Cvx Your Objective Function Is Not a Scalar

When working with optimization problems, especially in the con text of convex optimization using the CVX toolbox in MATLAB, you may encounter an error message that reads: “CVX Your objective function is not a scalar.” This error can seem puzzling at first, particularly if you’re not familiar with the specific requirements for defining objective functions in CVX. In this article, we’ll break down what this error means, why it occurs, and how to fix it.

What is CVX?

Before delving into the details of the error, let’s start by understanding what CVX is. CVX is a package for modeling and solving convex optimization problems in MATLAB. It allows users to express optimization problems in a natural and readable way, without needing to worry about the lower-level details of optimization algorithms. CVX supports a wide range of convex problems, from linear programming (LP) to semidefinite programming (SDP), and it automatically handles problem formulation and solution algorithms.

Objective Functions in Cvx Your Objective Function Is Not a Scalar

An optimization problem typically consists of two main components:

1. An objective function: The function you want to minimize or maximize.
2. Constraints: Conditions that the solution must satisfy.

In CVX, the objective function is expressed in terms of the variables that are being optimized. The objective is usually a scalar function, meaning it outputs a single real number for any given set of input values. For example, in a simple linear programming problem, the objective function might be a linear combination of variables that you aim to minimize or maximize.

The Scalar Requirement

CVX assumes that the objective function should be a scalar because optimization problems typically involve minimizing or maximizing a single real-valued quantity. For example, in a minimization problem, the goal is to find the values of the decision variables that yield the smallest possible scalar value for the objective function.

A typical objective function in CVX might look like this:

cvx_begin
    variable x(3)
    minimize( x(1) + 2*x(2) + 3*x(3) )
cvx_end

Here, x is a vector, and the objective is a scalar (the sum of weighted components of the vector x). This formulation is valid because the objective function produces a scalar value that we can minimize.

The Error: “CVX Your Objective Function Is Not a Scalar”

When you see the error “CVX Your objective function is not a scalar,” it means that the function you have defined as the objective is producing a non-scalar output, such as a vector or matrix, which CVX cannot handle as an objective. CVX requires that the objective function be scalar because it needs to minimize or maximize a single value. If the function outputs more than one value (i.e., a vector or matrix), the optimization problem becomes ill-defined.

Common Causes of the Error

There are several scenarios in which you might encounter this error. Below are some of the most common causes:

1. Vector or Matrix Objective: If you accidentally define the objective as a vector or matrix instead of a scalar, CVX will raise the error. For example:

   cvx_begin
       variable x(3)
       minimize( x )
   cvx_end
   

   Here, x is a vector, so the objective function is a vector itself, not a scalar.

2. Element-wise Operations: Sometimes, element-wise operations like .* or .^ can cause the output of an objective function to be a vector or matrix, even when you intended it to be scalar. For example:

   cvx_begin
       variable x(3)
       minimize( x .* x )  % This will result in a vector
   cvx_end
   

   The .* operation produces a vector, not a scalar, which leads to the error. You might want to sum the elements to obtain a scalar objective function:

   cvx_begin
       variable x(3)
       minimize( sum(x .* x) )  % This will result in a scalar
   cvx_end
   

3. Incorrect Use of Functions: Some built-in MATLAB functions or operations may return non-scalar outputs when applied to vector or matrix inputs. For example, using the norm of a vector as the objective is perfectly fine, but using an operation like det on a matrix will give you a scalar, while applying it incorrectly can return a non-scalar.

   cvx_begin
       variable A(3,3)
       minimize( det(A) )  % det is fine for scalar output
   cvx_end
   

   However, if you perform operations that return a matrix, like A * A, it could cause the objective to become non-scalar, leading to an error.

How to Fix the Error

Now that we understand why this error occurs, let’s look at how to fix it. The general rule is to ensure that the objective function evaluates to a scalar. Here are some steps you can take to resolve the issue:

Step 1: Verify Your Objective Function

Check the expression that defines your objective function. Make sure that it is designed to produce a scalar output. For example, if your objective involves a vector or matrix, you may need to sum its elements, compute a trace, or apply a similar operation that reduces the result to a scalar.

For instance, if your objective is to minimize the squared Euclidean norm of a vector x, make sure to use:

cvx_begin
    variable x(3)
    minimize( norm(x, 2)^2 )  % This is a scalar
cvx_end

Alternatively, if you’re minimizing a function of multiple variables, you can use operations like sum, norm, or trace to ensure the result is a scalar:

cvx_begin
    variable x(3)
    minimize( sum(x.^2) )  % Scalar result
cvx_end

Step 2: Check Element-wise Operations

If you’re using element-wise operations, be mindful of their effects. If you’re squaring a vector element-wise (e.g., x .* x), this will result in a vector, so use the sum() function to ensure a scalar output:

cvx_begin
    variable x(3)
    minimize( sum(x .* x) )  % Summing the squared elements results in a scalar
cvx_end

Step 3: Debug Your Problem Formulation

If you’re still unsure why your objective isn’t scalar, try breaking down the objective function into smaller parts and evaluate them separately. For example, you can print the size of intermediate expressions using size() to track down which part is returning a non-scalar result:

disp(size(x .* x));  % This will help identify if the output is a scalar or vector

Step 4: Use Built-in Functions Properly

Be cautious when using MATLAB’s built-in functions that may return vectors or matrices in certain cases. For example, norm(x) will return a scalar (the Euclidean norm), but x * x might give a matrix, depending on the dimensions of x.

Conclusion

The error “CVX Your objective function is not a scalar” occurs when the objective function you define in a CVX optimization problem returns a non-scalar value. Since CVX assumes the objective function is a scalar, the presence of a vector or matrix in the objective will result in this error.

To fix this, ensure that your objective function always evaluates to a scalar. You can achieve this by using operations like sum(), norm(), or trace() to reduce a vector or matrix to a scalar. Additionally, double-check that element-wise operations are not inadvertently turning your objective into a non-scalar. With these strategies, you can resolve the error and proceed with your optimization problem in CVX.