Reference guide — CVX Users' Guide

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# Reference guide

In this section we describe each operator, function, set, and command that you are likely to encounter in CVX. In some cases, limitations of the underlying solver place certain restrictions or caveats on their use:

- Functions marked with a dagger (†) are not supported natively by the solvers that CVX uses. They are handled using a successive approximation method which makes multiple calls to the underlying solver, achieving the same final precision. If you use one of these functions, you will be warned that successive approximation will be used. This technique is discussed further in The successive approximation method. As this section discusses, this is an experimental approach that works well in many cases, but cannot be guaranteed.
- Functions involving powers (e.g., `x^p`) and \(p\)-norms (e.g., `norm(x,p)`) are marked with a double dagger (‡). CVX represents these functions exactly when \(p\) is a rational number. For irrational values of `p`, a nearby rational is selected instead. See Power functions and p-norms for details on how both cases are handled.

## Arithmetic operators

Matlab’s standard arithmetic operations for addition `+`, subtraction `-`, multiplication, `*` `.*`, division `/` `./` `\` `.\`, and exponentiation `^` `.^` have been overloaded to work in CVX whenever appropriate—that is, whenever their use is consistent with both standard mathematical and Matlab conventions and the DCP ruleset. For example:

- Two CVX expressions can be added together if they are of the same dimension (or one is scalar) and have the same curvature (i.e., both are convex, concave, or affine).
- A CVX expression can be multiplied or divided by a scalar constant. If the constant is positive, the curvature is preserved; if it is negative, curvature is reversed.
- An affine column vector CVX expression can be multiplied by a constant matrix of appropriate dimensions; or it can be left-divided by a non-singular constant matrix of appropriate dimension.

Numerous other combinations are possible, of course. The use of the exponentiation operators `^` `.^` are somewhat limited; see the definitions of `power` in Nonlinear below.

Matlab’s basic matrix manipulation and arithmetic operations have been extended to work with CVX expressions as well, including:

- Concatenation: `[ A, B ; C, D ]`
- Indexing: `x(n+1:end)`, `X([3,4],:)`, etc.
- Indexed assignment, including deletion: `y(2:4) = 1`, `Z(1:4,:) = []`, etc.
- Transpose and conjugate transpose: `Z.'`, `y'`

## Built-in functions

### Linear

A number of Matlab’s basic linear and bilinear functions either work automatically with `cvx` expressions or have been extended to do so, including: `conj`, `conv`, `cumsum`, `diag`, `dot`, `find`, `fliplr`, `flipud`, `flipdim`, `horzcat`, `hankel`, `ipermute`, `kron`, `mean`, `permute`, `repmat`, `reshape`, `rot90`, `sparse`, `sum`, `trace`, `tril`, `triu`, `toeplitz`, `vertcat`.

Most should behave identically with CVX expressions as they do with numeric expressions. Those that perform some sort of summation, such as `cumsum`, `sum`, or multiplication, such as `conv`, `dot` or `kron`, can only be used in accordance with the disciplined convex programming rules. For example, `kron(X,Y)` is valid only if either `X` or `Y` is constant; and `trace(Z)` is valid only if the elements along the diagonal have the same curvature.

### Nonlinear

`abs`
  absolute value for real and complex arrays. Convex.
† `exp`
  exponential. Convex and nondecreasing.
† `log`
  logarithm. Concave and nondecreasing.
`max`
  maximum. Convex and nondecreasing.
`min`
  minimum. Concave and nondecreasing.
`norm`

  norms for real and complex vectors and matrices. Convex. Thus function follows the Matlab conventions closely. Thus the one-argument version `norm(x)` computes the 2-norm for vectors, and the 2-norm (maximum singular value) for matrices. The two-argument version `norm(x,p)` is supported as follows:

  - ‡ For vectors, all values \(p\geq 1\) are accepted.
  - For matrices, `p` must be `1`, `2`, `Inf`, or `'Fro'`.
`polyval`

  polynomial evaluation. `polyval(p,x)`, where `p` is a vector of length `n`, computes

      p(1) * x.^(n-1) + p(2) * x.^(n-2) + ... + p(n-1) * x + p(n)

