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 lsq_pos_l1_penalty(matrix, rhs, cost_multiplier, weights_0): """ min 2-norm (matrix*w - rhs)** + 1-norm(cost_multiplier*(w-w0)) s.t. e'w = 1 w >= 0 """ # define model with Model('lsqSparse') as model: # introduce n non-negative weight variables weights = model.variable("weights", matrix.shape[1], Domain.inRange(0.0, +np.infty)) # e'*w = 1 model.constraint(Expr.sum(weights), Domain.equalsTo(1.0)) # sum of squared residuals v = __l2_norm_squared(model, "2-norm(res)**", __residual(matrix, rhs, weights)) # \Gamma*(w - w0), p is an expression p = Expr.mulElm(cost_multiplier, Expr.sub(weights, weights_0)) cost = model.variable("cost", matrix.shape[1], Domain.unbounded()) model.constraint(Expr.sub(cost, p), Domain.equalsTo(0.0)) t = __l1_norm(model, 'abs(weights)', cost) # Minimise v + t model.objective(ObjectiveSense.Minimize, __sum_weighted(1.0, v, 1.0, t)) # solve the problem model.solve() return np.array(weights.level())
def lsq_pos_l1_penalty(matrix, rhs, cost_multiplier, weights_0): """ min 2-norm (matrix*w - rhs)** + 1-norm(cost_multiplier*(w-w0)) s.t. e'w = 1 w >= 0 """ # define model model = mModel.build_model('lsqSparse') # introduce n non-negative weight variables weights = mModel.weights(model, "weights", n=matrix.shape[1], lb=0.0) # e'*w = 1 mBound.equal(model, Expr.sum(weights), 1.0) # sum of squared residuals v = mMath.l2_norm_squared(model, "2-norm(res)**", __residual(matrix, rhs, weights)) print matrix.shape[1] print weights_0 print weights # \Gamma*(w - w0), p is an expression p = mMath.mat_vec_prod(cost_multiplier, Expr.sub(weights, weights_0)) t = mMath.l1_norm(model, 'abs(weights)', p) # Minimise v + t mModel.minimise(model, __sum_weighted(1.0, v, 1.0, t)) return np.array(weights.level())
def _update(self, clique_data, parent=None): for c in clique_data['cliques']: self.cliques[c['name']] = v = self.model.variable( c['name'], Domain.inRange(0.0, 1.0), Domain.isInteger() ) for img, cmd in product(c['images'], c['commands']): self.by_img_cmd[img,cmd].append(v) self._obj.append(Expr.mul(float(c['time']), v)) if parent is not None: self.model.constraint( 'c-%s-%s' % (c['name'], parent['name']), Expr.sub(v, self.cliques[parent['name']]), Domain.lessThan(0.0) ) for child in c['children']: self._update(child, c) for inter in clique_data['intersections']: self.model.constraint( 'iter-%d' % self._inter, Expr.add([self.cliques[i] for i in inter]), Domain.lessThan(1.0) ) self._inter += 1
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_2') as M: # variable X = M.variable('X', Domain.inPSDCone(n + 1)) t = M.variable() # objective function M.objective(ObjectiveSense.Maximize, t) # constraints for i in range(n + 1): M.constraint(f'c{i}{i}', X.index(i, i), Domain.equalsTo(1.)) if i == 0: continue M.constraint(f'c0,{i}', Expr.sub(X.index(0, i), t), Domain.greaterThan(0.)) for j in range(i + 1, n + 1): if A[i - 1, j - 1] == 0: M.constraint(f'c{i},{j}', X.index(i, j), Domain.equalsTo(0.)) # solution M.solve() X_sol = X.level() t_sol = t.level() t_sol = t_sol[0] theta = 1. / t_sol**2 return theta
def Markowitz(mu, covar, alpha, L=1.0): # defineerime kasutatava mudeli kasutades keskväärtusi model = mosek_model.build_model('meanVariance') # toome sisse n mitte-negatiivset kaalu weights = mosek_model.weights(model, "weights", n=covar.shape[1]) # leiame kaaludele vastavad finantsvõimendused lev = mosek_math.l1_norm(model, "leverage", weights) # nõuame, et oleks täidetud tingimus omega_1 + ... + omega_n = 1 mosek_bound.equal(model, Expr.sum(weights), 1.0) # nõuame, et oleks täidetud tingimus |omega_1| + ... + |omega_n| <= L mosek_bound.upper(model, lev, L) v = Expr.dot(mu.values, weights) # arvutame variatsiooni var = mosek_math.variance(model, "variance", weights, alpha*covar.values) # maksimeerime sihifunktsiooni mosek_model.maximise(model, Expr.sub(v, var)) # arvutame lõpuks kaalud ja tagastame need weights = pd.Series(data=np.array(weights.level()), index=stocks.keys()) return weights
