1 Introduction
1.1 What is cvx?
cvx is a modeling system for disciplined convex programming. Disciplined convex pro-
grams, or DCPs, are convex optimization problems that are described using a limited
set of construction rules, which enables them to be analyzed and solved efficiently.
cvx can solve standard problems such as linear programs (LPs), quadratic programs
(QPs), second-order cone programs (SOCPs), and semidefinite programs (SDPs);
but compared to directly using a solver for one or these types of problems, cvx can
greatly simplify the task of specifying the problem. cvx can also solve much more
complex convex optimization problems, including many involving nondifferentiable
functions, such as `
1
norms. You can use cvx to conveniently formulate and solve
constrained norm minimization, entropy maximization, determinant maximization,
and many other problems.
To use cvx effectively, you need to know at least a bit about convex optimiza-
tion. For background on convex optimization, see the book Convex Optimization
[BV04], available on-line at www.stanford.edu/
∼
boyd/cvxbook/, or the Stanford
course EE364A, available at www.stanford.edu/class/ee364a/.
cvx is implemented in Matlab [Mat04], effectively turning Matlab into an op-
timization 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 opti-
mization, such as a branch and bound method, or an engineering design framework.
cvx also provides special modes to simplify the construction of problems from two
specific problem classes. In SDP mode, cvx applies a matrix interpretation to the
inequality operator, so that linear matrix inequalities (LMIs) and SDPs may be ex-
pressed in a more natural form. In GP mode, cvx accepts all of the special functions
and combination rules of geometric programming, including monomials, posynomi-
als, and generalized posynomials, and transforms such problems into convex form
so that they can be solved efficiently. For background on geometric programming,
see the tutorial paper [BKVH05], available at www.stanford.edu/
∼
boyd/papers/
gp
tutorial.html.
cvx was designed by Michael Grant and Stephen Boyd, with input from Yinyu Ye;
and was implemented by Michael Grant [GBY06]. It incorporates ideas from earlier
work by L¨ofberg [L¨of05], Dahl and Vandenberghe [DV05], Crusius [Cru02], Wu and
Boyd [WB00], and many others. The modeling language follows the spirit of AMPL
[FGK99] or GAMS [BKMR98]; unlike thes e packages, however, cvx was designed
from the beginning to fully exploit convexity. The specific method for implementing
cvx in Matlab draws heavily from YALMIP [L¨of05]. We also hope to develop versions
of cvx for other platforms in the future.
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