  This function can be used in CVX in two ways:

  - If `p` is a variable and `x` is a constant, then `polyval(x,p)` computes a linear combination of the elements of `p`. The combination must satisfy the DCP rules for addition and scaling.
  - If `p` is a constant and `x` is a variable, then `polyval(x,p)` constructs a polynomial function of the variable `x`. The polynomial must be affine, convex, or concave, and `x` must be real and affine.
‡ `power(x,p)`

  `x^p` and `x.^p`, where `x` is a real variable and and `p` is a real constant. For `x^p`, both `x` and `p` must be scalars. Only those values of `p` which can reasonably and unambiguously interpreted as convex or concave are accepted:

  - \(p=0\). Constant. `x.^p` is treated as identically 1.
  - \(0 < p < 1\). Concave. The argument \(x\) must be concave (or affine), and is implicitly constrained to be nonnegative.
  - \(p = 1\). Affine. `x.^p` is simply `x`.
  - \(p \in \{2,4,6,8,...\}\). Convex. Argument \(x\) must be affine.
  - \(p > 1\), \(p\not\in\{2,3,4,5,...\}\). Convex. Argument \(x\) must be affine, and is implicitly constrained to be nonnegative.

  Negative and odd integral values of \(p\) are not permitted, but see the functions `pow_p`, `pow_pos`, and `pow_abs` in the next section for useful alternatives.

† `power(p,x)`

  `p.^x` and `p^x`, where `p` is a real constant and `x` is a real variable. For `p^x`, both `p` and `x` must be scalars. Valid values of `p` include:

  - \(p \in \{0,1\}\). Constant.
  - \(0 < p < 1\). Convex and nonincreasing; `x` must be concave.
  - \(p > 1\). Convex and nondecreasing; `x` must be convex.

  Negative values of `p` are not permitted.

`std`
  Standard deviation. Convex.
`sqrt`
  Square root. Implicitly constrains its argument to be nonnegative. Concave and nondecreasing.
`var`
  Variance. Convex.

## New functions

Even though these functions were developed specifically for CVX, they work outside of a CVX specification as well, when supplied with numeric arguments.

`avg_abs_dev`
  The average absolute deviation about the mean \(\mu(x)\) of \(x\). Convex.
    \[f_{\text{aad}}(x) = \frac{1}{n} \sum_{i=1}^n |x_i-\mu(x)| = \frac{1}{n} \sum_{i=1}^n \left| x_i - {\textstyle\frac{1}{n}\sum_i x_i}\right| = \frac{1}{n}\left\| (I-\tfrac{1}{n}\textbf{1}\textbf{1}^T)x \right\|_1.\]
`avg_abs_dev_med`
  The average absolute deviation about the median \(\mathop{\textrm{m}}(x)\) of \(x\). Convex.
    \[f_{\text{aadm}}(x) = \frac{1}{n} \sum_{i=1}^n |x_i-\mathop{\textrm{m}}(x)| = \inf_y \frac{1}{n} \sum_{i=1}^n |x_i-y|\]
`berhu(x,M)`
  The reversed Huber function (hence, Berhu), defined as
    \[\begin{split}f_{\text{berhu}}(x,M) \triangleq \begin{cases} |x| & |x| \leq M \\ (|x|^2+M^2)/2M & |x| \geq M \end{cases}\end{split}\]

  Convex. If \(M\) is omitted, \(M=1\) is assumed; but if supplied, it must be a positive constant. Also callable with three arguments as `berhu(x,M,t)`, which computes `t+t*berhu(x/t,M)`, useful for concomitant scale estimation (see [Owen06]).