def solve(x0, risk_alphas, loadings, srisk, cost_per_trade=DEFAULT_COST, max_risk=0.01): N = len(x0) # don't hold no risk data (likely dead) lim = np.where(srisk.isnull(), 0.0, 1.0) loadings = loadings.fillna(0) srisk = srisk.fillna(0) risk_alphas = risk_alphas.fillna(0) with Model() as m: w = m.variable(N, Domain.inRange(-lim, lim)) longs = m.variable(N, Domain.greaterThan(0)) shorts = m.variable(N, Domain.greaterThan(0)) gross = m.variable(N, Domain.greaterThan(0)) m.constraint( "leverage_consistent", Expr.sub(gross, Expr.add(longs, shorts)), Domain.equalsTo(0), ) m.constraint("net_consistent", Expr.sub(w, Expr.sub(longs, shorts)), Domain.equalsTo(0.0)) m.constraint("leverage_long", Expr.sum(longs), Domain.lessThan(1.0)) m.constraint("leverage_short", Expr.sum(shorts), Domain.lessThan(1.0)) buys = m.variable(N, Domain.greaterThan(0)) sells = m.variable(N, Domain.greaterThan(0)) gross_trade = Expr.add(buys, sells) net_trade = Expr.sub(buys, sells) total_gross_trade = Expr.sum(gross_trade) m.constraint( "net_trade", Expr.sub(w, net_trade), Domain.equalsTo(np.asarray(x0)), # cannot handle series ) # add risk constraint vol = m.variable(1, Domain.lessThan(max_risk)) stacked = Expr.vstack(vol.asExpr(), Expr.mulElm(w, srisk.values)) stacked = Expr.vstack(stacked, Expr.mul(loadings.values.T, w)) m.constraint("vol-cons", stacked, Domain.inQCone()) alphas = risk_alphas.dot(np.vstack([loadings.T, np.diag(srisk)])) gain = Expr.dot(alphas, net_trade) loss = Expr.mul(cost_per_trade, total_gross_trade) m.objective(ObjectiveSense.Maximize, Expr.sub(gain, loss)) m.solve() result = pd.Series(w.level(), srisk.index) return result
def markowitz(exp_ret, covariance_mat, aversion): # define model model = mModel.build_model("mean_var") # set of n weights (unconstrained) weights = mModel.weights(model, "weights", n=len(exp_ret)) mBound.equal(model, Expr.sum(weights), 1.0) # standard deviation induced by covariance matrix var = mMath.variance(model, "var", weights, covariance_mat) mModel.maximise(model=model, expr=Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var))) return np.array(weights.level())
def markowitz(exp_ret, covariance_mat, aversion): # define model with Model("mean var") as model: # set of n weights (unconstrained) weights = model.variable("weights", len(exp_ret), Domain.inRange(-np.infty, +np.infty)) model.constraint(Expr.sum(weights), Domain.equalsTo(1.0)) # standard deviation induced by covariance matrix var = __variance(model, "var", weights, covariance_mat) model.objective(ObjectiveSense.Maximize, Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var))) model.solve() #mModel.maximise(model=model, expr=Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var))) return np.array(weights.level())
def markowitz(exp_ret, covariance_mat, aversion): # define model with Model("mean var") as model: # set of n weights (unconstrained) weights = model.variable("weights", len(exp_ret), Domain.inRange(-np.infty, +np.infty)) model.constraint(Expr.sum(weights), Domain.equalsTo(1.0)) # standard deviation induced by covariance matrix var = __variance(model, "var", weights, covariance_mat) model.objective( ObjectiveSense.Maximize, Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var))) model.solve() #mModel.maximise(model=model, expr=Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var))) return np.array(weights.level())
