CVXPY的作用

与matlab中cvx的工具包类似，用于求解凸优化问题。cvx与cvxpy都是由CIT的Stephen Boyd教授课题组开发。cvx用于matlab的包，cvxpy是用于python的包。下载、安装及学习地址如下：
CVX
CVXPY

一、CVXPY入门 ## 简单例子 ## import cvxpy as cvx#定义优化变量x = cvx.Variable() y = cvx.Variable()# 定义约束条件constraints = [x + y == 1, x – y >= 1]# 定义优化问题obj = cvx.Minimize((x – y)**2)# 定义优化问题prob = cvx.Problem(obj, constraints)#求解问题prob.solve() #返回最优值 print(“status:”, prob.status) #求解状态 print(“optimal value”, prob.value) #目标函数优化值print(“optimal var”, x.value, y.value) #优化变量的值，相应变量加.value #运行结果status: optimaloptimal value 0.999999999761 #最优值视为 1（计算机精度问题）optimal var 1.00000000001 -1.19961841702e-11 （x=1,y=0） # 改变目标函数，利用已定义问题的约束，**问题被定义后，不可改变prob2 = cvx.Problem(cvx.Maximize(x + y), prob.constraints)print(“optimal value”, prob2.solve())# 改变约束constraints = [x + y <= 3] + prob.constraints[1:]prob2 = cvx.Problem(prob.objective, constraints)print(“optimal value”, prob2.solve()) #运行结果optimal value 1.0optimal value 3.00000000006 二、求解状态（XXX.status） 求解状态含义OPTIMAL最优解INFEASIBLE不可行UNBOUNDED无界OPTIMAL_INACCURATE不精确INFEASIBLE_INACCURATE不精确UNBOUNDED_INACCURATE不精确

对于opyimal情况，求解的就是最优解，对于infeasible与unbounded见2.1节。对于后三种情况，表明求解精度低（低于期望精度），如果求解器抛出异常（SolverError），可以尝试选择其他求解器。见2.2节。

2.1 不可行及无界情况 import cvxpy as cvxx = cvx.Variable()# 不可行问题prob = cvx.Problem(cvx.Minimize(x), [x >= 1, x <= 0])prob.solve()print(“status:”, prob.status)print(“optimal value”, prob.value)# 无界问题prob = cvx.Problem(cvx.Minimize(x))prob.solve()print(“status:”, prob.status)print(“optimal value”, prob.value) #运行结果status: infeasibleoptimal value infstatus: unboundedoptimal value -inf

对minimization问题，infeasible对应inf，unbounded对应-inf；maximization问题反之。

2.2 求解器选择

各求解器能求解的问题：

# Solving a problem with different solvers.x = cvx.Variable(2)obj = cvx.Minimize(x[0] + cvx.norm(x, 1))constraints = [x >= 2]prob = cvx.Problem(obj, constraints)# Solve with ECOS.prob.solve(solver=cvx.ECOS)print(“optimal value with ECOS:”, prob.value)# Solve with ECOS_BB.prob.solve(solver=cvx.ECOS_BB)print(“optimal value with ECOS_BB:”, prob.value)# Solve with CVXOPT.prob.solve(solver=cvx.CVXOPT)print(“optimal value with CVXOPT:”, prob.value)# Solve with SCS.prob.solve(solver=cvx.SCS)print(“optimal value with SCS:”, prob.value)# Solve with GLPK.prob.solve(solver=cvx.GLPK)print(“optimal value with GLPK:”, prob.value)# Solve with GLPK_MI.prob.solve(solver=cvx.GLPK_MI)print(“optimal value with GLPK_MI:”, prob.value)# Solve with GUROBI.prob.solve(solver=cvx.GUROBI)print(“optimal value with GUROBI:”, prob.value)# Solve with MOSEK.prob.solve(solver=cvx.MOSEK)print(“optimal value with MOSEK:”, prob.value)# Solve with Elemental.prob.solve(solver=cvx.ELEMENTAL)print(“optimal value with Elemental:”, prob.value)# Solve with CBC.prob.solve(solver=cvx.CBC)print(“optimal value with CBC:”, prob.value)optimal value with ECOS: 5.99999999551optimal value with ECOS_BB: 5.99999999551optimal value with CVXOPT: 6.00000000512optimal value with SCS: 6.00046055789optimal value with GLPK: 6.0optimal value with GLPK_MI: 6.0optimal value with GUROBI: 6.0optimal value with MOSEK: 6.0optimal value with Elemental: 6.0000044085242727optimal value with CBC: 6.0//Use the installed_solvers utility function to get a list of the solvers your installation of CVXPY //supports.print installed_solvers()[‘CBC’, ‘CVXOPT’, ‘MOSEK’, ‘GLPK’, ‘GLPK_MI’, ‘ECOS_BB’, ‘ECOS’, ‘SCS’ 变量类型 （Variable）

