# How to solve constraint qualification failure

• tanzl
In summary, the conversation discusses a problem involving the optimization of an integral with mixed and pure constraints, which can be solved using the Pontryagin minimum principle and necessary conditions. However, the necessary conditions require the constraint qualification (CQ) to hold, and the speaker is seeking suggestions for solving cases that violate this condition. They mention the possibility of changing the data to obtain full rank matrices.
tanzl
I am currently solving a problem (similar to optimal control theory) involving optimization of an integral with mixed and pure constraints. eg: $\int F(x,u,t) dt$ subject to x(t)$\geq$0 , u(t)$\geq$0.

The problem can be solved by Pontryagin minimum principle by introducing the Hamiltonian function and Langragian function and its corresponding necessary conditions. Solving the necessary conditions will yield optimal solutions for different cases.

However, the necessary conditions require the constraint qualification (CQ) to hold, ie: CQ matrix to be full rank. I have problem with some cases which they violate CQ (not full rank). Can anyone please suggest some techniques to solve the problem. Thanks.

If you change data a tiny bit, then you will get full rank matrices since those are a dense subset.

## 1. What is constraint qualification failure?

Constraint qualification failure refers to a situation in mathematical optimization where the constraints of a problem cannot be satisfied simultaneously, making it impossible to find a solution.

## 2. How can constraint qualification failure be detected?

Constraint qualification failure can be detected by analyzing the linear independence of the constraints and their gradients. If the gradients are not linearly independent, then the problem is said to have a constraint qualification failure.

## 3. What causes constraint qualification failure?

Constraint qualification failure can be caused by various factors, including a poorly formulated problem, incorrect modeling of constraints, or numerical errors in the optimization algorithm.

## 4. How can constraint qualification failure be solved?

Constraint qualification failure can be solved by reformulating the problem, adjusting the constraints, or using special optimization algorithms designed to handle such situations.

## 5. Can constraint qualification failure be avoided?

In some cases, constraint qualification failure can be avoided by carefully formulating the problem and using appropriate modeling techniques. However, in some cases, it may be an inherent issue due to the nature of the problem.

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