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 _update(self, clique_data, parent=None): for c in clique_data['cliques']: self.cliques[c['name']] = v = self.model.variable( c['name'], Domain.inRange(0.0, 1.0), Domain.isInteger()) for img, cmd in product(c['images'], c['commands']): self.by_img_cmd[img, cmd].append(v) self._obj.append(Expr.mul(float(c['time']), v)) if parent is not None: self.model.constraint( 'c-%s-%s' % (c['name'], parent['name']), Expr.sub(v, self.cliques[parent['name']]), Domain.lessThan(0.0)) for child in c['children']: self._update(child, c) for inter in clique_data['intersections']: self.model.constraint('iter-%d' % self._inter, Expr.add([self.cliques[i] for i in inter]), Domain.lessThan(1.0)) self._inter += 1
q_2_b = m.variable('q_2_b', *binary) q_2_c = m.variable('q_2_c', *binary) q_2_d = m.variable('q_2_d', *binary) q_3_b = m.variable('q_3_b', *binary) q_3_c = m.variable('q_3_c', *binary) q_3_d = m.variable('q_3_d', *binary) # Inclusion of an image and a command means that image must # use all command invocation from the clique. # For instance: # (1) z_1_a <= w_1 # (2) z_1_a <= y_a # (3) z_1_a >= w_1 + y_a - 1 m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0)) m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0)) m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])), Domain.greaterThan(-1.0)) m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0)) m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0)) m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])), Domain.greaterThan(-1.0)) m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])), Domain.greaterThan(-1.0)) m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])), Domain.greaterThan(-1.0)) m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0)) m.constraint('c_2_a_2', Expr.sub(z_2_a, y_a), Domain.lessThan(0.0)) m.constraint('c_2_a_3', Expr.sub(z_2_a, Expr.add([w_2, y_a])), Domain.greaterThan(-1.0)) m.constraint('c_2_b_1', Expr.sub(z_2_b, w_2), Domain.lessThan(0.0))
def solve(self, problem, saver): # Construct model. self.problem = problem self.model = model = Model() if self.time is not None: model.setSolverParam('mioMaxTime', 60.0 * int(self.time)) # x[1,c] = 1 if the master schedule has (null, c) in its first stage # x[s,c1,c2] = 1 if the master schedule has (c1, c2) in stage s > 1 x = {} for s in problem.all_stages: if s == 1: # First arc in the individual image path. for c in problem.commands: x[1,c] = model.variable( 'x[1,%s]' % c, 1, Domain.inRange(0.0, 1.0), Domain.isInteger() ) else: # Other arcs. for c1, c2 in product(problem.commands, problem.commands): if c1 == c2: continue x[s,c1,c2] = model.variable( 'x[%s,%s,%s]' % (s,c1,c2), 1, Domain.inRange(0.0, 1.0), Domain.isInteger() ) smax = max(problem.all_stages) obj = [0.0] # TODO: deal with images that do not have the same number of commands. # t[s,c] is the total time incurred at command c in stage s t = {} for s in problem.all_stages: for c in problem.commands: t[s,c] = model.variable( 't[%s,%s]' % (c,s), 1, Domain.greaterThan(0.0) ) if s == 1: model.constraint('t[1,%s]' % c, Expr.sub(t[1,c], Expr.mul(float(problem.commands[c]), x[1,c])), Domain.greaterThan(0.0) ) else: rhs = [0.0] for c1, coeff in problem.commands.items(): if c1 == c: continue else: rhs = Expr.add(rhs, Expr.mulElm(t[s-1,c1], x[s,c1,c])) model.constraint('t[%s,%s]' % (s,c), Expr.sub(t[1,c], rhs), Domain.greaterThan(0.0) ) # Objective function = sum of aggregate comand times if s == smax: obj = Expr.add(obj, t[s,c]) # y[i,1,c] = 1 if image i starts by going to c # y[i,s,c1,c2] = 1 if image i goes from command c1 to c2 in stage s > 1 y = {} for i, cmds in problem.images.items(): for s in problem.stages[i]: if s == 1: # First arc in the individual image path. for c in cmds: y[i,1,c] = model.variable( 'y[%s,1,%s]' % (i,c), 1, Domain.inRange(0.0, 1.0), Domain.isInteger() ) model.constraint('x_y[i%s,1,c%s]' % (i,c), Expr.sub(x[1,c], y[i,1,c]), Domain.greaterThan(0.0) ) else: # Other arcs. for c1, c2 in product(cmds, cmds): if c1 == c2: continue y[i,s,c1,c2] = model.variable( 'y[%s,%s,%s,%s]' % (i,s,c1,c2), 1, Domain.inRange(0.0, 