Advanced topics¶
Note
In this section we describe a number of the more advanced capabilities of CVX. We recommend that you skip this section at first, until you are comfortable with the basic capabilities described above.
Eliminating quadratic forms¶
One particular reformulation that we strongly encourage is to eliminate quadratic forms—that is, functions like sum_square, sum(square(.)) or quad_form—whenever it is possible to construct equivalent models using norm instead. Our experience tells us that quadratic forms often pose a numerical challenge for the underlying solvers that CVX uses.
We acknowledge that this advice goes against conventional wisdom: quadratic forms are the prototypical smooth convex function, while norms are nonsmooth and therefore unwieldy. But with the conic solvers that CVX uses, this wisdom is exactly backwards. It is the norm that is best suited for conic formulation and solution. Quadratic forms are handled by converting them to a conic form—using norms, in fact! This conversion process poses some interesting scaling challenges. It is better if the modeler can eliminate the need to perform this conversion.
For a simple example of such a change, consider the objective
minimize( sum_square( A * x - b ) )
In exact arithmetic, this is precisely equivalent to
minimize( square_pos( norm( A * x - b ) ) )
But equivalence is also preserved if we eliminate the square altogether:
minimize( norm( A * x - b ) )
The optimal value of x is identical in all three cases, but this last version is likely to produce more accurate results. Of course, if you need the value of the squared norm, you can always recover it by squaring the norm after the fact.
Conversions using quad_form can sometimes be a bit more difficult. For instance, consider
quad_form( A * x - b, Q ) <= 1
where Q is a positive definite matrix. The equivalent norm version is
norm( Qsqrt * ( A * x - b ) ) <= 1
where Qsqrt is an appropriate matrix square root of Q. One option is to compute the symmetric square root Qsqrt = sqrtm(Q), but this computation destroys sparsity. If Q is sparse, it is likely worth the effort to compute a sparse Cholesky-based square root:
[ Qsqrt, p, S ] = chol( Q );
Qsqrt = Qsqrt * S;
Sometimes an effective reformulation requires a practical understanding of what it means for problems to be equivalent. For instance, suppose we wanted to add an \(\ell_1\) regularization term to the objective above, weighted by some fixed, positive lambda:
minimize( sum_square( A * x - b ) + lambda * norm( x, 1 ) )
In this case, we typically do not care about the specific values of lambda; rather we are varying it over a range to study the tradeoff between the residual of A*x-b and the 1-norm of x. The same tradeoff can be studied by examining this modified model:
minimize( norm( A * x - b ) + lambda2 * norm( x, 1 ) )
This is not precisely the same model; setting lambda and lambda2 to the same value will not yield identical values of x. But both models do trace the same tradeoff curve—only the second form is likely to produce more accurate results.
Indexed dual variables¶
In some models, the number of constraints depends on the model parameters—not just their sizes. It is straightforward to build such models in CVX using, say, a Matlab for loop. In order to assign each of these constraints a separate dual variable, we must find a way to adjust the number of dual variables as well. For this reason, CVX supports indexed dual variables. In reality, they are simply standard Matlab cell arrays whose entries are CVX dual variable objects.
Let us illustrate by example how to declare and use indexed dual variables. Consider the following semidefinite program from the SeDuMi examples:
This problem minimizes a weighted sum of the main diagonal of a positive semidefinite matrix, while holding the sums along each diagonal constant. The parameters of the problem are the elements of the vector \(b\in\mathbf{R}^n\), and the optimization variable is a symmetric matrix \(X\in\mathbf{R}^{n\times n}\). The CVX version of this model is
cvx_begin
variable X( n, n ) symmetric
minimize( ( n - 1 : -1 : 0 ) * diag( X ) );
for k = 0 : n-1,
sum( diag( X, k ) ) == b( k+1 );
end
X == semidefinite(n);
cvx_end
If we wish to obtain dual information for the \(n\) simple equality constraints, we need a way to assign each constraint in the for loop a separate dual variable. This is accomplished as follows:
cvx_begin
variable X( n, n ) symmetric
dual variables y{n}
minimize( ( n - 1 : -1 : 0 ) * diag( X ) );
for k = 0 : n-1,
sum( diag( X, k ) ) == b( k+1 ) : y{k+1};
end
X == semidefinite(n);
cvx_end
The statement dual variables y{n} allocates a cell array of \(n\) dual variables, and stores the result in the Matlab variable Z. The equality constraint in the for loop has been augmented with a reference to y{k+1}, so that each constraint is assigned a separate dual variable. When the cvx_end command is issued, CVX will compute the optimal values of these dual variables, and deposit them into an \(n\)-element cell array y.
