Affine Variational Inequalities

DAQP can find a solution to the affine variational inequality (AVI):

\[\text{find } x^\star \in C \text{ such that } \langle Hx^\star + f,\, y - x^\star\rangle \geq 0 \quad \forall\, y \in C,\]

where \(C = \lbrace x \mid b_l \leq Ax \leq b_u \rbrace\). AVI generalizes both QPs and complementarity problems.

C

#include "api.h"

int n = 2, m = 2, ms = 0;
double H[4]     = {1, 1.75, 0, 1};   // row-major, asymmetric
double f[2]     = {2.0, 2.0};
double A[4]     = {1, 0, 0, 1};
double bupper[2]= {1.0, 1.0};
double blower[2]= {-1.0, -1.0};
int    sense[2] = {0, 0};

// Set problem_type = 1 to indicate AVI
DAQPProblem qp = {n, m, ms, H, f, A, bupper, blower, sense, NULL, 0, 1};

double x[2], lam[2];
DAQPResult result = {x, lam};
daqp_avi(&result, &qp, NULL);

Julia

using DAQP

H = [1.0 1.75; 0.0 1.0];   # asymmetric
f = [2.0; 2.0];
A = [1.0 0.0; 0.0 1.0];
bupper = [1.0; 1.0];
blower = [-1.0; -1.0];

x, _, exitflag, info = DAQP.avi(H, f, A, bupper, blower)

MATLAB

H = [1 1.75; 0 1];
f = [2; 2];
A = eye(2);
bupper = [1; 1];
blower = [-1; -1];
sense  = int32(zeros(2, 1));

[x, fval, exitflag, info] = daqp.avi(H, f, A, bupper, blower, sense);

Python

import daqp, numpy as np
from ctypes import c_double

H      = np.array([[1, 1.75], [0, 1]], dtype=c_double)
f      = np.array([2, 2], dtype=c_double)
A      = np.eye(2, dtype=c_double)
bupper = np.array([1, 1], dtype=c_double)
blower = np.array([-1, -1], dtype=c_double)

x, fval, exitflag, info = daqp.solve(H, f, A, bupper, blower, is_avi=True)