1.0), Domain.isInteger() ) model.constraint('x_y[i%s,s%s,c%s,c%s]' % (i,s,c1,c2), Expr.sub(x[s,c1,c2], y[i,s,c1,c2]), Domain.greaterThan(0.0) ) for c in cmds: # Each command is an arc destination exactly once. arcs = [y[i,1,c]] for c1 in cmds: if c1 == c: continue arcs.extend([y[i,s,c1,c] for s in problem.stages[i][1:]]) model.constraint('y[i%s,c%s]' % (i,c), Expr.add(arcs), Domain.equalsTo(1.0) ) # Network balance equations (stages 2 to |stages|-1). # Sum of arcs in = sum of arcs out. for s in problem.stages[i][:len(problem.stages[i])-1]: if s == 1: arcs_in = [y[i,1,c]] else: arcs_in = [y[i,s,c1,c] for c1 in cmds if c1 != c] arcs_out = [y[i,s+1,c,c2] for c2 in cmds if c2 != c] model.constraint('y[i%s,s%s,c%s]' % (i,s,c), Expr.sub(Expr.add(arcs_in), Expr.add(arcs_out)), Domain.equalsTo(0.0) ) model.objective('z', ObjectiveSense.Minimize, Expr.add(x.values())) # model.objective('z', ObjectiveSense.Minimize, obj) model.setLogHandler(sys.stdout) model.acceptedSolutionStatus(AccSolutionStatus.Feasible) model.solve() # Create optimal schedule. schedule = defaultdict(list) for i, cmds in problem.images.items(): for s in problem.stages[i]: if s == 1: # First stage starts our walk. for c in cmds: if y[i,s,c].level()[0] > 0.5: schedule[i].append(c) break else: # After that we know what our starting point is. for c2 in cmds: if c2 == c: continue if y[i,s,c,c2].level()[0] > 0.5: schedule[i].append(c2) c = c2 break saver(schedule)
z_2_a = m.variable('z_2_a', *binary) z_2_b = m.variable('z_2_b', *binary) z_2_c = m.variable('z_2_c', *binary) z_2_d = m.variable('z_2_d', *binary) z_3_b = m.variable('z_3_b', *binary) z_3_c = m.variable('z_3_c', *binary) z_3_d = m.variable('z_3_d', *binary) # Inclusion of an image and a command means that image must # use all command invocation from the clique. # For instance: # (1) z_1_a <= w_1 # (2) z_1_a <= y_a # (3) z_1_a >= w_1 + y_a - 1 m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0)) m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0)) m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])), Domain.greaterThan(-1.0)) m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0)) m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0)) m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])), Domain.greaterThan(-1.0)) m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])), Domain.greaterThan(-1.0)) m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])), Domain.greaterThan(-1.0)) m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0))
m.constraint('c_2_a', Expr.add([x_2_a, x_12_a]), Domain.equalsTo(1.0)) m.constraint('c_2_b', Expr.add([x_2_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_2_c', Expr.add([x_2_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_2_d', Expr.add([x_2_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_3_b', Expr.add([x_3_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_3_c', Expr.add([x_3_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_3_d', Expr.add([x_3_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) # Add dependency constraints for sub-cliques. m.constraint('d_123_b_23_cd', Expr.sub(x_123_b, x_123_b_23_cd), Domain.greaterThan(0.0)) # Eliminated intersections between cliques. m.constraint('e1', Expr.add([x_23_bcd, x_123_b]), Domain.lessThan(1.0)) m.constraint('e2', Expr.add([x_12_a, x_123_b]), Domain.lessThan(1.0)) m.constraint('e3', Expr.add([x_12_a, x_23_bcd]), Domain.lessThan(1.0)) # Minimize resources required to construct all images. obj = [ Expr.mul(c, x) for c, x in [ # Individual image/command pairs (r['A'], x_1_a), (r['B'], x_1_b), (r['A'], x_2_a), (r['B'], x_2_b),
