Operations¶
Convex.jl currently supports the following functions. These functions may be composed according to the DCP composition rules to form new convex, concave, or affine expressions. Convex.jl transforms each problem into an equivalent cone program in order to pass the problem to a specialized solver. Depending on the types of functions used in the problem, the conic constraints may include linear, secondorder, exponential, or semidefinite constraints, as well as any binary or integer constraints placed on the variables. Below, we list each function available in Convex.jl organized by the (most complex) type of cone used to represent that function, and indicate which solvers may be used to solve problems with those cones. Problems mixing many different conic constraints can be solved by any solver that supports every kind of cone present in the problem.
Linear Program Representable Functions¶
An optimization problem using only these functions can be solved by any LP solver.
operation  description  vexity  slope  implicit constraint / notes 

x+y or x.+y 
addition  affine  increasing  none 
xy or x.y 
subtraction  affine  increasing in \(x\) decreasing in \(y\) 
none none 
x*y 
multiplication  affine  increasing if constant term \(\ge 0\) decreasing if constant term \(\le 0\) not monotonic otherwise 
one term is constant 
x/y 
division  affine  increasing  \(y\) is a scalar constant 
x .* y 
elemwise multiplication  affine  increasing  one term is constant 
x[1:4, 2:3] 
indexing and slicing  affine  increasing  none 
diag(x, k) 
\(k\)th diagonal of a matrix  affine  increasing  none 
diagm(x) 
turn vector into diagonal matrix  affine  increasing  \(x\) is a vector 
x' 
transpose  affine  increasing  none 
x'*y or dot(x,y) 
\(x' y\)  affine  increasing  one term is constant 
vec(x) 
vector representation  affine  increasing  none 
reshape(x, m, n) 
reshape into \(m \times n\)  affine  increasing  none 
minimum(x) 
\(\min(x)\)  concave  increasing  none 
maximum(x) 
\(\max(x)\)  convex  increasing  none 

stacking  affine  increasing  none 
trace(x) 
\(\mathrm{tr} \left(X \right)\)  affine  increasing  none 
conv(h,x) 
\(h \in \mathbb{R}^m\) \(x \in \mathbb{R}^m\) \(h*x \in \mathbb{R}^{m+n1}\) entry \(i\) is given by \(\sum_{j=1}^m h_jx_{ij}\) 
affine  increasing if \(h\ge 0\) decreasing if \(h\le 0\) not monotonic otherwise 
\(h\) is constant 
min(x,y) 
\(\min(x,y)\)  concave  increasing  none 
max(x,y) 
\(\max(x,y)\)  convex  increasing  none 
pos(x,y) 
\(\max(x,0)\)  convex  increasing  none 
neg(x,y) 
\(\max(x,0)\)  convex  decreasing  none 
inv_pos(x) 
\(1/\max(x,0)\)  convex  decreasing  \(x>0\) 
abs(x) 
\(\leftx\right\)  convex  increasing on \(x \ge 0\) decreasing on \(x \le 0\) 
none 
SecondOrder Cone Representable Functions¶
An optimization problem using these functions can be solved by any SOCP solver (including ECOS, SCS, Mosek, Gurobi, and CPLEX). Of course, if an optimization problem has both LP and SOCP representable functions, then any solver that can solve both LPs and SOCPs can solve the problem.
operation  description  vexity  slope  implicit constraint 

norm(x, p) 
\((\sum x_i^p)^{1/p}\)  convex  increasing on \(x \ge 0\) decreasing on \(x \le 0\) 
p >= 1 
vecnorm(x, p) 
\((\sum x_{ij}^p)^{1/p}\)  convex  increasing on \(x \ge 0\) decreasing on \(x \le 0\) 
p >= 1 
quad_form(x, P) 
\(x^T P x\)  convex in \(x\) affine in \(P\) 
increasing on \(x \ge 0\) decreasing on \(x \le 0\) increasing in \(P\) 
either \(x\) or \(P\) must be constant 
quad_over_lin(x, y) 
\(x^T x/y\)  convex  increasing on \(x \ge 0\) decreasing on \(x \le 0\) decreasing in \(y\) 
\(y > 0\) 
sum_squares(x) 
\(\sum x_i^2\)  convex  increasing on \(x \ge 0\) decreasing on \(x \le 0\) 
none 
sqrt(x) 
\(\sqrt{x}\)  convex  decreasing  \(x>0\) 
square(x), x^2 
\(x^2\)  convex  increasing on \(x \ge 0\) decreasing on \(x \le 0\) 
none 
geo_mean(x, y) 
\(\sqrt{xy}\)  concave  increasing  \(x\ge0\), \(y\ge0\) 

\(\begin{cases} x^2 &x \leq M \\ 2Mx  M^2 &x > M \end{cases}\)  convex  increasing on \(x \ge 0\) decreasing on \(x \le 0\) 
\(M>=1\) 
Exponential Cone Representable Functions¶
An optimization problem using these functions can be solved by any exponential cone solver (SCS).
operation  description  vexity  slope  implicit constraint 

logsumexp(x) 
\(\log(\sum_i \exp(x_i))\)  convex  increasing  none 
exp(x) 
\(\exp(x)\)  convex  increasing  none 
log(x) 
\(\log(x)\)  concave  increasing  \(x>0\) 
entropy(x) 
\(\sum_{ij} x_{ij} \log (x_{ij})\)  concave  not monotonic  \(x>0\) 
logistic_loss(x) 
\(\log(1 + \exp(x_i))\)  convex  increasing  none 
Semidefinite Program Representable Functions¶
An optimization problem using these functions can be solved by any SDP solver (including SCS and Mosek).
operation  description  vexity  slope  implicit constraint 

nuclear_norm(x) 
sum of singular values of \(x\)  convex  not monotonic  none 
operator_norm(x) 
max of singular values of \(x\)  convex  not monotonic  none 
lambda_max(x) 
max eigenvalue of \(x\)  convex  increasing  x is positive semidefinite 
lambda_min(x) 
min eigenvalue of \(x\)  concave  increasing  x is positive semidefinite 
matrix_frac(x, P) 
\(x^TP^{1}x\)  convex  not monotonic  P is positive semidefinite 
Exponential + SDP representable Functions¶
An optimization problem using these functions can be solved by any solver that supports exponential constraints and semidefinite constraints simultaneously (SCS).
operation  description  vexity  slope  implicit constraint 

logdet(x) 
log of determinant of \(x\)  concave  increasing  x is positive semidefinite 
Promotions¶
When an atom or constraint is applied to a scalar and a higher dimensional variable, the scalars are promoted. For example, we can do max(x, 0)
gives an expression with the shape of x
whose elements are the maximum of the corresponding element of x
and 0
.