Model Matrix Least Squares (Frobenius Norm) Problem with pyomo? - pyomo

Question

How to model the following problem with pyomo and Gurobi solver?
This is a non-convex problem, and can not model as a QCQP problem. So I want to solve it with pyomo and Gurobi solver. The most difficult part is matrix algebra (pyomo do not support yet).
Many thanks!

Answer 1

Here is my own answer. Please let me know if there are any mistakes. Thanks!
import pyomo.environ as pyo
## get your param
L, Delta =
K, P =
O, l =
ones3 = np.ones((3, 1))
model = pyo.ConcreteModel()
model.r = pyo.RangeSet(0,2) # 3
model.s = pyo.RangeSet(0,0) # 1
model.n = pyo.RangeSet(0,len(L)-1) # n
model.m = pyo.RangeSet(0,len(K)-1) # m
model.q = pyo.RangeSet(0,len(O)-1) # q
def param_rule(I, J, mat):
return dict(((i,j), mat[i,j]) for i in I for j in J)
model.L = pyo.Param(model.n, model.n, initialize=param_rule(model.n, model.n, L)) # L
model.Delta = pyo.Param(model.n, model.r, initialize=param_rule(model.n, model.r, Delta)) # Delta
model.K = pyo.Param(model.m, model.n, initialize=param_rule(model.m, model.n, K)) # K
model.P = pyo.Param(model.m, model.r, initialize=param_rule(model.m, model.r, P)) # P
model.O = pyo.Param(model.q, model.n, initialize=param_rule(model.q, model.n, O)) # O
model.ones3 = pyo.Param(model.r, model.s, initialize=param_rule(model.r, model.s, ones3)) # ones3
model.l = pyo.Param(model.q, initialize=dict(((i),l[i,0]) for i in model.q)) # l
model.V = pyo.Var(model.n, model.r, initialize=param_rule(model.n, model.r, Delta)) # V
def ObjRule(model):
return 0.5*sum((sum(model.L[i,k]*model.V[k,j] for k in model.n) - model.Delta[i,j])**2 for i in model.n for j in model.r)
model.obj = pyo.Objective(rule=ObjRule, sense=pyo.minimize)
model.constraints = pyo.ConstraintList()
# constraints 1
[model.constraints.add(sum(model.K[i,k] * model.V[k,j] for k in model.n) == model.P[i,j]) for i in model.m for j in model.r]
# constraint 2
for i in model.q:
model.constraints.add(sum(sum(model.O[i,k] * model.V[k,j] for k in model.n)**2 for j in model.r) == model.l[i])
opt = pyo.SolverFactory('gurobi')
opt.options['nonconvex'] = 2
solution = opt.solve(model)
V_opt = np.array([pyo.value(model.V[i,j]) for i in model.n for j in model.r]).reshape(len(model.n),len(model.r))
obj_values = pyo.value(model.obj)
print("optimum point: \n {} ".format(V_opt))
print("optimal objective: {}".format(obj_values))