x_123_b = m.variable('x_123_b', *binary) x_123_b_23_cd = m.variable('x_123_b_23_cd', *binary) # Each command must be run once for each image. m.constraint('c_1_a', Expr.add([x_1_a]), Domain.equalsTo(1.0)) m.constraint('c_1_b', Expr.add([x_1_b, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_2_a', Expr.add([x_2_a]), Domain.equalsTo(1.0)) m.constraint('c_2_b', Expr.add([x_2_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_2_c', Expr.add([x_2_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_2_d', Expr.add([x_2_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_3_b', Expr.add([x_3_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_3_c', Expr.add([x_3_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_3_d', Expr.add([x_3_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) # Add dependency constraints for sub-cliques. m.constraint('d_123_b_23_cd', Expr.sub(x_123_b, x_123_b_23_cd), Domain.greaterThan(0.0)) # Eliminated intersections between cliques. m.constraint('e1', Expr.add([x_23_bcd, x_123_b]), Domain.lessThan(1.0)) # Minimize resources required to construct all images. obj = [Expr.mul(c, x) for c, x in [ # Individual image/command pairs (r['A'], x_1_a), (r['B'], x_1_b), (r['A'], x_2_a), (r['B'], x_2_b), (r['C'], x_2_c), (r['D'], x_2_d), (r['B'], x_3_b), (r['C'], x_3_c), (r['D'], x_3_d), # Cliques (r['B'] + r['C'] + r['D'], x_23_bcd), (r['B'], x_123_b), (r['C'] + r['D'], x_123_b_23_cd),
# clique or incur its own cost. q_2_b = m.variable('q_2_b', *binary) q_2_c = m.variable('q_2_c', *binary) q_2_d = m.variable('q_2_d', *binary) q_3_b = m.variable('q_3_b', *binary) q_3_c = m.variable('q_3_c', *binary) q_3_d = m.variable('q_3_d', *binary) # Inclusion of an image and a command means that image must # use all command invocation from the clique. # For instance: # (1) z_1_a <= w_1 # (2) z_1_a <= y_a # (3) z_1_a >= w_1 + y_a - 1 m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0)) m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0)) m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])), Domain.greaterThan(-1.0)) m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0)) m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0)) m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])), Domain.greaterThan(-1.0)) m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])), Domain.greaterThan(-1.0)) m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])), Domain.greaterThan(-1.0)) m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0)) m.constraint('c_2_a_2', Expr.sub(z_2_a, y_a), Domain.lessThan(0.0)) m.constraint('c_2_a_3', Expr.sub(z_2_a, Expr.add([w_2, y_a])), Domain.greaterThan(-1.0)) m.constraint('c_2_b_1', Expr.sub(z_2_b, w_2), Domain.lessThan(0.0))
x_12_a = m.variable('x_12_a', *binary) # Each command must be run once for each image. m.constraint('c_1_a', Expr.add([x_1_a, x_12_a]), Domain.equalsTo(1.0)) m.constraint('c_1_b', Expr.add([x_1_b, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_2_a', Expr.add([x_2_a, x_12_a]), Domain.equalsTo(1.0)) m.constraint('c_2_b', Expr.add([x_2_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_2_c', Expr.add([x_2_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_2_d', Expr.add([x_2_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_3_b', Expr.add([x_3_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0)) m.constraint('c_3_c', Expr.add([x_3_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) m.constraint('c_3_d', Expr.add([x_3_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0)) # Add dependency constraints for sub-cliques. m.constraint('d_123_b_23_cd', Expr.sub(x_123_b, x_123_b_23_cd), Domain.greaterThan(0.0)) # Eliminated intersections between cliques. m.constraint('e1', Expr.add([x_23_bcd, x_123_b]), Domain.lessThan(1.0)) m.constraint('e2', Expr.add([x_12_a, x_123_b]), Domain.lessThan(1.0)) m.constraint('e3', Expr.add([x_12_a, x_23_bcd]), Domain.lessThan(1.0)) # Minimize resources required to construct all images. obj = [Expr.mul(c, x) for c, x in [ # Individual image/command pairs (r['A'], x_1_a), (r['B'], x_1_b), (r['A'], x_2_a), (r['B'], x_2_b), (r['C'], x_2_c), (r['D'], x_2_d), (r['B'], x_3_b), (r['C'], x_3_c), (r['D'], x_3_d), # Cliques (r['B'] + r['C'] + r['D'], x_23_bcd),
