Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints

Lei Guo () and Gaoxi Li ()
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Lei Guo: East China University of Science and Technology
Gaoxi Li: Chongqing Technology and Business University

Journal of Optimization Theory and Applications, 2024, vol. 202, issue 3, No 17, 1445 pages

Abstract: Abstract We consider how to solve a class of non-Lipschitz mathematical programs with equilibrium constraints (MPEC) where the objective function involves a non-Lipschitz sparsity-inducing function and other functions are smooth. Solving the non-Lipschitz MPEC is highly challenging since the standard constraint qualifications fail due to the existence of equilibrium constraints and the subdifferential of the objective function is unbounded due to the existence of the non-Lipschitz function. On the one hand, for tackling the non-Lipschitzness of the objective function, we introduce a novel class of locally Lipschitz approximation functions that consolidate and unify a diverse range of existing smoothing techniques for the non-Lipschitz function. On the other hand, we use the Kanzow and Schwartz regularization scheme to approximate the equilibrium constraints since this regularization can preserve certain perpendicular structure as in equilibrium constraints, which can induce better convergence results. Then an approximation method is proposed for solving the non-Lipschitz MPEC and its convergence is established under weak conditions. In contrast with existing results, the proposed method can converge to a better stationary point under weaker qualification conditions. Finally, a computational study on the sparse solutions of linear complementarity problems is presented. The numerical results demonstrate the effectiveness of the proposed method.

Keywords: Mathematical program with equilibrium constraints; Nonconvex optimization; Non-Lipschitz continuity; Sparsity-inducing penalty; Smoothing function; Regularization method; 90C33; 90C30; 90C59 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02475-6

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