Setting the initial conditions for the optimization.

[dlcfort]

Does this function found in "x_dense_qp_sol.c" set the optimization variables initial conditions for the dense qp optimization?

void DENSE_QP_SOL_SET_V(REAL *v, struct DENSE_QP_SOL *qp_sol)
	{
	int nv = qp_sol->dim->nv;
	PACK_VEC(nv, v, 1, qp_sol->v, 0);
	}

[giaf]

Yes you can set the initial guess for the optimization variables in the solution struct like that.
Then if you set warm_start arg to 1 (or 2 if also the dual variables are set) the solver would read the value of the primal variables out of there when setting up their value before the first IPM iteration.

[dlcfort]

I ran some tests using the dense qcqp optimizer and set warm start =0, 1 , and 2 over three test cases. When I run the optimization with warm_start = 2 and set "v" equal to the solution, I get the fastest performance. However when I set warm start equal to 1 it has unpredictable effects. My question is, why does warm start = 1 have worse timing than warm start = 0 when we start near the solution?

My Results and Code are below running "example_dense_qcqp_getting_started.py".

START CONDITIONS EQUAL TO SOLUTION
0 Warm start: ipm iter = 20
1 Warm start: ipm iter = 21
2 Warm start: ipm iter = 3

START CONDITIONS NEAR SOLUTION
0 Warm start: ipm iter = 20
1 Warm start: ipm iter = 22
2 Warm start: ipm iter = 4

START CONDITIONS FAR FROM SOLUTION
0 Warm start: ipm iter = 20
1 Warm start: ipm iter = 12
2 Warm start: ipm iter = 30

nv = 2  # number of variables
nq = 1  # number of quadratic inequality constraints
dim = hpipm_dense_qcqp_dim()
dim.set('nv', nv)
dim.set('nq', nq)

H = np.array([[1,0],
             [0,1]])
g = np.array([[0],[0]])
Hq = np.array([[2,0],[0,2]])
gq = np.array([[-2],[-2]])
uq = -1

# qp
qcqp = hpipm_dense_qcqp(dim)
# data
qcqp.set('H', H)
qcqp.set('g', g)
qcqp.set('Hq', Hq)
qcqp.set('gq', gq)
qcqp.set('uq', uq)

# qp sol
qcqp_sol = hpipm_dense_qcqp_sol(dim)
# set up solver arg
# mode = 'speed'
mode = 'robust'
# create and set default arg based on mode
arg = hpipm_dense_qcqp_solver_arg(dim, mode)
# create and set default arg based on mode
arg.set('mu0', 1e4)
arg.set('iter_max', 30)
arg.set('tol_stat', 1e-5)
arg.set('tol_eq', 1e-5)
arg.set('tol_ineq', 1e-5)
arg.set('tol_comp', 1e-5)
arg.set('reg_prim', 1e-12)

# if warm_start=1, then the primal variable is initialized from qp_sol
warm_start = 0; # set to 1 to warm-start the primal variable
# warm_start = 1; # set to 1 to warm-start the primal variable
# warm_start = 2; # set to 1 to warm-start the primal variable
arg.set('warm_start', warm_start)
qcqp_sol.set('v', np.array([-4.4929, 2.2929]))
# qcqp_sol.set('v', np.array([0.39289321, 0.19289321]))
# qcqp_sol.set('v', np.array([0.29289321, 0.29289321])) ### SOLUTION

# set up solver
solver = hpipm_dense_qcqp_solver(dim, arg)
start_time = time.time()
solver.solve(qcqp, qcqp_sol)
end_time = time.time()
v = qcqp_sol.get('v')

# get solver statistics
status = solver.get('status')
iters = solver.get('iter')
stat = solver.get('stat')
status = solver.get('status')
print('v      = {}'.format(v.flatten()))
print('ipm iter = ', iters)
print('solve time {:e}'.format(end_time-start_time))