This example admittedly is a bit simplistic. With a bit of careful arrangement, it is possible to rewrite this model so that the \(n\) equality constraints can be combined into a single vector constraint, which in turn would require only a single vector dual variable. [3] For a more complex example that is not amenable to such a simplification, see the file
examples/cvxbook/Ch07_statistical_estim/cheb.m
in the CVX distribution. In that problem, each constraint in the for loop is a linear matrix inequality, not a scalar linear equation; so the indexed dual variables are symmetric matrices, not scalars.
The successive approximation method¶
Note
If you were referred to this web page by CVX’s warning message: welcome! Please read this section carefully to fully understand why using functions like log, exp, etc. within CVX models requires special care.
Prior to version 1.2, the functions exp, log, log_det, and other functions from the exponential family could not be used within CVX. Unfortunately, CVX utilizes symmetric primal/dual solvers that simply cannot support those functions natively [4], and a variety of factors prevent us from incorporating other types of solvers into CVX.
Nevertheless, support for these functions was requested quite frequently. For this reason, we constructed a successive approximation heuristic that allows the symmetric primal/dual solvers to support the exponential family of functions. A precise description of the approach is beyond the scope of this text, but roughly speaking, the method proceeds as follows:
- Choose an initial approximation centerpoint \(x_c=0\).
- Construct a polynomial approximation for each log/exp/entropy term which is accurate in the neighborhood of \(x_c\).
- Solve this approximate model to obtain its optimal point \(\bar{x}\).
- If \(\bar{x}\) satisfies the optimality conditions for the orignal model to sufficient precision, exit.
- Otherwise, shift \(x_c\) towards \(\bar{x}\), and repeat steps 2-5.
Again, this is a highly simplified description of the approach; for instance, we actually employ both the primal and dual solutions to guide our judgements for shifting \(x_c\) and terminating.
This approach has proven surprisingly effective for many problems. However, as with many heuristic approaches, it is not perfect. It will sometimes fail to converge even for problems known to have solutions. Even when it does converge, it is several times slower than the standard solver, due to its iterative approach. Therefore, it is best to use it sparingly and carefully. Here are some specific usage tips:
First, confirm that the log/exp/entropy terms are truly necessary for your model. In many cases, an exactly equivalent model can be constructed without them, and that should always be preferred. For instance, the constraint
sum_log(x) >= 10
can be expressed in terms of the geo_mean function as
geo_mean(x) >= log(10)^(1/length(x))
Many determinant maximization problems are commonly written using log_det, but in fact that is often unnecessary. For instance, consider the objective
minimize( log_det(X) )
CVX actually converts this internally to this:
minimize( n*log(det_rootn(X)) )
So what you can do instead is simply remove the logarithm, and solve this instead:
minimize( det_rootn(X) )
The value of log_det(X) can simply be computed after the model is completed. Unfortunately, this only works if log_det is the only term in the objective; so, for instance, this function cannot, unfortunately, be converted in a similar fashion:
minimize( log_det(X) + trace(C*X) )
Second, try different solvers. For instance, SeDuMi and MOSEK tend to be more effective with the successive approximation method than SDPT3. So if the default solver choice fails to give a solution to your model, try switching to one of these solvers.
Third, try smaller instances of your problem. If they succeed where the larger instance fails, then at least you can confirm if the model is behaving as you hope before considering alternative options like a different solver.
The bottom line, unfortunately, is that we cannot guarantee that the successive approximation approach will successfully handle your specific models. If you encounter problems, you are invited to submit a bug report, but we will not be able to promise a fix.
Suppressing the warning¶
Because of all of these caveats, we believe that it is necessary to issue a warning when it is used so that users understand its experimental nature. This warning appears the first time you attempt to specify a model in CVX that uses an function that requires the successive approximation method. In fact, that warning may very well have brought you to this section of the manual.
If you wish to suppress this warning in the future, simply issue the command
cvx_expert true
before you construct your model. If you wish to suppress this message for all future sessions of MATLAB, follow this command with the cvx_save_prefs command.
Power functions and p-norms¶
In order to implement the convex or concave branches of the power function \(x^p\) and \(p\)-norms \(\|x\|_p\) for general values of \(p\), CVX uses an enhanced version of the SDP/SOCP conversion method described by [AG00]. This approach is exact—as long as the exponent \(p\) is rational. To determine integral values \(p_n,p_d\) such that \(p_n/p_d=p\), CVX uses Matlab’s rat function with its default tolerance of \(10^{-6}\). There is currently no way to change this tolerance. See the MATLAB documentation for the rat function for more details.
