CVXMOD – Built-in atoms

Introduction

Atoms in CVXMOD are used to manipulate expressions and generate functions. All atoms take one or more expressions as arguments, and generate a new expression. CVXMOD supplies several common atoms. Here they are split into two groups: LP representable atoms, which can be used within linear programs, and other atoms, which cannot.

LP representable atoms

• sum(expr)
  Sum of the elements of expr.
  Affine and increasing in expr.

• max(expr)
  Maximum element of expr.
  Convex and increasing in expr.

• max((expr1, expr2, ...))
  Maximum element of a list of (scalar) expressions.
  Convex and increasing in expr.

• min(expr)
  Minimum element of expr.
  Concave and increasing in expr.

• min((expr1, expr2, ...))
  Minimum element of a list of (scalar) expressions.
  Concave and increasing in expr.

• abs(expr)
  Elementwise absolute value of expr.
  Convex in expr.

• pos(expr)
  Elementwise positive portion of expr. Equivalent to max(expr, 0).
  Positive; convex and increasing in expr.

• neg(expr)
  Elementwise negative portion of expr. It is equivalent to min(expr, 0).
  Negative; concave and increasing in expr.

• norm1(expr)
  1-norm of expr. Equivalent to sum(abs(expr)).
  Positive; convex in expr.

• norminf(expr)
  Infinity-norm of expr. Equivalent to max(abs(expr)).
  Positive; convex in expr.

Other atoms

• norm2(expr)
  2-norm (euclidean norm) of expr. Mathematically equivalent to sqrt(sum(square(expr))).
  Positive; convex in expr.

• entropy(expr)
  Entropy of expr. Mathematically equivalent to the sum over i of -expr_i*log(expr_i).
  Concave in expr.

• exp(expr)
  Elementwise exponentiation of expr.
  Positive; convex and increasing in expr.

• huber(expr, M=1)
  Huber penalty function of expr, with halfwidth M. Defined as 2*M*abs(x) - M**2 for abs(x) >= M, and abs(x)**2 for abs(x) <= M. M must be a positive constant.
  Convex in expr.

• kl(u, v)
  Kullback Leibler divergence between u and v. Defined as the sum over i of u_i*log(u_i/v_i).
  Jointly convex in u and v.

• log(expr)
  Elementwise logarithm of expr.
  Concave and increasing in expr.

• lse(expr)
  Logarithm of the sum of the elementwise exponentiation of expr. (Often called log-sum-exp.) Equivalent to log(sum(exp(expr))).
  Convex and increasing in expr.

• power(expr, p)
  Elementwise pth power.
  Positive; convex and increasing in expr if p is a positive even integer.
  Positive; convex and increasing in expr if p > 1 and expr is known positive.
  Positive; concave and increasing in expr if 0 < p < 1 and expr is known positive.
  

• quadform(expr, Q)
  Quadratic form. Mathematically equivalent to transpose(expr)*Q*expr.
  Positive; convex in expr if Q is known positive semidefinite (PSD).
  Negative; concave in expr if Q is known negative semidefinite (NSD).
  Monotonic when the sign of expr is known.

• square(expr)
  Elementwise square of expr.
  Positive; convex in expr.
  Monotonic when the sign of expr is known.

• sqrt(expr)
  Concave; increasing in expr, with the domain constraint expr >= 0.