Ansary–Panda (AP)

This family comprises the AP1, AP2, AP3, and AP4 constructors. All four problems are drawn from “A modified Quasi-Newton method for vector optimization problem” [2].

Overview

The table below shows the default dimensions and componentwise variable bounds for each constructor.

Analytical Jacobians and Hessians are registered for all four constructors. The catalog metadata classifies the strict convexity of the objective functions as follows:

• AP1: [:not_strictly_convex, :strictly_convex, :strictly_convex]
• AP2: [:strictly_convex, :strictly_convex]
• AP3: [:not_strictly_convex, :not_strictly_convex]
• AP4: [:not_strictly_convex, :strictly_convex, :strictly_convex]

Mathematical formulations

The formulas below describe the objective functions implemented by the constructors. For each problem, let $F:\mathbb{R}^n \to \mathbb{R}^m$ be defined by $F(x)=(f_1(x),\ldots,f_m(x))$.

AP1

The objectives are

\[\begin{aligned} f_1(x) &= \frac{1}{4} \left( (x_1 - 1)^4 + 2(x_2 - 2)^4 \right), \\ f_2(x) &= \exp\left( \frac{x_1 + x_2}{2} \right) + x_1^2 + x_2^2, \\ f_3(x) &= \frac{1}{6} \left( \exp(-x_1) + 2\exp(-x_2) \right). \end{aligned}\]

where $x = (x_1, x_2) \in \mathbb{R}^2$.

AP2

The objectives are

\[\begin{aligned} f_1(x) &= x_1^2 - 4, \\ f_2(x) &= (x_1 - 1)^2. \end{aligned}\]

where $x \in \mathbb{R}^1$.

AP3

The objectives are

\[\begin{aligned} f_1(x) &= \frac{1}{4} \left( (x_1 - 1)^4 + 2(x_2 - 2)^4 \right), \\ f_2(x) &= (x_2 - x_1^2)^2 + (1 - x_1)^2. \end{aligned}\]

where $x = (x_1, x_2) \in \mathbb{R}^2$.

AP4

The objectives are

\[\begin{aligned} f_1(x) &= \frac{1}{9} \left( (x_1 - 1)^4 + 2(x_2 - 2)^4 + 3(x_3 - 3)^4 \right), \\ f_2(x) &= \exp\left( \frac{x_1 + x_2 + x_3}{3} \right) + x_1^2 + x_2^2 + x_3^2, \\ f_3(x) &= \frac{1}{12} \left( 3\exp(-x_1) + 4\exp(-x_2) + 3\exp(-x_3) \right). \end{aligned}\]

where $x = (x_1, x_2, x_3) \in \mathbb{R}^3$.

Usage

using MOProblems

prob = AP1()
x = [0.0, 0.0]

values = eval_f(prob, x)
J = eval_jacobian(prob, x)
H = eval_hessian(prob, x)

Constructor reference

AP1()

AP2()

AP3()

AP4()