# JuliaOpt/Convex.jl

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# 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, second-order, 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.

In the notes column in the tables below, we denote implicit constraints imposed on the arguments to the function by IC, and parameter restrictions that the arguments must obey by PR. (Convex.jl will automatically impose ICs; the user must make sure to satisfy PRs.) Elementwise means that the function operates elementwise on vector arguments, returning a vector of the same size.

## Linear Program Representable Functions

An optimization problem using only these functions can be solved by any LP solver.

operation description vexity slope notes
x+y or x.+y addition affine increasing none
x-y 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

PR: one argument is constant
x/y division affine increasing PR: y is scalar constant
dot(*)(x, y) elementwise multiplication affine increasing PR: one argument is constant
dot(/)(x, y) elementwise division affine increasing PR: one argument 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) construct diagonal matrix affine increasing PR: x is a vector
x' transpose affine increasing none
vec(x) vector representation affine increasing none
dot(x,y) \sum_i x_i y_i affine increasing PR: one argument is constant
kron(x,y) Kronecker product affine increasing PR: one argument is constant
vecdot(x,y) dot(vec(x),vec(y)) affine increasing PR: one argument is constant
sum(x) \sum_{ij} x_{ij} affine increasing none
sum(x, k)

sum elements across

dimension k

affine increasing none
sumlargest(x, k)

sum of k largest

elements of x

convex increasing none
sumsmallest(x, k)

sum of k smallest

elements of x

concave increasing none
dotsort(a, b) dot(sort(a),sort(b)) convex increasing PR: one argument is constant
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

[x y] or [x; y]

hcat(x, y) or

vcat(x, y)

stacking affine increasing none
trace(x) \mathrm{tr} \left(X \right) affine increasing none
partialtrace(x,sys,dims) Partial trace affine increasing none
conv(h,x)

h \in \mathbb{R}^m

x \in \mathbb{R}^m

h*x \in \mathbb{R}^{m+n-1}

entry i is given by

\sum_{j=1}^m h_jx_{i-j}

affine

increasing if h\ge 0

decreasing if h\le 0

not monotonic

otherwise

PR: h is constant
min(x,y) \min(x,y) concave increasing none
max(x,y) \max(x,y) convex increasing none
pos(x) \max(x,0) convex increasing none
neg(x) \max(-x,0) convex decreasing none
invpos(x) 1/x convex decreasing IC: x>0
abs(x) \left|x\right| convex

increasing on x \ge 0

decreasing on x \le 0

none

## Second-Order 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 notes
norm(x, p) (\sum x_i^p)^{1/p} convex

increasing on x \ge 0

decreasing on x \le 0

PR: p >= 1
vecnorm(x, p) (\sum x_{ij}^p)^{1/p} convex

increasing on x \ge 0

decreasing on x \le 0

PR: p >= 1
quadform(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

PR: either x or P

must be constant; if x is not constant, then P must be symmetric and positive semidefinite

quadoverlin(x, y) x^T x/y convex

increasing on x \ge 0

decreasing on x \le 0

decreasing in y

IC: y > 0
sumsquares(x) \sum x_i^2 convex

increasing on x \ge 0

decreasing on x \le 0

none
sqrt(x) \sqrt{x} concave decreasing IC: x>0
square(x), x^2 x^2 convex

increasing on x \ge 0

decreasing on x \le 0

PR : x is scalar
dot(^)(x,2) x.^2 convex increasing on x \ge 0 decreasing on x \le 0 elementwise
geomean(x, y) \sqrt{xy} concave increasing IC: x\ge0, y\ge0
huber(x, M=1) \begin{cases} x^2 &|x| \leq M \\ 2M|x| - M^2 &|x| > M \end{cases} convex

increasing on x \ge 0

decreasing on x \le 0

PR: 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 notes
logsumexp(x) \log(\sum_i \exp(x_i)) convex increasing none
exp(x) \exp(x) convex increasing none
log(x) \log(x) concave increasing IC: x>0
entropy(x) \sum_{ij} -x_{ij} \log (x_{ij}) concave not monotonic IC: x>0
logisticloss(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 notes
nuclearnorm(x) sum of singular values of x convex not monotonic none
operatornorm(x) max of singular values of x convex not monotonic none
lambdamax(x) max eigenvalue of x convex not monotonic none
lambdamin(x) min eigenvalue of x concave not monotonic none
matrixfrac(x, P) x^TP^{-1}x convex not monotonic IC: 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 notes
logdet(x) log of determinant of x concave increasing IC: 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.