CVX is a modeling system for constructing and solving
*disciplined convex programs* (DCPs).
CVX supports a number of standard problem types, including linear and
quadratic programs (LPs/QPs), second-order cone programs (SOCPs), and
semidefinite programs (SDPs). CVX can also solve much more complex convex
optimization problems, including many involving nondifferentiable
functions, such as \(\ell_1\) norms. You can use CVX to conveniently
formulate and solve constrained norm minimization, entropy maximization,
determinant maximization, and many other convex programs. As of version
2.0, CVX also solves *mixed integer disciplined convex programs* (MIDCPs)
as well, with an appropriate integer-capable solver.

To use CVX effectively, you need to know at least a bit about convex optimization. For background on convex optimization, see the book Convex Optimization [BV04] or the Stanford course EE364A.

CVX is implemented in Matlab, effectively turning Matlab into an optimization modeling language. Model specifications are constructed using common Matlab operations and functions, and standard Matlab code can be freely mixed with these specifications. This combination makes it simple to perform the calculations needed to form optimization problems, or to process the results obtained from their solution. For example, it is easy to compute an optimal trade-off curve by forming and solving a family of optimization problems by varying the constraints. As another example, CVX can be used as a component of a larger system that uses convex optimization, such as a branch and bound method, or an engineering design framework.

CVX provides special modes to simplify the construction of problems
from two specific problem classes. In
*semidefinite programming (SDP) mode*,
CVX applies a matrix interpretation to the
inequality operator, so that linear matrix inequalities (LMIs) and
SDPs may be expressed in a more natural form. In
*geometric programming (GP) mode*,
CVX accepts all of the special functions and
combination rules of geometric programming, including monomials,
posynomials, and generalized posynomials, and transforms such problems
into convex form so that they can be solved efficiently. For background
on geometric programming, see this
tutorial paper [BKVH05].

Previous versions of CVX supported two free SQLP solvers,
SeDuMi [Stu99] and
SDPT3 [TTT03]. These
solvers are included with the CVX distribution. Starting with version 2.0,
CVX supports two *commercial* solvers as well,
Gurobi and MOSEK. For
more information, see *Solvers*.

The ability
to use CVX with commercial solvers is a capability that we have decided
to include under a new CVX Professional license model. Academic users will
be able to utilize these features at no charge, but commercial users will require
a paid CVX Professional license. For more details, see *Licensing*.

Recent developments in solver technology are expanding CVX’s capability to solve problems. In particular, the commercial solver Mosek has added native support for the exponential cone; and this allows CVX to solve problems involving logarithms, exponentials, and entropy functions withouth reliance on its successive approximation heuristic.

*Disciplined convex programming* is a methodology for constructing
convex optimization problems proposed by
Michael Grant, Stephen Boyd, and Yinyu Ye [GBY06], [Gra04].
It is meant to support the formulation and construction of optimization
problems that the user intends *from the outset* to be convex.

Disciplined convex programming
imposes a set of conventions or rules, which we call *the DCP ruleset*.
Problems which adhere to the ruleset can be rapidly and automatically
verified as convex and converted to solvable form. Problems that violate
the ruleset are rejected—even when the problem is convex. That is not
to say that such problems cannot be solved using DCP; they just need to
be rewritten in a way that conforms to the DCP ruleset.

A detailed description of the DCP ruleset is given in *The DCP ruleset*.
It is extremely important for anyone who intends to
actively use CVX to understand it. The ruleset is simple to learn, and
is drawn from basic principles of convex analysis. In return for
accepting the restrictions imposed by the ruleset, we obtain
considerable benefits, such as automatic conversion of problems to
solvable form, and full support for nondifferentiable functions. In
practice, we have found that disciplined convex programs closely
resemble their natural mathematical forms.

With version 2.0, CVX now supports *mixed integer* disciplined convex programs (MIDCPs).
A MIDCP is a model that obeys the same convexity rules as standard DCPs, except
that one or more of its variables is constrained to take on integral values. In other
words, if the integer constraints are removed, the result is a standard DCP.

Unlike a true DCP, a mixed integer problem is *not* convex. Finding the global optimum
requires the combination of a traditional convex optimization algorithm with an exhaustive
search such as a branch-and-bound algorithm. Some CVX solvers do not include this second
piece and therefore do not support MIDCPs; see *Solvers* for more information.
What is more, even the best solvers cannot
guarantee that every moderately-sized MIDCP can be solved in a reasonable amount of time.

Mixed integer disciplined convex programming represents new territory for the CVX modeling framework—and for the supporting solvers as well. While solvers for mixed integer linear and quadratic programs (MILP/MIQP) are reasonably mature, support for more general convex nonlinearities is a relatively new development. We anticipate that MIDCP support will improve over time.

CVX is *not* meant to be a tool for checking if your problem is convex.
You need to know a bit about convex optimization to effectively use CVX;
otherwise you are the proverbial monkey at the typewriter, hoping to
(accidentally) type in a valid disciplined convex program. If you are
not certain that your problem is convex *before* you enter it into CVX,
you are using the tool improperly, and your efforts will likely fail.

CVX is *not* meant for very large problems, so if your problem is very
large (for example, a large image processing or machine learning problem), CVX is unlikely
to work well (or at all). For such problems you will likely need to
directly call a solver, or to develop your own methods, to get the
efficiency you need.

For such problems CVX can play an important role, however. Before starting to develop a specialized large-scale method, you can use CVX to solve scaled-down or simplified versions of the problem, to rapidly experiment with exactly what problem you want to solve. For image reconstruction, for example, you might use CVX to experiment with different problem formulations on \(50 \times 50\) pixel images.

CVX *will* solve many medium and large scale problems, provided they
have exploitable structure (such as sparsity), and you avoid `for`
loops, which can be slow in Matlab. If you encounter difficulties in
solving large problem instances, consider posting your model to the
CVX Forum; the CVX community
may be able to suggest an equivalent formulation that CVX
can process more efficiently.

CVX is free for use in both academic and commercial settings when paired with a free solver—including the versions of SeDuMi and SDPT3 that are included with the package.

With version 2.0, we have added the ability to connect CVX to *commercial* solvers as well.
This new functionality is released under a *CVX Professional* product tier
which we intend to license to commercial users for a fee, and offer to academic users
at no charge. The licensing structure is as follows:

*All users*are free to use the standard features of CVX*at no charge*. This includes the ability to construct and solve any of the models supported by the free solvers SeDuMi and SDPT3.*Commercial users*who wish to solve CVX models using Gurobi or MOSEK will need to purchase a CVX Professional license. Please send an email to CVX Research for inquiries. for an availability schedule and pricing details.*Academic users*may utilize the CVX Professional capability*at no charge*. To do so, it is necessary to obtain licenses for Mosek and/or Gurobi directly from the vendors.

The bulk of CVX remains open source under a slightly modified version of the GPL Version
2 license. A small number of files that support the CVX Professional functionality remain
closed source. If those files are removed, the modified package remains *fully functional*
using the free solvers, SeDuMi and SDPT3. Users
may freely modify, augment, and redistribute this free version of CVX, as long as all
modifications are themselves released under the same license. This includes adding support
for new solvers released under a free software license such as the GPL.
For more details, please see the full *Licensing* section.