# Convex.jl - Convex Optimization in Julia

Convex.jl is a Julia package for Disciplined Convex Programming (DCP). Convex.jl makes it easy to describe optimization problems in a natural, mathematical syntax, and to solve those problems using a variety of different (commercial and open-source) solvers. Convex.jl can solve

• linear programs
• mixed-integer linear programs and mixed-integer second-order cone programs
• dcp-compliant convex programs including
• second-order cone programs (SOCP)
• exponential cone programs
• semidefinite programs (SDP)

Convex.jl supports many solvers, including COSMO, Mosek, Gurobi, ECOS, SCS and GLPK, through MathOptInterface.

Note that Convex.jl was previously called CVX.jl. This package is under active development; we welcome bug reports and feature requests. For usage questions, please contact us via the Julia Discourse.

## Extended formulations and the DCP ruleset

Convex.jl works by transforming the problem—which possibly has nonsmooth, nonlinear constructions like the nuclear norm, the log determinant, and so forth—into a linear optimization problem subject to conic constraints. This reformulation often involves adding auxiliary variables, and is called an "extended formulation", since the original problem has been extended with additional variables. These formulations rely on the problem being modelled by combining Convex.jl's "atoms" or primitives according to certain rules which ensure convexity, called the disciplined convex programming (DCP) ruleset. If these atoms are combined in a way that does not ensure convexity, the extended formulations are often invalid. As a simple example, consider the problem

minimize(abs(x), x >= 1, x <= 2)

Obviously, the optimum occurs at x=1, but let us imagine we want to solve this problem via Convex.jl using a linear programming (LP) solver. Since abs is a nonlinear function, we need to reformulate the problem to pass it to the LP solver. We do this by introducing an auxiliary variable t and instead solving

minimize(t, x >= 1, x <= 2, t >= x, t >= -x)

That is, we add the constraints t >= x and t >= -x, and replace abs(x) by t. Since we are minimizing over t and the smallest possible t satisfying these constraints is the absolute value of x, we get the right answer. That is, this reformulation worked because we were minimizing abs(x), and that is a valid way to use the primitive abs.

If we were maximizing abs, Convex.jl would print

Warning: Problem not DCP compliant: objective is not DCP

Why? Well, let us consider the same reformulation for a maximization problem. The original problem is now

maximize(abs(x), x >= 1, x <= 2)

and trivially the optimum is 2, obtained at x=2. If we do the same replacements as above, however, we arrive at the problem

maximize(t, x >= 1, x <= 2, t >= x, t >= -x)

whose solution is infinity. In other words, we got the wrong answer by using the reformulation, since the extended formulation was only valid for a minimization problem. Convex.jl always performs these reformulations, but they are only guaranteed to be valid when the DCP ruleset is followed. Therefore, Convex.jl programatically checks the whether or not these rules were satisfied and warns if they were not. One should not take these DCP warnings lightly!