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MacMPEC Collection

This library contains some selected MPCC problems from the MacMPEC Collection originally built in AMPL by Sven Leyffer. For this library, these problems were written in Julia as a JuMP model and they are described below along with a link to the .jl file in the github repository.

Minimum weight design problem from M.C. Ferris and F. Tin-Loi, On the solution of a minimumWeight elastoplastic problem involving displacement and complementarity constraints, Comp. Meth. in Appl. Mech & Engng, 174:107-120, 1999.

Goal: minimize the volume of a struture with fixed topology that has to resist certain specified loads while keeping displacements within a specified limit.

\[ \begin{aligned} \min \quad &L^Ta \\ \text{s.t.} \quad &Q = SCw - SNz \\ \quad &F = C^TQ \\ \quad &w = -N^TQ + Hz + r \\ \quad &S_i = Ea_i/L_i \text{ for } i = 1,\ldots,nm \\ \quad &H_{i,jk} = 0.125S_j \text{ for } j = k \text{ and } i = 1,\ldots,nm \\ \quad &H_{i,jk} = 0 \text{ for } j \neq k \text{ and } i = 1,\ldots,nm \\ \quad &r_{i,j} = \sigma a_i \text{ for } j = 1,\ldots,ny \text{ and } i = 1,\ldots,nm \\ \quad &Ta = 0 \\ \quad &a \ge 0 \\ \quad &-u^b \le u \le u^b \\ \quad &0 \le w \perp z \ge 0 \end{aligned}\]

where the variables are

$u \in \mathbb{R}^{nd}$ - displacements
$z \in \mathbb{R}^{ny}$ - plastic multipliers
$a \in \mathbb{R}^{nm}$ - cross-sectional areas
$Q \in \mathbb{R}^{nm}$ - natural generalized stresses
$S \in \mathbb{R}^{nm}$ - element stiffness
$H_i \in \mathbb{R}^{ny \times ny}$ - hardening models
$r_i \in \mathbb{R}^{ny}$ - yield limits
$w_i(Q(z),z): \mathbb{R}^{ny} \rightarrow \mathbb{R}^{ny}$ - linear yield function

and parameters are

\[ L = \begin{bmatrix} 500 & 400 & 500 \end{bmatrix}^T, \quad F = \begin{bmatrix} 400 & 600 \end{bmatrix}^T, \quad C = \begin{bmatrix} 0.6 & 0.0 & -0.6 \\ 0.8 & 1.0 & 0.8 \end{bmatrix}^T, \quad N = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \end{bmatrix}^T \]

$E = 2e4$
$\sigma = 50$.

$nm$ is the number of elements, $nd$ is the number of structure degrees of freedom and $ny$ is the number of yield functions per element.
($Ta = 0$ corresponds to technoogical constraints on the design variables $a$)

Two-level optimization toy model presented in J.F. Bard, Convex two-level optimization, Mathematical Programming 40(1), 15-27, 1988.

\[ \begin{aligned} \min_{x \ge 0} \quad &(x - 5)^2 + (2y + 1)^2 \\ \text{s.t.} \quad &\min_{y \ge 0} \quad (y - 1)^2 - 1.5xy \\ &\text{s.t.} \quad 3x - y \ge 3 \\ &\quad \quad -x + 0.5y \ge 4 \\ &\quad \quad -x -y \ge -7 \end{aligned}\]

It can be reformulated as a MPCC problem by using the KKT conditions of the inner problem as constraints for the outer problem.

\[ \begin{aligned} \min_{x,y \ge 0} \quad &(x - 5)^2 + (2y + 1)^2 \\ \text{s.t.} \quad &2(y - 1) - 1.5x + \lambda_1 - 0.5\lambda_2 + \lambda_3 = 0 \\ &0 \le 3x - y -3 \perp \lambda_1 \ge \\ &0 \le -x + 0.5y + 4 \perp \lambda_2 \ge 0 \\ &0 \le -x -y + 7 \perp \lambda_ 3 \ge 0 \end{aligned}\]

where $\lambda_i$ is the Lagrange multiplier corresponding to the $i^\text{th}$ inequality of the inner problem.

Toy model

\[ \begin{aligned} \min_{x,y} \quad &x^2 + y^2 - 4xy \\ \text{s.t.} \quad &0 \le x \perp y \ge 0 \end{aligned}\]

Toy model

\[ \begin{aligned} \min_{x,y} \quad &(100x - 1)^2 + (y - 1)^2 \\ \text{s.t.} \quad &0 \le x \perp y \ge 0 \end{aligned}\]