`det_inv`
  determinant of inverse of a symmetric (or Hermitian) positive definite matrix, \(\det X^{-1}\), which is the same as the product of the inverses of the eigenvalues. When used inside a CVX specification, `det_inv` constrains the matrix to be symmetric (if real) or Hermitian (if complex) and positive semidefinite. When used with numerical arguments, `det_inv` returns `+Inf` if these constraints are not met. Convex.
`det_rootn`
  \(n\)-th root of the determinant of a semidefinite matrix, \((\det X)^{1/n}\). When used inside a CVX specification, `det_rootn` constrains the matrix to be symmetric (if real) or Hermitian (if complex) and positive semidefinite. When used with numerical arguments, `det_rootn` returns `-Inf` if these constraints are not met. Concave.
`det_root2n`
  the \(2n\)-th root of the determinant of a semidefinite matrix; i.e., `det_root2n(X)=sqrt(det_rootn(X))`. Concave. Maintained solely for back-compatibility purposes.
† `entr`
  the elementwise entropy function: `entr(x)=-x.*log(x)`. Concave. Returns `-Inf` when called with a constant argument that has a negative entry.
`geo_mean`
  the geometric mean of a vector, \(\left( \prod_{k=1}^n x_k \right)^{1/n}\). When used inside a CVX specification, `geo_mean` constrains the elements of the vector to be nonnegative. When used with numerical arguments, `geo_mean` returns `-Inf` if any element is negative. Concave and increasing.
`huber(x,M)`
  The Huber function, defined as
    \[\begin{split}f_{\text{huber}}(x,M) \triangleq \begin{cases} |x|^2 & |x| \leq M \\ 2M|x|-M^2 & |x| \geq M \end{cases}\end{split}\]

  Convex. If $x$ is a vector or array, the function is applied on an elementwise basis. If $M$ is omitted, then $M=1$ is assumed; but if it supplied, it must be a positive constant. Also callable as `huber(x,M,t)`, which computes `t+t*huber(x/t,M)`, useful for concomitant scale estimation (see [Owen06]).

`huber_circ(x,M)`
  The circularly symmetric Huber function, defined as
    \[\begin{split}f_{\text{huber\_circ}}(x,M) \triangleq \begin{cases} \|x\|_2^2 & \|x\|_2 \leq M \\ 2M\|x\|_2-M^2 & \|x\|_2 \geq M \end{cases}\end{split}\]

  Convex. Same (and implemented) as `huber_pos(norm(x),M)`.

`huber_pos(x,M)`
  The same as the Huber function for nonnegative `x`; zero for negative `x`. Convex and nondecreasing.
`inv_pos`
  The inverse of the positive portion, \(1/\max\{x,0\}\). Inside CVX specification, imposes constraint that its argument is positive. Outside CVX specification, returns \(+\infty\) if \(x\leq 0\). Convex and decreasing.
† `kl_div`

  Kullback-Leibler distance:

  \[\begin{split}f_{\text{kl}}(x,y) \triangleq \begin{cases} x\log(x/y)-x+y & x,y>0 \\ 0 & x=y=0 \\ +\infty & \text{otherwise} \end{cases}\end{split}\]

  Convex. Outside CVX specification, returns \(+\infty\) if arguments aren’t in the domain.

`lambda_max`
  maximum eigenvalue of a real symmetric or complex Hermitian matrix. Inside CVX, imposes constraint that its argument is symmetric (if real) or Hermitian (if complex). Convex.
`lambda_min`
  minimum eigenvalue of a real symmetric or complex Hermitian matrix. Inside CVX, imposes constraint that its argument is symmetric (if real) or Hermitian (if complex). Concave.
`lambda_sum_largest(X,k)`
  sum of the largest \(k\) values of a real symmetric or complex Hermitian matrix. Inside CVX, imposes constraint that its argument is symmetric (if real) or Hermitian (if complex). Convex.
`lambda_sum_smallest(X,k)`
  sum of the smallest \(k\) values of a real symmetric or complex Hermitian matrix. Inside CVX, imposes constraint that its argument is symmetric (if real) or Hermitian (if complex). Concave.
`log_det`
  log of determinant of a positive definite matrix, \(\log \det(X)\). When used inside a CVX specification, `log_det` constrains its argument to be symmetric (if real) or Hermitian (if complex) and positive definite. With numerical argument, `log_det` returns `-Inf` if these constraints are not met. Concave.
‡ `log_normcdf(x)`
  logarithm of cumulative distribution function of standard normal random variable. Concave and increasing. The current implementation is a fairly crude SDP-representable approximation, with modest accuracy over the interval \([-4,4]\); we intend to replace it with a much better approximation at some point.
† `log_prod(x)`
  \(\log\prod_i x_i\) if when \(x\) is positive; \(-\infty\) otherwise. Concave and nonincreasing. Equivalent to `sum_log(x)`.
† `log_sum_exp(x)`
  the logarithm of the sum of the elementwise exponentials of `x`. Convex and nondecreasing.
`logsumexp_sdp`
  a polynomial approximation to the log-sum-exp function with global absolute accuracy. This can be used to estimate the log-sum-exp function without using the successive approximation method.
`matrix_frac(x,Y)`
  matrix fractional function, \(x^TY^{-1}x\). In CVX, imposes constraint that \(Y\) is symmetric (or Hermitian) and positive definite; outside CVX, returns \(+\infty\) unless \(Y=Y^T\succ 0\). Convex.
`norm_largest(x,k)`
  For real and complex vectors, returns the sum of the largest `k` magnitudes in the vector `x`. Convex.
`norm_nuc(X)`
  The sum of the singular values of a real or complex matrix `X`. (This is the dual of the usual spectral matrix norm, i.e., the largest singular value.) Convex.
‡ `norms(x,p,dim)`, `norms_largest(x,k,dim)`
  Computes vector norms along a specified dimension of a matrix or N-d array. Useful for sum-of-norms and max-of-norms problems. Convex.
`poly_env(p,x)`

