Expressions

Expressions in the DCP analyzer and quiz are formed from the variables x, y, z, u, v, w, the parameters a, b, c, d, e, f, numerical constants such as 3 and -2.44, the standard arithmetic operators +, -, *, /, and a library of functions. For simplicity, all expressions are scalar. Some examples are

3.69 + b/3
x - 4*u
sqrt(x) - min(u, v - a)
max(2.66 - sqrt(u), square(x + 2*y))

Sign

Each (sub)expression is flagged as positive (non-negative), negative (non-positive), or unknown.

The signs of larger expressions are determined from the signs of their subexpressions. For example, the sign of the expression expr1*expr2 is

• Positive if expr1 and expr2 have the same (known) sign.
• Negative if expr1 and expr2 have opposite (known) signs.
• Unknown if either expression has unknown sign.

The sign given to an expression is always correct. But DCP sign analysis may flag an expression as unknown sign when the sign could be figured out through more complex analysis. For instance, x*x is positive but has unknown sign by the rules above.

Curvature

Each (sub)expression is flagged as one of the following curvatures

Curvature	Meaning
constant	$ f(x) $ independent of $ x $
affine	$ f(\theta x + (1-\theta)y) = \theta f(x) + (1-\theta)f(y), \; \forall x, \; y,\; \theta \in [0,1] $
convex	$ f(\theta x + (1-\theta)y) \leq \theta f(x) + (1-\theta)f(y), \; \forall x, \; y,\; \theta \in [0,1] $
concave	$ f(\theta x + (1-\theta)y) \geq \theta f(x) + (1-\theta)f(y), \; \forall x, \; y,\; \theta \in [0,1] $
unknown	DCP analysis cannot determine the curvature

using the curvature rules given below. As with sign analysis, the conclusion is always correct, but the simple analysis can flag expressions as unknown even when they are convex or concave. Note that any constant expression is also affine, and any affine expression is convex and concave.

Curvature Rules

DCP analysis is based on applying a general composition theorem from convex analysis to each (sub)expression.

$f(expr_1, expr_2, ..., expr_n)$ is convex if $\text{ } f$ is a convex function and for each $expr_{i}$ one of the following conditions holds:

• $f$ is increasing in argument $i$ and $expr_{i}$ is convex.
• $f$ is decreasing in argument $i$ and $expr_{i}$ is concave.
• $expr_{i}$ is affine.

$f(expr_1, expr_2, ..., expr_n)$ is concave if $\text{ } f$ is a concave function and for each $expr_{i}$ one of the following conditions holds:

• $f$ is increasing in argument $i$ and $expr_{i}$ is concave.
• $f$ is decreasing in argument $i$ and $expr_{i}$ is convex.
• $expr_{i}$ is affine.

$f(expr_1, expr_2, ..., expr_n)$ is affine if $\text{ } f$ is an affine function and each $expr_{i}$ is affine.

If none of the three rules apply, the expression $\text{ } f(expr_1, expr_2, ..., expr_n)$ is marked as having unknown curvature.

Whether a function is increasing or decreasing in an argument may depend on the sign of the argument. For instance, square is increasing for positive arguments and decreasing for negative arguments.

Infix Operators

The infix operators +, -, *, / are treated exactly like functions. The infix operators + and - are affine, so the rules above are used to flag the curvature. For example, expr1 + expr2 is flagged as convex if expr1 and expr2 are convex.

expr1*expr2 is allowed only when one of the expressions is constant. expr1/expr2 is allowed only when expr2 is constant. The curvature rules above apply. For example, expr1/expr2 is convex when expr1 is concave and expr2 is negative and constant.

Example 1

DCP analysis breaks expressions down into subexpressions. The tree visualization below shows how this works for the expression 2*square(x) + 3. Each subexpression is shown in a blue box. We mark its curvature on the left and its sign on the right.

Example 2

We'll walk through the application of the DCP rules to the expression sqrt(1 + square(x)).

The variable x has affine curvature and unknown sign. The square function is convex and non-monotone for arguments of unknown sign. It can take the affine expression x as an argument; the result square(x) is convex.

The arithmetic operator + is affine and increasing, so the composition 1 + square(x) is convex by the curvature rule for convex functions. The sqrt function is concave and increasing, which means it can only take a concave argument. Since 1 + square(x) is convex, sqrt(1 + square(x)) violates the DCP rules and cannot be verified as convex.

In fact, sqrt(1 + square(x)) is a convex function of x, but the DCP rules are not able to verify convexity. If the expression is written as norm2(1, x), the L2 norm of the vector $[1,x]$, which has the same value as sqrt(1 + square(x)), then it will be certified as convex using the DCP rules.