The complexity of the resulting model depends roughly on the size of the values \(p_n\) and \(p_d\). Let us introduce a more precise measure of this complexity. For \(p=2\), a constraint \(x^p\leq y\) can be represented with exactly one \(2\times 2\) LMI:
For other values of \(p=p_n/p_d\), CVX generates a number of \(2\times 2\) LMIs that depends on both \(p_n\) and \(p_d\); we denote this number by \(k(p_n,p_d)\). (In some cases additional linear constraints are also generated, but we ignore them for this analysis.) For instance, for \(p=3/1\), we have
So \(k(3,1)=2\). An empirical study has shown that for \(p=p_n/p_d>1\), we have
where the \(\alpha(p_n)\) term grows very slowly compared to the \(\log_2\) term. Indeed, for \(p_n\leq 4096\), we have verified that \(\alpha(p_n)\) is usually 1 or 2, but occasionally 0 or 3. Similar results are obtained for \(0 < p < 1\) and \(p < 0\).
The cost of this SDP representation is relatively small for nearly all useful values of \(p\). Nevertheless, CVX issues a warning whenever \(k(p_n,p_d)>10\) to insure that the user is not surprised by any unexpected slowdown. In the event that this threshold does not suit you, you may change it using the command cvx_power_warning(thresh), where thresh is the desired cutoff value. Setting the threshold to Inf disables it completely. As with the command cvx_precision, you can place a call to cvx_power_warning within a model to change the threshold for a single model; or outside of a model to make a global change. The command always returns the previous value of the threshold, so you can save it and restore it upon completion of your model, if you wish. You can query the current value by calling cvx_power_warning with no arguments.
Overdetermined problems¶
The status message Overdetermined commonly occurs when structure in a variable or set is not properly recognized. For example, consider the problem of finding the smallest diagonal addition to a matrix \(W\in\mathbf{R}^{n\times n}\) to make it positive semidefinite:
In CVX, this problem might be expressed as follows:
n = size(W,1);
cvx_begin
variable D(n,n) diagonal;
minimize( trace( D ) );
subject to
W + D == semidefinite(n);
cvx_end
If we apply this specification to the matrix W=randn(5,5), a warning is issued,
Warning: Overdetermined equality constraints;
problem is likely infeasible.
and the variable cvx_status is set to Overdetermined.
What has happened here is that the unnamed variable returned by statement semidefinite(n) is symmetric, but \(W\) is fixed and unsymmetric. Thus the problem, as stated, is infeasible. But there are also \(n^2\) equality constraints here, and only \(n+n*(n+1)/2\) unique degrees of freedom—thus the problem is overdetermined. We can correct this problem by replacing the equality constraint with
sym( W ) + D == semidefinite(n);
sym is a function we have provided that extracts the symmetric part of its argument; that is, sym(W) equals 0.5 * ( W + W' ).
Log convexity¶
A few CVX users have stumbled across a construction that should be rejected, but is not: usually something similar to this expression:
log( exp(x) + 1 )
This expression is indeed convex, but it does not comply with the DCP composition rules. After all, log is concave and increasing, and therefore can accept a concave argument; but exp(x) + 1 is convex! What’s going on here?
The answer lies in CVX’s support for geometric programming. Geometric programs in CVX must obey a different set of rules than standard DCP models. But underneath the hood, CVX processes both models in a unified manner. In short, every geometric variable x is related to an internal variable, say x_, by an exponential; that is, x = exp(x_). Most geometric programming rules have a corresponding rule in the linear/DCP space.
The net effect is that CVX has a set of rules that govern three additional kinds of curvature in CVX: log-convex, log-concave, and log-affine. As the name suggests, a log-concave expression is one whose logarithm is convex; so, for example, \(e^{x^2}\) is log-convex (and convex). Similarly, a log-concave expression has a concave logarithm, and a log-affine expression has an affine logarithm.
For more information about log-convexity, see Convex Optimization, Section 3.5. Not surprisingly, CVX does not recognize all instances of log-convexity, just those that adhere to a finite set of rules:
- A valid log-convex expression is
- exp( expr ), where expr is convex;
- a log-convex expression multiplied by a positive constant;
- the inverse of a log-concave expression;
- a log-convex expression raised to a positive power;
- a log-concave expression raised to a negative power;
- the sum of log-convex and/or log-affine expressions.
- A valid log-concave expression is
- exp( expr ), where expr is concave;
- a log-concave expression multiplied by a positive constant;
- the inverse of a log-convex expression;
- a log-concave expression raised to a positive power;
- a log-convex expression raised to a negative power.
