Das–Dennis (DD)

The DD1 constructor implements the numerical biobjective example from Indraneel Das and J. E. Dennis, “Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems” [4].

Overview

DD1 has five variables, two objectives, two equality constraints, and one inequality constraint. It has no explicit variable bounds. The constraint mapping follows the convention

\[l_c \leq c(x) \leq u_c.\]

Analytical Jacobians and Hessians are registered for both objectives and all constraints. The catalog metadata classifies the first objective as strictly convex (:strictly_convex) and the second as not strictly convex (:not_strictly_convex).

recommended_bounds returns $[-20,20]^5$, a box recommended by the package developers for experiments that need a bounded region. It is not added to prob.bounds and does not replace the three general constraints: the box was chosen so that feasible points exist inside it, not so that every point of it is feasible.

Mathematical formulation

Let $F:\mathbb{R}^5 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$. The objectives are

\[\begin{aligned} f_1(x) &= x_1^2+x_2^2+x_3^2+x_4^2+x_5^2,\\ f_2(x) &= 3x_1+2x_2-\frac{x_3}{3}+0.01(x_4-x_5)^3. \end{aligned}\]

The traditional equality and inequality constraint functions are

\[\begin{aligned} h_1(x) &= x_1+2x_2-x_3-0.5x_4+x_5-2 = 0,\\ h_2(x) &= 4x_1-2x_2+0.8x_3+0.6x_4+0.5x_5^2 = 0,\\ g_1(x) &= x_1^2+x_2^2+x_3^2+x_4^2+x_5^2-10 \leq 0. \end{aligned}\]

In the evaluation API, these functions are stored as

\[c(x)=\begin{bmatrix}h_1(x) & h_2(x) & g_1(x)\end{bmatrix}^{\mathsf T}, \qquad l_c=\begin{bmatrix}0&0&-\infty\end{bmatrix}^{\mathsf T}, \qquad u_c=\begin{bmatrix}0&0&0\end{bmatrix}^{\mathsf T}.\]

Usage

The point below satisfies the two equalities and the inequality.

julia> using MOProblems

julia> prob = DD1();

julia> x = [1 / 3, 0.0, -5 / 3, 0.0, 0.0];

julia> objectives = eval_f(prob, x);

julia> constraints = eval_c(prob, x);

julia> all(prob.lcon .- 1e-12 .<= constraints .<= prob.ucon .+ 1e-12)
true

julia> objective_jacobian = eval_jacobian(prob, x);

julia> constraint_jacobian = eval_constraint_jacobian(prob, x);

julia> objective_hessians = eval_hessian(prob, x);

julia> constraint_hessians = eval_constraint_hessian(prob, x);

julia> (length(objectives), length(constraints), size(objective_jacobian),
           size(constraint_jacobian), length(objective_hessians),
           size(objective_hessians[1]), length(constraint_hessians),
           size(constraint_hessians[1]))
(2, 3, (2, 5), (3, 5), 2, (5, 5), 3, (5, 5))

The constraint values are interpreted together with prob.lcon and prob.ucon.

Constructor reference

DD1()