# Discplined Convex Programming

Disciplined convex programming (DCP) is a system for constructing mathematical expressions with known curvature from a given library of base functions. `CVXR`

uses DCP to ensure that the specified optimization problems are convex.

This section of the tutorial explains the rules of DCP and how they are applied by `CVXR`

.

Visit dcp.stanford.edu for a more interactive introduction to DCP.

## Expressions

Expressions in `CVXR`

are formed from variables, numerical constants such as R vectors and matrices, the standard arithmetic operators `+, -, *, %*%, /`

, and a library of functions. Here are some examples of `CVXR`

expressions:

```
suppressWarnings(suppressMessages(library(CVXR)))
## Create variables.
x <- Variable()
y <- Variable()
## Examples of cvxr expressions.
3.69 + x / 3
```

`## AddExpression(c("Expression(AFFINE, UNKNOWN, 1)", "Expression(AFFINE, UNKNOWN, 1)"), Constant(CONSTANT, POSITIVE, (1,1)))`

`x - 4 * y`

`## AddExpression(Variable(1, 1), c("Expression(AFFINE, UNKNOWN, 1)", "Expression(AFFINE, UNKNOWN, 1)"))`

`sqrt(x) - min_elemwise(y, x - 2.5)`

`## AddExpression(c("Expression(CONCAVE, POSITIVE, 1)", "Expression(CONCAVE, POSITIVE, 1)"), c("Expression(CONVEX, UNKNOWN, 1)", "Expression(CONVEX, UNKNOWN, 1)"))`

`max_elemwise(2.66 - sqrt(y), abs(x + 2 * y))`

`## MaxElemwise(c("Expression(CONVEX, UNKNOWN, 1)", "Expression(CONVEX, UNKNOWN, 1)"), c("Expression(CONVEX, POSITIVE, 1)", "Expression(CONVEX, POSITIVE, 1)"))`

Expressions can be scalars, vectors, or matrices. The dimensions of an expression can be obtained using the function `size`

. `CVXR`

will raise an exception if an expression is used in a way that doesn’t make sense given its dimensions, e.g. adding matrices of different sizes.

```
Z <- Variable(5, 4)
A <- matrix(1, nrow = 2, ncol = 5)
## Use size to get the dimensions.
cat("dimensions of Z:", size(Z), "\n")
```

`## dimensions of Z: 5 4`

`cat("dimensions of sum_entries(Z):", size(sum_entries(Z)), "\n")`

`## dimensions of sum_entries(Z): 1 1`

`cat("dimensions of A %*% Z:", size(A %*% Z), "\n")`

`## dimensions of A %*% Z: 2 4`

Error raised for invalid dimensions.

`tryCatch(A + Z, error = function(e) geterrmessage())`

`## [1] "Incompatible dimensions"`

`CVXR`

uses DCP analysis to determine the sign and curvature of each expression.

## Sign

Each (sub)expression is flagged as *positive* (non-negative), *negative* (non-positive), *zero*, 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

- Zero if either expression has sign zero.
- 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. However, 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.

`CVXR`

determines the sign of constants by looking at their value. For scalar constants, this is straightforward. Vector and matrix constants with all positive (negative) entries are marked as positive (negative). Vector and matrix constants with both positive and negative entries are marked as unknown sign.

The sign of an expression is obtained via the `sign`

function:

```
x <- Variable()
a <- Constant(-1)
c <- matrix(c(1, -1), ncol = 1)
cat("sign of x:", sign(x), "\n")
```

`## sign of x: UNKNOWN`

`cat("sign of a:", sign(a), "\n")`

`## sign of a: NEGATIVE`

`cat("sign of abs(x):", sign(abs(x)), "\n")`

`## sign of abs(x): POSITIVE`

`cat("sign of c*a:", sign(c * a), "\n")`

`## sign of c*a: UNKNOWN`

## Curvature

Each (sub)expression is flagged as one of the following curvatures (with respect to its variables)

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.

\(f(e_1, e_2, \ldots, e_n)\) is convex if \(f\) is a convex function and for each \(e_{i}\), one of the following conditions holds:

- \(f\) is increasing in argument \(i\) and \(e_i\) is convex.
- \(f\) is decreasing in argument \(i\) and \(e_i\) is concave.
- \(e_i\) is affine or constant.

\(f(e_1, e_2,\ldots, e_n)\) is concave if \(f\) is a concave function and for each \(e_i\) one of the following conditions holds:

- \(f\) is increasing in argument \(i\) and \(e_i\) is concave.
- \(f\) is decreasing in argument \(i\) and \(e_i\) is convex.
- \(e_i\) is affine or constant.

\(f(e_1, e_2, \ldots, e_n)\) is affine if \(f\) is an affine function and each \(e_i\) is affine.

If none of the three rules apply, the expression \(f(e_1, e_2,\ldots, e_n)\) 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, `abs`

is increasing for positive arguments and decreasing for negative arguments.

