An Active-Set Algorithm for Convex Quadratic Programming Subject to Box Constraints with Applications in Non-Linear Optimization and Machine Learning

Konstantinos Vogklis () and Isaac E. Lagaris
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Konstantinos Vogklis: Department of Tourism, Ionian University, 49100 Kerkira, Greece
Isaac E. Lagaris: Department of Computer Science and Engineering, University of Ioannina, 45110 Ioannina, Greece

Mathematics, 2025, vol. 13, issue 9, 1-37

Abstract: A quadratic programming problem with positive definite Hessian subject to box constraints is solved, using an active-set approach. Convex quadratic programming (QP) problems with box constraints appear quite frequently in various real-world applications. The proposed method employs an active-set strategy with Lagrange multipliers, demonstrating rapid convergence. The algorithm, at each iteration, modifies both the minimization parameters in the primal space and the Lagrange multipliers in the dual space. The algorithm is particularly well suited for machine learning, scientific computing, and engineering applications that require solving box constraint QP subproblems efficiently. Key use cases include Support Vector Machines (SVMs), reinforcement learning, portfolio optimization, and trust-region methods in non-linear programming. Extensive numerical experiments demonstrate the method’s superior performance in handling large-scale problems, making it an ideal choice for contemporary optimization tasks. To encourage and facilitate its adoption, the implementation is available in multiple programming languages, ensuring easy integration into existing optimization frameworks.

Keywords: convex quadratic programming; machine learning; optimization; active set; Lagrange multipliers; practical applications (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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