  Computes the value of the convex or concave envelope of the polynomial described by `p` (in the `polyval` sense). `p` must be a real constant vector whose length `n` is 0, 1, 2, 3, or some other odd length; and `x` must be real and affine. The sign of the first nonzero element of `p` determines whether a convex (positive) or concave (negative) envelope is constructed. For example, consider the function \(p(x)\triangleq (x^2-1)^2=x^4-2x^2+1\), depicted along with its convex envelope in the figure below.

  The two coincide when \(|x|\geq 1\), but deviate when \(|x|<1\). Attempting to call `polyval([1,0,2,0,1],x)` in a CVX model would yield an error, but a call to `poly_env([1,0,2,0,1],x)` yields a valid representation of the envelope. For convex or concave polynomials, this function produces the same result as `polyval`.

  The polynomial function \(p(x)=x^4-2x^2+1\) and its convex envelope.

`pos(x)`
  \(\max\{x,0\}\), for real \(x\). Convex and increasing.
‡ `pow_abs(x,p)`
  \(|x|^p\) for \(x\in\mathbf{R}\) or \(x\in\mathbf{C}\) and \(p\geq 1\). Convex.
‡ `pow_pos(x,p)`
  \(\max\{x,0\}^p\) for \(x\in\mathbf{R}\) and \(p\geq 1\). Convex and nondecreasing.
‡ `pow_p(x,p)`

  for \(x\in\mathbf{R}\) and real constant \(p\), computes nonnegative convex and concave branches of the power function:

  \[\begin{split}\begin{array}{ccl} p\leq 0 & f_p(x) \triangleq \begin{cases} x^p & x > 0 \\ +\infty & x \leq 0 \end{cases} & \text{convex, nonincreasing} \\ 0 < p \leq 1 & f_p(x) \triangleq \begin{cases} x^p & x \geq 0 \\ -\infty & x < 0 \end{cases} & \text{concave, nondecreasing} \\ p \geq 1 & f_p(x) \triangleq \begin{cases} x^p & x \geq 0 \\ +\infty & x < 0 \end{cases} & \text{convex, nonmonotonic} \end{array}\end{split}\]
`prod_inv(x)`
  \(\prod_i x_i^{-1}\) when \(x\) is positive; \(+\infty\) otherwise. Convex and nonincreasing.
`quad_form(x,P)`
  \(x^TPx\) for real \(x\) and symmetric \(P\), and \(x^HPx\) for complex \(x\) and Hermitian \(P\). Convex in \(x\) for \(P\) constant and positive semidefinite; concave in \(x\) for \(P\) constant and negative semidefinite.

Note

Quadratic functions such as `quad_form`, `sum_square` can often be replaced by the `norm` function without sacrificing equivalence. For numerical reasons, this alternate formulation is preferred. Please see Eliminating quadratic forms for more information.