def solve(self, problem, saver): # Construct model. self.problem = problem self.model = model = Model() if self.time is not None: model.setSolverParam('mioMaxTime', 60.0 * int(self.time)) # x[i,s,c] = 1 if image i runs command c during stage s, 0 otherwise. self.x = x = {} for i, cmds in problem.images.items(): for s, c in product(problem.stages[i], cmds): x[i,s,c] = model.variable( 'x[%s,%s,%s]' % (i,s,c), 1, Domain.inRange(0.0, 1.0), Domain.isInteger() ) # y[ip,iq,s,c] = 1 if images ip & iq have a shared path through stage # s by running command c during s, 0 otherwise. y = {} for (ip, iq), cmds in problem.shared_cmds.items(): for s, c in product(problem.shared_stages[ip, iq], cmds): y[ip,iq,s,c] = model.variable( 'y[%s,%s,%s,%s]' % (ip,iq,s,c), 1, Domain.inRange(0.0, 1.0), Domain.isInteger() # Domain.inRange(0.0, 1.0) ) # TODO: need to remove presolved commands so the heuristic doesn't try them. # TODO: Add a heuristic initial solution. # TODO: Presolving # Each image one command per stage, and each command once. for i in problem.images: for s in problem.stages[i]: model.constraint('c1[%s,%s]' % (i,s), Expr.add([x[i,s,c] for c in problem.images[i]]), Domain.equalsTo(1.0) ) for c in problem.images[i]: model.constraint('c2[%s,%s]' % (i,c), Expr.add([x[i,s,c] for s in problem.stages[i]]), Domain.equalsTo(1.0) ) # Find shared paths among image pairs. for (ip, iq), cmds in problem.shared_cmds.items(): for s in problem.shared_stages[ip,iq]: for c in cmds: model.constraint('c3[%s,%s,%s,%s]' % (ip,iq,s,c), Expr.sub(y[ip,iq,s,c], x[ip,s,c]), Domain.lessThan(0.0) ) model.constraint('c4[%s,%s,%s,%s]' % (ip,iq,s,c), Expr.sub(y[ip,iq,s,c], x[iq,s,c]), Domain.lessThan(0.0) ) if s > 1: lhs = Expr.add([y[ip,iq,s,c] for c in cmds]) rhs = Expr.add([y[ip,iq,s-1,c] for c in cmds]) model.constraint('c5[%s,%s,%s,%s]' % (ip,iq,s,c), Expr.sub(lhs, rhs), Domain.lessThan(0.0) ) if y: obj = Expr.add(y.values()) else: obj = 0.0 model.objective('z', ObjectiveSense.Maximize, obj) model.setLogHandler(sys.stdout) model.acceptedSolutionStatus(AccSolutionStatus.Feasible) model.solve() # Create optimal schedule. schedule = defaultdict(list) for i, stages in problem.stages.items(): for s in stages: for c in problem.images[i]: if x[i,s,c].level()[0] > 0.5: schedule[i].append(c) break saver(schedule)
def __residual(matrix, rhs, expr): """ Introduce the residual matrix*expr - rhs """ return Expr.sub(__mat_vec_prod(matrix, expr), rhs)
z_2_a = m.variable('z_2_a', *binary) z_2_b = m.variable('z_2_b', *binary) z_2_c = m.variable('z_2_c', *binary) z_2_d = m.variable('z_2_d', *binary) z_3_b = m.variable('z_3_b', *binary) z_3_c = m.variable('z_3_c', *binary) z_3_d = m.variable('z_3_d', *binary) # Inclusion of an image and a command means that image must # use all command invocation from the clique. # For instance: # (1) z_1_a <= w_1 # (2) z_1_a <= y_a # (3) z_1_a >= w_1 + y_a - 1 m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0)) m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0)) m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])), Domain.greaterThan(-1.0)) m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0)) m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0)) m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])), Domain.greaterThan(-1.0)) m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])), Domain.greaterThan(-1.0)) m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])), Domain.greaterThan(-1.0)) m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0)) m.constraint('c_2_a_2', Expr.sub(z_2_a, y_a), Domain.lessThan(0.0)) m.constraint('c_2_a_3', Expr.sub(z_2_a, Expr.add([w_2, y_a])), Domain.greaterThan(-1.0)) m.constraint('c_2_b_1', Expr.sub(z_2_b, w_2), Domain.lessThan(0.0))
