Mathematical analyses of quasi active set strategy for quadratic programming using affine mapping

Zhaocheng Xuan

doi:10.1117/12.2692216

23 August 2023 Mathematical analyses of quasi active set strategy for quadratic programming using affine mapping

Zhaocheng Xuan

Proceedings Volume 12784, Second International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2023); 127841M (2023) https://doi.org/10.1117/12.2692216
Event: 2023 2nd International Conference on Applied Statistics, Computational Mathematics and Software Engineering (ASCMSE 2023), 2023, Kaifeng, China

Abstract

The quasi-active set method that is an iteration procedure has been used to solve quadratic programming problems, a feature of iteration is that subproblems are constructed by violating the constraints of the unconstrained solution and using the solution at each step as the starting point for the next step, and the surrogate dual of each subproblem has been shown to have no gap between the primal and the dual. In this paper, we present a theoretical analysis of the convexity of the dual objective function in the quasi-active set method, by constructing superlevel sets in terms of the objective function and using an inverse image of the second-order convex cone under an affine mapping. We prove that the fractional objective function of the surrogate duality problem is quasi-concave under condition λ^T (AK⁻¹t − g) ≥ 0, thus primal quadratic programming problems are transformed into a dual problem with a fraction as the objective, and a simplex and a single constraint as the constraints. An illustrative example of a contact problem is given to show the effect of the choice of potential contact nodes on the convexity of the dual objective function. The quasi-concavity of the objective function is guaranteed if suitable potential contact nodes are chosen. Theoretically, the optimal solution can be found in only one step.

Citation Download Citation

Zhaocheng Xuan "Mathematical analyses of quasi active set strategy for quadratic programming using affine mapping", Proc. SPIE 12784, Second International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2023), 127841M (23 August 2023); https://doi.org/10.1117/12.2692216

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