- A valid log-affine expression is
- any positive constant;
- exp( expr ), where expr is affine;
- a log-affine expression multiplied by a positive constant;
- the inverse of a log-affine expression;
- a log-affine expression raised to a constant power.
In these rules above we show how exp is used to “elevate” expressons with “standard” curvature to log-curvature; not surprisingly, log reverses the effect:
- If expr is log-convex, log-concave, or log-affine, then log( expr ) is convex, concave, or affine, respectively.
Log-convex/concave/affine expressions can be employed in objectives and constraints. Acceptable log-convex objectives include
- minimize( expr ), where expr is log-convex; and
- maximize( expr ), where expr is log-concave.
Acceptable log-convex constraints include
- An equality constraint, using ==, where both sides are log-affine.
- A less-than inequality constraint, using <=, where the left side is log-convex and the right side is log-concave.
- A greater-than inequality constraint, using >=, where the left side is log-concave and the right side is log-convex.
So there you have it: the hidden log-convexity rules of CVX. Again, we do not recommend that you actually use these, as they are meant simply to support the geometric programming capability of CVX. In fact, because log-convexity requires the successive approximation approach to solve, its use is unsupported.
Adding new functions to the atom library¶
CVX allows new convex and concave functions to be defined and added to the atom library, in two ways, described in this section. The first method is simple, and can (and should) be used by many users of CVX, since it requires only a knowledge of the basic DCP ruleset. The second method is very powerful, but a bit complicated, and should be considered an advanced technique, to be attempted only by those who are truly comfortable with convex analysis, disciplined convex programming, and CVX in its current state.
Please let us know if you have implemented a convex or concave function that you think would be useful to other users; we will be happy to incorporate it in a future release.
New functions via the DCP ruleset¶
The simplest way to construct a new function that works within CVX is to construct it using expressions that fully conform to the DCP ruleset. Consider, for instance, the deadzone function
To implement this function in CVX, simply create a file deadzone.m containing
function y = deadzone( x )
y = max( abs( x ) - 1, 0 )
This function works just as you expect it would outside of CVX — that is, when its argument is numerical. But thanks to Matlab’s operator overloading capability, it will also work within CVX if called with an affine argument. CVX will properly conclude that the function is convex, because all of the operations carried out conform to the rules of DCP: abs is recognized as a convex function; we can subtract a constant from it, and we can take the maximum of the result and 0, which yields a convex function. So we are free to use deadzone anywhere in a CVX specification that we might use abs, for example, because CVX knows that it is a convex function.
Let us emphasize that when defining a function this way, the expressions you use must conform to the DCP ruleset, just as they would if they had been inserted directly into a CVX model. For example, if we replace max with min above; e.g.,
function y = deadzone_bad( x )
y = min( abs( x ) - 1, 0 )
then the modified function fails to satisfy the DCP ruleset. The function will work outside of a CVX specification, happily computing the value \(\min \{|x|-1,0\}\) for a numerical argument \(x\). But inside a CVX specification, invoked with a nonconstant argument, it will generate an error.
New functions via partially specified problems¶
A more advanced method for defining new functions in CVX relies on the following basic result of convex analysis. Suppose that \(S\subset\mathbf{R}^n\times\mathbf{R}^m\) is a convex set and \(g:(\mathbf{R}^n\times\mathbf{R}^m)\rightarrow(\mathbf{R}\cup+\infty)\) is a convex function. Then
is also a convex function. (This rule is sometimes called the partial minimization rule.) We can think of the convex function \(f\) as the optimal value of a family of convex optimization problems, indexed or parametrized by \(x\),
with optimization variable \(y\).
One special case should be very familiar: if \(m=1\) and \(g(x,y)\triangleq y\), then
gives the classic epigraph representation of \(f\):
where \(0 \in \mathbf{R}^n\).
In CVX you can define a convex function in this very manner, that is, as the optimal value of a parameterized family of disciplined convex programs. We call the underlying convex program in such cases an incomplete specification—so named because the parameters (that is, the function inputs) are unknown when the specification is constructed. The concept of incomplete specifications can at first seem a bit complicated, but it is very powerful mechanism that allows CVX to support a wide variety of functions.
Let us look at an example to see how this works. Consider the unit-halfwidth Huber penalty function \(h(x)\):
We can express the Huber function in terms of the following family of convex QPs, parameterized by \(x\):
with scalar variables \(v\) and \(w\). The optimal value of this simple QP is equal to the Huber penalty function of \(x\). We note that the objective and constraint functions in this QP are (jointly) convex in \(v\), \(w\) and \(x\).
