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
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