Cvx Without Objective Function

MIster jashon

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Oct 01, 2024

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CVX Without Objective Function: A Guide to Constraints-Only Optimization

CVX, a powerful MATLAB toolbox for convex optimization, is renowned for its ability to solve problems with both objective functions and constraints. However, there are scenarios where you might find yourself working with a problem that solely involves constraints, with no explicit objective function to minimize or maximize.

Why Would You Need CVX Without an Objective Function?

While seemingly unusual, solving problems with only constraints arises in various practical situations:

• Feasibility Problems: You need to determine if a set of constraints is feasible, meaning if there exists a solution that satisfies all the constraints. This is often crucial in engineering design, where ensuring a design meets specific requirements is paramount.
• Finding the "Best" Feasible Solution: If multiple feasible solutions exist, you might want to find one that meets certain criteria, such as minimizing the distance to a specific point or maximizing a certain variable, but without explicitly defining this as an objective function.
• Initializing Optimization Algorithms: In some optimization algorithms, you might need to find a feasible starting point before proceeding with the optimization process. This initial point can be obtained by solving a constraint-only problem.

How to Use CVX Without an Objective Function

CVX provides a way to handle these situations by utilizing its powerful constraint handling capabilities. Here's how:

1. Define your constraints: Use the standard CVX syntax to define your constraints. For example, if you have linear constraints, you can express them using <=, >=, or ==.

2. Use the optimize Command: Instead of using minimize or maximize, use the optimize command. This tells CVX to find a feasible solution that satisfies all the defined constraints.

Example: A Simple Feasibility Problem

Let's illustrate with a simple example. Suppose we want to find a solution x that satisfies the following constraints:

• x >= 1
• x <= 5

cvx_begin
  variable x
  x >= 1;
  x <= 5;
  optimize;
cvx_end

Running this code will return a feasible value for x that satisfies both constraints. In this case, any value between 1 and 5 will work.

Additional Tips and Considerations

• Empty Objective Function: If you want to explicitly state that there's no objective function, you can use the empty function in CVX. For instance:

cvx_begin
  variable x
  x >= 1;
  x <= 5;
  minimize(empty(x)); % This explicitly states no objective function
  optimize;
cvx_end

• Handling Non-Convex Constraints: CVX is designed for convex optimization problems. If your constraints are non-convex, CVX might not be able to guarantee finding a feasible solution or might even return a solution that does not satisfy all constraints.

• Alternative Approaches: If you're working with non-convex constraints, you may need to explore alternative optimization methods like genetic algorithms or simulated annealing.

Conclusion

Although CVX is primarily known for objective-function driven optimization, its constraint handling capabilities allow you to solve problems without an explicit objective function. This opens up possibilities for addressing feasibility problems, finding "best" feasible solutions, and initializing optimization algorithms. By understanding the optimize command and employing CVX's constraint syntax, you can effectively leverage CVX's power even in scenarios where a traditional objective function is not present.