We can implement the Huber penalty function in CVX as follows:
function cvx_optval = huber( x )
cvx_begin
variables w v;
minimize( w^2 + 2 * v );
subject to
abs( x ) <= w + v;
w <= 1; v >= 0;
cvx_end
If huber is called with a numeric value of x, then upon reaching the cvx_end statement, CVX will find a complete specification, and solve the problem to compute the result. CVX places the optimal objective function value into the variable cvx_optval, and function returns that value as its output. Of course, it’s very inefficient to compute the Huber function of a numeric value \(x\) by solving a QP. But it does give the correct value (up to the core solver accuracy).
What is most important, however, is that if huber is used within a CVX specification, with an affine CVX expression for its argument, then CVX will do the right thing. In particular, CVX will recognize the Huber function, called with affine argument, as a valid convex expression. In this case, the function huber will contain a special Matlab object that represents the function call in constraints and objectives. Thus the function huber can be used anywhere a traditional convex function can be used, in constraints or objective functions, in accordance with the DCP ruleset.
There is a corresponding development for concave functions as well. Given a convex set \(S\) as above, and a concave function \(g:(\mathbf{R}^n\times\mathbf{R}^m)\rightarrow(\mathbf{R}\cup-\infty)\), the function
is concave. If \(g(x,y)\triangleq y\), then
gives the hypograph representation of \(f\):
In CVX, a concave incomplete specification is simply one that uses a maximize objective instead of a minimize objective; and if properly constructed, it can be used anywhere a traditional concave function can be used within a CVX specification.
For an example of a concave incomplete specification, consider the function
Its hypograph can be represented using a single linear matrix inequality:
So we can implement this function in CVX as follows:
function cvx_optval = lambda_min_symm( X )
n = size( X, 1 );
cvx_begin
variable y;
maximize( y );
subject to
X + X' - y * eye( n ) == semidefinite( n );
cvx_end
If a numeric value of X is supplied, this function will return min(eig(X+X')) (to within numerical tolerances). However, this function can also be used in CVX constraints and objectives, just like any other concave function in the atom library.
There are two practical issues that arise when defining functions using incomplete specifications, both of which we will illustrate using our huber example above. First of all, as written the function works only with scalar values. To apply it (elementwise) to a vector requires that we iterate through the elements in a for loop—a very inefficient enterprise, particularly in CVX. A far better approach is to extend the huber function to handle vector inputs. This is, in fact, rather simple to do: we simply create a multiobjective version of the problem:
function cvx_optval = huber( x )
sx = size( x );
cvx_begin
variables w( sx ) v( sx );
minimize( w .^ 2 + 2 * v );
subject to
abs( x ) <= w + v;
w <= 1; v >= 0;
cvx_end
This version of huber will in effect create sx “instances” of the problem in parallel; and when used in a CVX specification, will be handled correctly.
The second issue is that if the input to huber is numeric, then direct computation is a far more efficient way to compute the result than solving a QP. (What is more, the multiobjective version cannot be used with numeric inputs.) One solution is to place both versions in one file, with an appropriate test to select the proper version to use:
function cvx_optval = huber( x )
if isnumeric( x ),
xa = abs( x );
flag = xa < 1;
cvx_optval = flag .* xa.^2 + (~flag) * (2*xa-1);
else,
sx = size( x );
cvx_begin
variables w( sx ) v( sx );
minimize( w .^ 2 + 2 * v );
subject to
abs( x ) <= w + v;
w <= 1; v >= 0;
cvx_end
end
Alternatively, you can create two separate versions of the function, one for numeric input and one for CVX expressions, and place the CVX version in a subdirectory called @cvx. (Do not include this directory in your Matlab path; only include its parent.) Matlab will automatically call the version in the @cvx directory when one of the arguments is a CVX variable. This is the approach taken for the version of huber found in the CVX atom library.
One good way to learn more about using incomplete specifications is to examine some of the examples already in the CVX atom library. Good choices include huber, inv_pos, lambda_min, lambda_max, matrix_frac, quad_over_lin, sum_largest, and others. Some are a bit difficult to read because of diagnostic or error-checking code, but these are relatively simple.
| [3] | Indeed, a future version of CVX will support the use of the Matlab function spdiags, which will reduce the entire for loop to the single constraint spdiags(X,0:n-1)==b. |
| [4] | Technically there are a couple of exceptions here. First of all, SDPT3 does, in fact, support the existence of logarithms and log_det terms in the objective function. However, it doesn’t support such terms within constraints. Unfortunately, because CVX does not differentiate between objective terms and constraint terms internally, it is not able to utilize this capability of SDPT3. Secondly, this section was written before the inclusion of MOSEK support in CVX, and CVX does indeed provide support for smooth nonlinearities in its solver. But this capability is not easy to use in MATLAB. |