def solve(self, problem, saver): # Construct model. self.problem = problem self.model = model = Model() if self.time is not None: model.setSolverParam('mioMaxTime', 60.0 * int(self.time)) # x[1,c] = 1 if the master schedule has (null, c) in its first stage # x[s,c1,c2] = 1 if the master schedule has (c1, c2) in stage s > 1 x = {} for s in problem.all_stages: if s == 1: # First arc in the individual image path. for c in problem.commands: x[1, c] = model.variable('x[1,%s]' % c, 1, Domain.inRange(0.0, 1.0), Domain.isInteger()) else: # Other arcs. for c1, c2 in product(problem.commands, problem.commands): if c1 == c2: continue x[s, c1, c2] = model.variable('x[%s,%s,%s]' % (s, c1, c2), 1, Domain.inRange(0.0, 1.0), Domain.isInteger()) smax = max(problem.all_stages) obj = [0.0] # TODO: deal with images that do not have the same number of commands. # t[s,c] is the total time incurred at command c in stage s t = {} for s in problem.all_stages: for c in problem.commands: t[s, c] = model.variable('t[%s,%s]' % (c, s), 1, Domain.greaterThan(0.0)) if s == 1: model.constraint( 't[1,%s]' % c, Expr.sub(t[1, c], Expr.mul(float(problem.commands[c]), x[1, c])), Domain.greaterThan(0.0)) else: rhs = [0.0] for c1, coeff in problem.commands.items(): if c1 == c: continue else: rhs = Expr.add( rhs, Expr.mulElm(t[s - 1, c1], x[s, c1, c])) model.constraint('t[%s,%s]' % (s, c), Expr.sub(t[1, c], rhs), Domain.greaterThan(0.0)) # Objective function = sum of aggregate comand times if s == smax: obj = Expr.add(obj, t[s, c]) # y[i,1,c] = 1 if image i starts by going to c # y[i,s,c1,c2] = 1 if image i goes from command c1 to c2 in stage s > 1 y = {} for i, cmds in problem.images.items(): for s in problem.stages[i]: if s == 1: # First arc in the individual image path. for c in cmds: y[i, 1, c] = model.variable('y[%s,1,%s]' % (i, c), 1, Domain.inRange(0.0, 1.0), Domain.isInteger()) model.constraint('x_y[i%s,1,c%s]' % (i, c), Expr.sub(x[1, c], y[i, 1, c]), Domain.greaterThan(0.0)) else: # Other arcs. for c1, c2 in product(cmds, cmds): if c1 == c2: continue y[i, s, c1, c2] = model.variable( 'y[%s,%s,%s,%s]' % (i, s, c1, c2), 1, Domain.inRange(0.0, 1.0), Domain.isInteger()) model.constraint( 'x_y[i%s,s%s,c%s,c%s]' % (i, s, c1, c2), Expr.sub(x[s, c1, c2], y[i, s, c1, c2]), Domain.greaterThan(0.0)) for c in cmds: # Each command is an arc destination exactly once. arcs = [y[i, 1, c]] for c1 in cmds: if c1 == c: continue arcs.extend( [y[i, s, c1, c] for s in problem.stages[i][1:]]) model.constraint('y[i%s,c%s]' % (i, c), Expr.add(arcs), Domain.equalsTo(1.0)) # Network balance equations (stages 2 to |stages|-1). # Sum of arcs in = sum of arcs out. for s in problem.stages[i][:len(problem.stages[i]) - 1]: if s == 1: arcs_in = [y[i, 1, c]] else: arcs_in = [y[i, s, c1, c] for c1 in cmds if c1 != c] arcs_out = [y[i, s + 1, c, c2] for c2 in cmds if c2 != c] model.constraint( 'y[i%s,s%s,c%s]' % (i, s, c), Expr.sub(Expr.add(arcs_in), Expr.add(arcs_out)), Domain.equalsTo(0.0)) model.objective('z', ObjectiveSense.Minimize, Expr.add(x.values())) # model.objective('z', ObjectiveSense.Minimize, obj) model.setLogHandler(sys.stdout) model.acceptedSolutionStatus(AccSolutionStatus.Feasible) model.solve() # Create optimal schedule. schedule = defaultdict(list) for i, cmds in problem.images.items(): for s in problem.stages[i]: if s == 1: # First stage starts our walk. for c in cmds: if y[i, s, c].level()[0] > 0.5: schedule[i].append(c) break else: # After that we know what our starting point is. for c2 in cmds: if c2 == c: continue if y[i, s, c, c2].level()[0] > 0.5: schedule[i].append(c2) c = c2 break saver(schedule)
def __residual(matrix, rhs, expr): """ Introduce the residual matrix*expr - rhs """ return Expr.sub(mMath.mat_vec_prod(matrix, expr), rhs)
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