How to solve constraint qualification failure

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SUMMARY

This discussion focuses on solving constraint qualification (CQ) failures in optimization problems involving mixed and pure constraints, specifically using the Pontryagin minimum principle. The Hamiltonian and Lagrangian functions are essential for deriving necessary conditions for optimal solutions. However, issues arise when the CQ matrix is not full rank, leading to difficulties in finding solutions. A suggested approach is to slightly alter the data, as full rank matrices form a dense subset in this context.

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  • Understanding of Pontryagin minimum principle
  • Familiarity with Hamiltonian and Lagrangian functions
  • Knowledge of constraint qualification (CQ) concepts
  • Basic principles of optimization theory
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  • Explore techniques for ensuring full rank in constraint qualification matrices
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  • Investigate perturbation methods to achieve full rank matrices
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Researchers and practitioners in optimization theory, control engineers, and anyone dealing with constraint qualifications in mathematical programming.

tanzl
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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)\geq0 , u(t)\geq0.

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

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