\]

AAS2

The objectives are

\[\begin{aligned} f_1(x) &= \frac{\mu_1}{p_1}\lVert D_1(x-c_1)\rVert_{p_1}^{p_1}, & f_2(x) &= \frac{\mu_2}{p_2}\lVert D_2(x-c_2)\rVert_{p_2}^{p_2}. \end{aligned}\]

The implemented parameters are

\[\begin{aligned} p_1 &= 1.003, & \mu_1 &= 1.2, & D_1 &= \begin{bmatrix}1.2 & -0.3\\0.4 & 1.5\end{bmatrix}, & c_1 &= \begin{bmatrix}1.5\\-1.0\end{bmatrix},\\[6pt] p_2 &= 1.002, & \mu_2 &= 0.8, & D_2 &= \begin{bmatrix}1.8 & 0.5\\-0.2 & 1.1\end{bmatrix}, & c_2 &= \begin{bmatrix}-1.2\\0.8\end{bmatrix}. \end{aligned}\]

Usage

using MOProblems

prob = AAS1()
x = [0.0, 0.0]

values = eval_f(prob, x)

Constructor reference

AAS1()

AAS2()