`quad_over_lin(x,y)`
  \(x^Tx/y\) for \(x \in \mathbf{R}^n\), \(y >0\); for \(x \in \mathbf{C}^n\), \(y>0\), \(x^*x/y\). In CVX specification, adds constraint that \(y>0\). Outside CVX specification, returns \(+\infty\) if \(y\leq 0\). Convex, and decreasing in \(y\).
`quad_pos_over_lin(x,y)`
  `sum_square_pos( x )/y` for \(x\in\mathbf{R}^n\), \(y>0\). Convex, increasing in \(x\), and decreasing in \(y\).
† `rel_entr(x)`
  Scalar relative entropy; `rel_entr(x,y)=x.*log(x/y)`. Convex.
`sigma_max`
  maximum singular value of real or complex matrix. Same as `norm`. Convex.
`square`
  \(x^2\) for \(x \in \mathbf{R}\). Convex.
`square_abs`
  \(|x|^2\) for \(x\in\mathbf{R}\) or \(x\in\mathbf{C}\).
`square_pos`
  \(\max\{x,0\}^2\) for \(x\in\mathbf{R}\). Convex and increasing.
`sum_largest(x,k)`
  sum of the largest \(k\) values, for real vector \(x\). Convex and nondecreasing.
† `sum_log(x)`
  \(\sum_i\log(x_i)\) when \(x\) is positive; \(-\infty\) otherwise. Concave and nondecreasing.
`sum_smallest(x,k)`
  sum of the smallest \(k\) values, i.e., equivalent to `-sum_largest(-x,k)`. Concave and nondecreasing.
`sum_square`
  Equivalent to `sum(square(x))`, but more efficient. Convex. Works only for real values.
`sum_square_abs`
  Equivalent to `sum(square_abs(x))`, but more efficient. Convex.
`sum_square_pos`
  Equivalent to `sum(square_pos(x))`, but more efficient. Works only for real values. Convex and increasing.
`trace_inv(X)`
  trace of the inverse of an SPD matrix `X`, which is the same as the sum of the inverses of the eigenvalues. Convex. Outside of CVX, returns `+Inf` if argument is not positive definite.
`trace_sqrtm(X)`
  trace of the matrix square root of a positive semidefinite matrix `X`. which is the same as the sum of the squareroots of the eigenvalues. Concave. Outside of CVX, returns `+Inf` if argument is not positive semidefinite.

## Sets

CVX currently supports the following sets; in each case, `n` is a positive integer constant.

`nonnegative(n)`
  \[R^n_+ \triangleq \left\{\,x\in\mathbf{R}^n\,~|~\,x_i\geq 0,~i=1,2,\dots,n\,\right\}\]
`simplex(n)`
  \[R^n_{1+} \triangleq \left\{\,x\in\mathbf{R}^n\,~|~\,x_i\geq 0,~i=1,2,\dots,n,~\textstyle\sum_ix_i=1\,\right\}\]
`lorentz(n)`
  \[\mathbf{Q}^n \triangleq \left\{\,(x,y)\in\mathbf{R}^n\times\mathbf{R}\,~|~\,\|x\|_2\leq y\,\right\}\]
`rotated_lorentz(n)`
  \[\mathbf{Q}^n_r \triangleq \left\{\,(x,y,z)\in\mathbf{R}^n\times\mathbf{R}\times\mathbf{R}\,~|~\,\|x\|_2\leq \sqrt{yz},~y,z\geq 0\,\right\}\]
`complex_lorentz(n)`
  \[\mathbf{Q}^n_c \triangleq \left\{\,(x,y)\in\mathbf{C}^n\times\mathbf{R}\,~|~\,\|x\|_2\leq y\,\right\}\]
`rotated_complex_lorentz(n)`
  \[\mathbf{Q}^n_{rc} \triangleq \left\{\,(x,y,z)\in\mathbf{C}^n\times\mathbf{R}\times\mathbf{R}\,~|~\,\|x\|_2\leq \sqrt{yz},~y,z\geq 0\,\right\}\]
`semidefinite(n)`
  \[\mathbf{S}^n_+ \triangleq \left\{\,X\in\mathbf{R}^{n\times n}\,~|~\,X=X^T,~X\succeq 0\,\right\}\]
`hermitian_semidefinite(n)`
  \[\mathbf{H}^n_+ \triangleq \left\{\,Z\in\mathbf{C}^{n\times n}\,~|~\,Z=Z^H,~Z\succeq 0\,\right\}\]
`nonneg_poly_coeffs(n)`