The curvature of an expression is determined using the `curvature`

function:

```
x <- Variable()
a <- Constant(5)
cat("curvature of x:", curvature(x), "\n")
```

`## curvature of x: AFFINE`

`cat("curvature of a:", curvature(a), "\n")`

`## curvature of a: CONSTANT`

`cat("curvature of abs(x):", curvature(abs(x)), "\n")`

`## curvature of abs(x): CONVEX`

`cat("curvature of sqrt(x):", curvature(sqrt(x)), "\n")`

`## curvature of sqrt(x): CONCAVE`

## 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`

and `expr1 %*% expr2`

are allowed only when one of the expressions is constant. If both expressions are non-constant, `CVXR`

will raise an exception. `expr1/expr2`

is allowed only when `expr2`

is a scalar 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`

. Here `square`

is synonymous with squaring each element of the input `x`

. Each subexpression is displayed 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 function `sqrt`

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 `p_norm(vstack(1, x), 2)`

, 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.

`cat("sqrt(1 + square(x)) curvature:", curvature(sqrt(1 + square(x))), "\n")`

`## sqrt(1 + square(x)) curvature: UNKNOWN`

`cat("p_norm(vstack(1, x), 2) curvature:", curvature(p_norm(vstack(1, x), 2)), "\n")`

`## p_norm(vstack(1, x), 2) curvature: CONVEX`

## DCP problems

A problem is constructed from an objective and a list of constraints. If a problem follows the DCP rules, it is guaranteed to be convex and solvable by `CVXR`

. The DCP rules require that the problem objective have one of two forms:

- Minimize(convex)
- Maximize(concave)

The only valid constraints under the DCP rules are

- affine == affine
- convex <= concave
- concave >= convex

You can check that a problem, constraint, or objective satisfies the DCP rules by calling `is_dcp(object)`

. Here are some examples of DCP and non-DCP problems:

```
x <- Variable()
y <- Variable()
## DCP problems.
prob1 <- Problem(Minimize((x - y)^2), list(x + y >= 0))
prob2 <- Problem(Maximize(sqrt(x - y)),
list(2 * x - 3 == y,
x^2 <= 2))
cat("prob1 is DCP:", is_dcp(prob1), "\n")
```

`## prob1 is DCP: TRUE`

`cat("prob2 is DCP:", is_dcp(prob2), "\n")`

`## prob2 is DCP: TRUE`

```
## Non-DCP problems.
## A non-DCP objective.
prob3 <- Problem(Maximize(x^2))
cat("prob3 is DCP:", is_dcp(prob3), "\n")
```

`## prob3 is DCP: FALSE`

`cat("Maximize(x^2) is DCP:", is_dcp(Maximize(x^2)), "\n")`

`## Maximize(x^2) is DCP: FALSE`

```
## A non-DCP constraint.
prob4 <- Problem(Minimize(x^2), list(sqrt(x) <= 2))
cat("prob4 is DCP:", is_dcp(prob4), "\n")
```

`## prob4 is DCP: FALSE`

`cat("sqrt(x) <= 2 is DCP:", is_dcp(sqrt(x) <= 2), "\n")`

`## sqrt(x) <= 2 is DCP: FALSE`

`CVXR`

will raise an exception if you call `solve()`

on a non-DCP problem.

```
## A non-DCP problem.
prob <- Problem(Minimize(sqrt(x)))
tryCatch(solve(prob), error = function(e) geterrmessage())
```

`## [1] "Problem does not follow DCP rules."`

## Session Info

`sessionInfo()`

```
## R version 3.4.2 (2017-09-28)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS High Sierra 10.13.1
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] methods stats graphics grDevices datasets utils base
##
## other attached packages:
## [1] kableExtra_0.6.0 CVXR_0.94-4
##
## loaded via a namespace (and not attached):
## [1] gmp_0.5-13.1 Rcpp_0.12.13 highr_0.6
## [4] plyr_1.8.4 compiler_3.4.2 R.methodsS3_1.7.1
## [7] R.utils_2.6.0 tools_3.4.2 digest_0.6.12
## [10] bit_1.1-12 viridisLite_0.2.0 evaluate_0.10.1
## [13] tibble_1.3.4 lattice_0.20-35 rlang_0.1.2
## [16] Matrix_1.2-11 yaml_2.1.14 blogdown_0.1.7
## [19] Rmpfr_0.6-1 ECOSolveR_0.3-2 stringr_1.2.0
## [22] httr_1.3.1 knitr_1.17 xml2_1.1.1
## [25] hms_0.3 rprojroot_1.2 bit64_0.9-7
## [28] grid_3.4.2 R6_2.2.2 rmarkdown_1.6
## [31] bookdown_0.5 readr_1.1.1 magrittr_1.5
## [34] backports_1.1.1 scales_0.5.0 htmltools_0.3.6
## [37] scs_1.1-1 rvest_0.3.2 colorspace_1.3-2
## [40] stringi_1.1.5 munsell_0.4.3 R.oo_1.21.0
```