def solve_sdp_program(W): assert W.ndim == 2 assert W.shape[0] == W.shape[1] W = W.copy() n = W.shape[0] W = expand_matrix(W) with Model('gw_max_3_cut') as M: W = Matrix.dense(W / 3.) J = Matrix.ones(3*n, 3*n) # variable Y = M.variable('Y', Domain.inPSDCone(3*n)) # objective function M.objective(ObjectiveSense.Maximize, Expr.dot(W, Expr.sub(J, Y))) # constraints for i in range(3*n): M.constraint(f'c_{i}{i}', Y.index(i, i), Domain.equalsTo(1.)) for i in range(n): M.constraint(f'c_{i}^01', Y.index(i*3, i*3+1), Domain.equalsTo(-1/2.)) M.constraint(f'c_{i}^02', Y.index(i*3, i*3+2), Domain.equalsTo(-1/2.)) M.constraint(f'c_{i}^12', Y.index(i*3+1, i*3+2), Domain.equalsTo(-1/2.)) for j in range(i+1, n): for a, b in product(range(3), repeat=2): M.constraint(f'c_{i}{j}^{a}{b}-0', Y.index(i*3 + a, j*3 + b), Domain.greaterThan(-1/2.)) M.constraint(f'c_{i}{j}^{a}{b}-1', Expr.sub(Y.index(i*3 + a, j*3 + b), Y.index(i*3 + (a + 1) % 3, j*3 + (b + 1) % 3)), Domain.equalsTo(0.)) M.constraint(f'c_{i}{j}^{a}{b}-2', Expr.sub(Y.index(i*3 + a, j*3 + b), Y.index(i*3 + (a + 2) % 3, j*3 + (b + 2) % 3)), Domain.equalsTo(0.)) # solution M.solve() Y_opt = Y.level() return np.reshape(Y_opt, (3*n,3*n))
def minimum_variance(matrix): # Given the matrix of returns a (each column is a series of returns) this method # computes the weights for a minimum variance portfolio, e.g. # min 2-Norm[a*w]^2 # s.t. # w >= 0 # sum[w] = 1 # This is the left-most point on the efficiency frontier in the classic Markowitz theory # build the model with Model("Minimum Variance") as model: # introduce the weight variable weights = model.variable("weights", matrix.shape[1], Domain.inRange(0.0, 1.0)) # sum of weights has to be 1 model.constraint(Expr.sum(weights), Domain.equalsTo(1.0)) # returns r = Expr.mul(Matrix.dense(matrix), weights) # compute l2_norm squared of those returns # minimize this l2_norm model.objective(ObjectiveSense.Minimize, __l2_norm_squared(model, "2-norm^2(r)", expr=r)) # solve the problem model.solve() # return the series of weights return np.array(weights.level())
def solve_sdp_program(A): assert A.ndim == 2 assert A.shape[0] == A.shape[1] A = A.copy() n = A.shape[0] with Model('theta_1') as M: A = Matrix.dense(A) # variable X = M.variable('X', Domain.inPSDCone(n)) # objective function M.objective(ObjectiveSense.Maximize, Expr.sum(Expr.dot(Matrix.ones(n, n), X))) # constraints M.constraint(f'c1', Expr.sum(Expr.dot(X, A)), Domain.equalsTo(0.)) M.constraint(f'c2', Expr.sum(Expr.dot(X, Matrix.eye(n))), Domain.equalsTo(1.)) # solution M.solve() sol = X.level() return sum(sol)
def solve_sdp_program(W): assert W.ndim == 2 assert W.shape[0] == W.shape[1] W = W.copy() n = W.shape[0] with Model('gw_max_cut') as M: W = Matrix.dense(W / 4.) J = Matrix.ones(n, n) # variable Y = M.variable('Y', Domain.inPSDCone(n)) # objective function M.objective(ObjectiveSense.Maximize, Expr.dot(W, Expr.sub(J, Y))) # constraints for i in range(n): M.constraint(f'c_{i}', Y.index(i, i), Domain.equalsTo(1.)) # solve M.solve() # solution Y_opt = Y.level() return np.reshape(Y_opt, (n, n))
def solve_sdp_program(W, k): assert W.ndim == 2 assert W.shape[0] == W.shape[1] W = W.copy() n = W.shape[0] with Model('fj_max_k_cut') as M: W = Matrix.dense((k - 1) / (2 * k) * W) J = Matrix.ones(n, n) # variable Y = M.variable('Y', Domain.inPSDCone(n)) # objective function M.objective(ObjectiveSense.Maximize, Expr.dot(W, Expr.sub(J, Y))) # constraints for i in range(n): M.constraint(f'c_{i}', Y.index(i, i), Domain.equalsTo(1.)) for j in range(i + 1, n): M.constraint(f'c_{i},{j}', Y.index(i, j), Domain.greaterThan(-1 / (k - 1))) # solution M.solve() Y_opt = Y.level() return np.reshape(Y_opt, (n, n))