变量可以是标量、向量以及矩阵

# 标量a = cvx.Variable()# 向量x = cvx.Variable(5)# 矩阵大小为(5, 1).x = cvx.Variable((5, 1))# 矩阵大小为(4, 7).A = cvx.Variable((4, 7))

cvxpy中可以做常数使用的用：
– NumPy ndarrays
– NumPy matrices
– SciPy sparse matrices

# Solves a bounded least-squares problem.import cvxpy as cvximport numpy# Problem data.m = 10n = 5numpy.random.seed(1)A = numpy.random.randn(m, n)b = numpy.random.randn(m)# Construct the problem.x = cvx.Variable(n)objective = cvx.Minimize(cvx.sum_squares(A*x – b))constraints = [0 <= x, x <= 1]prob = cvx.Problem(objective, constraints)print(“Optimal value”, prob.solve())print(“Optimal var”)print(x.value) # A numpy ndarray.** #运行结果 Optimal value 4.14133859146 Optimal var [ -5.11480673e-21 6.30625742e-21 1.34643668e-01 1.24976681e-01-4.79039542e-21] 三、约束（constraint）

可以使用 ==, <=，>= ，不能使用< ，>（没有意义，不接收）。也不能使用0 <= x <= 1（与cvx不同） or x == y == 2.（不能识别）
约束：[0 <= x , x <= 1]，意味着x的每个元素都在0，1之间。

四、参数（parameters）

parameters可以理解为参数求解问题里的一个常数，可以是标量、向量、矩阵。在没有求解问题前（xxx.solve()），其允许你改变其值。【有点难理解，看一下下面的例子。】

#paraters的定义与赋值# Positive scalar parameter.m = cvx.Parameter(nonneg=True)# Column vector parameter with unknown sign (by default).c = cvx.Parameter(5)# Matrix parameter with negative entries.G = cvx.Parameter((4, 7), nonpos=True)# Assigns a constant value to G.G.value = -numpy.ones((4, 7))# Create parameter, then assign value.rho = cvx.Parameter(nonneg=True)rho.value = 2# Initialize parameter with a value.rho = cvx.Parameter(nonneg=True, value=2) #使用实例import cvxpy as cvximport numpyimport matplotlib.pyplot as plt# Problem data.n = 15m = 10numpy.random.seed(1)A = numpy.random.randn(n, m)b = numpy.random.randn(n)# gamma must be nonnegative due to DCP rules.gamma = cvx.Parameter(nonneg=True)# Construct the problem.x = cvx.Variable(m)error = cvx.sum_squares(A*x – b)obj = cvx.Minimize(error + gamma*cvx.norm(x, 1))prob = cvx.Problem(obj)# Construct a trade-off curve of ||Ax-b||^2 vs. ||x||_1sq_penalty = []l1_penalty = []x_values = []gamma_vals = numpy.logspace(-4, 6)for val in gamma_vals: gamma.value = val prob.solve() # Use expr.value to get the numerical value of # an expression in the problem. sq_penalty.append(error.value) l1_penalty.append(cvx.norm(x, 1).value) x_values.append(x.value)plt.rc(‘text’, usetex=True)plt.rc(‘font’, family=’serif’)plt.figure(figsize=(6,10))# Plot trade-off curve.plt.subplot(211)plt.plot(l1_penalty, sq_penalty)plt.xlabel(r’\|x\|_1′, fontsize=16)plt.ylabel(r’\|Ax-b\|^2′, fontsize=16)plt.title(‘Trade-Off Curve for LASSO’, fontsize=16)# Plot entries of x vs. gamma.plt.subplot(212)for i in range(m): plt.plot(gamma_vals, [xi[i] for xi in x_values])plt.xlabel(r’\gamma’, fontsize=16)plt.ylabel(r’x_{i}’, fontsize=16)plt.xscale(‘log’)plt.title(r’\text{Entries of x vs. }\gamma’, fontsize=16)plt.tight_layout()plt.show()from multiprocessing import Pool# Assign a value to gamma and find the optimal x.def get_x(gamma_value): gamma.value = gamma_value result = prob.solve() return x.value# Parallel computation (set to 1 process here).pool = Pool(processes = 1)x_values = pool.map(get_x, gamma_vals)

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