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An SQP method for mathematical programs with vanishing constraints with strong convergence properties.
1Institute of Computational Mathematics, Johannes Kepler University Linz, 4040 Linz, Austria.
Computational Optimization and Applications
|May 9, 2017
Summary
We introduce a Sequential Quadratic Programming (SQP) algorithm for mathematical programs with vanishing constraints. This method ensures limit points are M-stationary or the stronger [Formula: see text]-stationary.
Area of Science:
- Optimization Theory
- Mathematical Programming
Background:
- Mathematical programs with vanishing constraints present unique challenges in optimization.
- Existing algorithms may struggle with the convergence properties at stationary points.
Purpose of the Study:
- To develop a novel Sequential Quadratic Programming (SQP) algorithm tailored for mathematical programs with vanishing constraints.
- To establish theoretical convergence guarantees for the proposed SQP method.
Main Methods:
- The proposed SQP algorithm solves a quadratic program with linear vanishing constraints at each iteration.
- The method leverages the concept of [Formula: see text]-stationarity, recently introduced in optimization literature.
- Demonstration of obtaining [Formula: see text]-stationary solutions for the subproblem.
Main Results:
- All limit points generated by the basic SQP method are proven to be at least M-stationary.
- An extension of the SQP method guarantees the stronger property of [Formula: see text]-stationarity for limit points.
- The algorithm effectively handles vanishing constraints through specialized subproblem formulations.
Conclusions:
- The developed SQP algorithm provides a robust framework for solving mathematical programs with vanishing constraints.
- The theoretical analysis confirms improved convergence properties, reaching at least M-stationarity and potentially [Formula: see text]-stationarity.
- This work advances the field of optimization by offering a principled approach to a challenging problem class.
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