Skip to content

CVXPY

Basics

import cvxpy as cp
x = cp.Variable()
y = cp.Variable()

constraints = [x + y == 1,
               x - y >= 1]

obj = cp.Minimize((x-y)**2)

prob = cp.Problem(obj, constraints)
prob.solve()

print(prob.status)
print(prob.value)
print(x.value, y.value)

  • Status:

    • optimal (cp.OPTIMAL)

      solved successfully.

    • infeasible (cp.INFEASIBLE)

      eg. Minimize(x), [x>=1, x<=0]

    • unbounded (cp.UNBOUNDED)

      eg. only Minimize(x)

    • *_inaccurate

      if only get low accuracy result than desired.

    • SolverError

      eg. not a DCP problem.

      should try other solvers.

  • Variables

    dimensions can be 0,1,2.

    scalar = cp.Variable()
vector = cp.Variable(5) # [5,]
matrix = cp.Variable((2,3)) # [2,3]

    Numpy ndarray, Scipy sparse matrix is supported.

    # Problem data.
m = 10
n = 5
numpy.random.seed(1)
A = numpy.random.randn(m, n)
b = numpy.random.randn(m)

# Construct the problem.
x = cp.Variable(n)
objective = cp.Minimize(cp.sum_squares(A*x - b))
constraints = [0 <= x, x <= 1]
prob = cp.Problem(objective, constraints)

print("Optimal value", prob.solve())
print("Optimal var")
print(x.value) # A numpy ndarray.

  • Constraints

    use ==, >=, <= for constraints.

    semidefinite

  • Parameters

    Parameter is used to change the value of a constant without rebuilding the Problem.

    Can be attributed as nonpos, nonneg, ... (for DCP use)

    # Positive scalar parameter.
m = cp.Parameter(nonneg=True)

# Column vector parameter with unknown sign (by default).
c = cp.Parameter(5)

# Matrix parameter with negative entries.
G = cp.Parameter((4, 7), nonpos=True)

# Assigns a constant value to G.
G.value = -numpy.ones((4, 7))

# Create parameter, then assign value.
rho = cp.Parameter(nonneg=True)
rho.value = 2

# Initialize parameter with a value.
rho = cp.Parameter(nonneg=True, value=2)

DCP

DCP is used to ensure the optimization problem is convex. (sufficient but not necessary)

  • Expressions

    formed by

    • Variable, Parameter, numerical constants (support ndarray)
    • +,-,*,/
    • cvxpy functions

    Properties:shape, size, ndim, sign, curvature

    • shape, size, ndim

    • sign

      ZERO, UNKNOWN, NONPOSITIVE, NONNEGATIVE

    • curvature

      AFFINE, CONSTANT, CONVEX, CONCAVE, UNKNOWN

  • DCP tree analysis

    ../../_images/example2.png

  • DCP rules

    • Objectice
      • Minimize(Convex)
      • Maximize(Concave)
    • Constraints
      • affine == affine
      • convex <= concave
      • concave >= convex

    prob.is_dcp()

Atomic functions

Index and slice and transpose and power

like numpy

expr.T

expr**2 or power(expr, 2)

scalar functions
  • max() convex

  • min() concave

    consider min(concave, concave) is concave.

  • norm(X) default is 2-norm.

    norm(X, 1), norm(X, "inf")

  • sum_squares(X) convex
  • sum(X) affine
  • quad_over_lin(X, y)
  • trace(X) affine
  • ...
element-wise functions
  • abs(x) convex

  • entr(x)

    entropy, -xlogx, concave

  • exp(x), log(x)
  • neg(x), pos(x), both convex
  • sqrt(x), power(x)
vector/matrix functions
  • reshape() affine
  • diag() affine