  The cone of all coefficients of nonnegative polynomials of degree \(n\); \(n\) must be even:

  \[\mathbf{P}_{+,n} \triangleq \left\{\,p\in\mathbf{R}^n[n+1]\,~|~\,\sum_{i=0}^n p_{i+1} x^{n-i} \geq 0 ~ \forall x\in\mathbf{R}\,\right\}\]
`convex_poly_coeffs(n)`

  The cone of all coefficients of convex polynomials of degree \(n\); \(n\) must be even:

  \[\mathbf{P}_{+,n} \triangleq \left\{\,p\in\mathbf{R}^n[n+1]\,~|~\,\sum_{i=0}^{n-2} (n-i)(n-i-1) p_{i+1} x^{n-i-2} \geq 0 ~ \forall x\in\mathbf{R}\,\right\}\]
`exp_cone`
  \[\begin{split}\mathbf{E} \triangleq \text{cl}\left\{\,(x,y,z)\in\mathbf{R}\times\mathbf{R}\times\mathbf{R}\,~|~\,y>0,~ye^{x/y}\leq z\,\right\}\end{split}\]
`geo_mean_cone(n)`
  \[\mathbf{G}_n \triangleq \text{cl}\left\{\,(x,y)\in\mathbf{R}^n\times\mathbf{R}^n\times\mathbf{R}^n\,~|~\,x\geq 0,~(\prod_{i=1}^n x_i)^{1/n} \geq y\,\right\}\]

## Commands

`cvx_begin`
  Begins a new CVX model. If a model is already in progress, it will issue a warning and clear it. See cvx_begin and cvx_end for a full description, including the modifying keywords that control solver output, SDP mode, GDP mode, etc.
`cvx_clear`
  Clears any model being constructed. Useful when an error has been made and it is necessary to start from the beginning. Whereas `cvx_begin` issues a warning if called with a model in progress, `cvx_clear` is silent.
`cvx_end`
  Signals the end of a CVX model. In typical use, this instructs CVX to begin the solution process. See cvx_begin and cvx_end.
`cvx_expert`
  Controls the issuance of warnings when models requiring the use of successive approximation are employed; see The successive approximation method more details.
`cvx_power_warning`
  Controls if and when CVX issues warnings during the construction of models involving rational power functions (i.e., `x^p`, where `x` is a variable and `p` is a constant); see Power functions and p-norms.
`cvx_precision`
  Controls solver precision; see Controlling precision.
`cvx_quiet`
  Enables or disables screen output during the solution process; see Controlling screen output. Also see cvx_begin and cvx_end for the newer, preferred syntax `cvx_begin quiet`.
`cvx_save_prefs`
  Saves the current states for `cvx_expert`, `cvx_power_warning`, `cvx_precision`, and `cvx_solver` to disk, so that their values are retained when quitting and re-starting MATLAB. The file is saved in MATLAB’s preference directory, which can be located by typing the `prefdir` command.
`cvx_setup`
  The setup script used to install and configure CVX; see Installation.
`cvx_solver`
  Selects the solver to be employed when solving CVX models; see Selecting a solver.
`cvx_solver_settings`
  Allows the user to deliver advanced, solver-specific settings to the solver that CVX does not otherwise support; see Advanced solver settings.
`cvx_version`
  Prints information about the current versions of CVX, Matlab, and the operating system. When submitting bug reports, please include the output of this command.
`cvx_where`
  Returns the directory where CVX is installed.
`dual variable`, `dual variables`
  Creates one or more dual variables to be connected to constraints in the current model; see Dual variables.
`expression`, `expressions`
  Creates one or more expression holders; see Assignment and expression holders.
`maximise`, `maximize`
  Specifies a maximization objective; see Objective functions.
`minimise`, `minimize`
  Specifies a minimization objective; see Objective functions.
`variable`, `variables`
  Creates one or more variables for use in the current CVX model; see Variables.

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