def __init__(self,name, foods, nutrients, daily_allowance, nutritive_value): Model.__init__(self,name) finished = False try: self.foods = [ str(f) for f in foods ] self.nutrients = [ str(n) for n in nutrients ] self.dailyAllowance = numpy.array(daily_allowance, float) self.nutrientValue = Matrix.dense(nutritive_value).transpose() M = len(self.foods) N = len(self.nutrients) if len(self.dailyAllowance) != N: raise ValueError("Length of daily_allowance does not match " "the number of nutrients") if self.nutrientValue.numColumns() != M: raise ValueError("Number of rows in nutrient_value does not " "match the number of foods") if self.nutrientValue.numRows() != N: raise ValueError("Number of columns in nutrient_value does " "not match the number of nutrients") self.__dailyPurchase = self.variable('Daily Purchase', StringSet(self.foods), Domain.greaterThan(0.0)) self.__dailyNutrients = \ self.constraint('Nutrient Balance', StringSet(nutrients), Expr.mul(self.nutrientValue,self.__dailyPurchase), Domain.greaterThan(self.dailyAllowance)) self.objective(ObjectiveSense.Minimize, Expr.sum(self.__dailyPurchase)) finished = True finally: if not finished: self.__del__()
def __mat_vec_prod(matrix, expr): return Expr.mul(Matrix.dense(matrix), expr)
def Build_Co_Model(self): r = len(self.roads) mu, sigma = self.mu, self.sigma m, n, r = self.m, self.n, len(self.roads) f, h = self.f, self.h M, N = m + n + r, 2 * m + 2 * n + r A = self.__Construct_A_Matrix() A_Mat = Matrix.dense(A) b = self.__Construct_b_vector() # ---- build Mosek Model COModel = Model() # -- Decision Variable Z = COModel.variable('Z', m, Domain.inRange(0.0, 1.0)) I = COModel.variable('I', m, Domain.greaterThan(0.0)) Alpha = COModel.variable('Alpha', M, Domain.unbounded()) # M by 1 vector Beta = COModel.variable('Beta', M, Domain.unbounded()) # M by 1 vector Theta = COModel.variable('Theta', N, Domain.unbounded()) # N by 1 vector # M1_matrix related decision variables ''' [tau, xi^T, phi^T M1 = xi, eta, psi^t phi, psi, w ] ''' # no-need speedup variables Psi = COModel.variable('Psi', [N, n], Domain.unbounded()) Xi = COModel.variable('Xi', n, Domain.unbounded()) # n by 1 vector Phi = COModel.variable('Phi', N, Domain.unbounded()) # N by 1 vector # has the potential to speedup Tau, Eta, W = self.__Declare_SpeedUp_Vars(COModel) # M2 matrix decision variables ''' [a, b^T, c^T M2 = b, e, d^t c, d, f ] ''' a_M2 = COModel.variable('a_M2', 1, Domain.greaterThan(0.0)) b_M2 = COModel.variable('b_M2', n, Domain.greaterThan(0.0)) c_M2 = COModel.variable('c_M2', N, Domain.greaterThan(0.0)) e_M2 = COModel.variable('e_M2', [n, n], Domain.greaterThan(0.0)) d_M2 = COModel.variable('d_M2', [N, n], Domain.greaterThan(0.0)) f_M2 = COModel.variable('f_M2', [N, N], Domain.greaterThan(0.0)) # -- Objective Function obj_1 = Expr.dot(f, Z) obj_2 = Expr.dot(h, I) obj_3 = Expr.dot(b, Alpha) obj_4 = Expr.dot(b, Beta) obj_5 = Expr.dot([1], Expr.add(Tau, a_M2)) obj_6 = Expr.dot([2 * mean for mean in mu], Expr.add(Xi, b_M2)) obj_7 = Expr.dot(sigma, Expr.add(Eta, e_M2)) COModel.objective( ObjectiveSense.Minimize, Expr.add([obj_1, obj_2, obj_3, obj_4, obj_5, obj_6, obj_7])) # Constraint 1 _expr = Expr.sub(Expr.mul(A_Mat.transpose(), Alpha), Theta) _expr = Expr.sub(_expr, Expr.mul(2, Expr.add(Phi, c_M2))) _expr_rhs = Expr.vstack(Expr.constTerm([0.0] * n), Expr.mul(-1, I), Expr.constTerm([0.0] * M)) COModel.constraint('constr1', Expr.sub(_expr, _expr_rhs), Domain.equalsTo(0.0)) del _expr, _expr_rhs # Constraint 2 _first_term = Expr.add([ Expr.mul(Beta.index(row), np.outer(A[row], A[row]).tolist()) for row in range(M) ]) _second_term = Expr.add([ Expr.mul(Theta.index(k), Matrix.sparse(N, N, [k], [k], [1])) for k in range(N) ]) _third_term = Expr.add(W, f_M2) _expr = Expr.sub(Expr.add(_first_term, _second_term), _third_term) COModel.constraint('constr2', _expr, Domain.equalsTo(0.0)) del _expr, _first_term, _second_term, _third_term # Constraint 3 _expr = Expr.mul(-2, Expr.add(Psi, d_M2)) _expr_rhs = Matrix.sparse([[Matrix.eye(n)], [Matrix.sparse(N - n, n)]]) COModel.constraint('constr3', Expr.sub(_expr, _expr_rhs), Domain.equalsTo(0)) del _expr, _expr_rhs # Constraint 4: I <= M*Z COModel.constraint('constr4', Expr.sub(Expr.mul(20000.0, Z), I), Domain.greaterThan(0.0)) # Constraint 5: M1 is SDP COModel.constraint( 'constr5', Expr.vstack(Expr.hstack(Tau, Xi.transpose(), Phi.transpose()), Expr.hstack(Xi, Eta, Psi.transpose()), Expr.hstack(Phi, Psi, W)), Domain.inPSDCone(1 + n + N)) return COModel