Abstract
In this paper, a primal–dual interior point QP-free algorithm for mathematical programs with complementarity constraints is presented. Firstly, based on Fischer–Burmeister function and smoothing techniques, the investigated problem is approximated by a smooth nonlinear constrained optimization problem. Secondly, combining with an effective penalty function technique and working set, a QP-free algorithm is proposed to solve the smooth constrained optimization problem. At each iteration, only two reduced linear equations with the same coefficient matrix are solved to obtain the search direction. Under some mild conditions, the proposed algorithm possesses global convergence. Finally, some numerical results are reported.
Highlights
In this paper, we discuss the following mathematical programming problem with complementarity constraints (MPCC for short): min f (x, y) s.t. g(x, y) ≤ 0, (1)0 ≤ F(x, y) ⊥ y ≥ 0, where f : Rn+m → R, g = (g1, . . . , gmg )T : Rn+m → Rmg, F = (F1, . . . , Fm)T : Rn+m → Rm are continuously differentiable functions
It is well known that the QP-free method is one of the efficient methods for nonlinear programming
Motivated by the ideas of the algorithms in [21, 24,25,26,27] and combining with smoothing techniques, we propose a primal–dual interior point QP-free algorithm for the MPCC (1)
Summary
We discuss the following mathematical programming problem with complementarity constraints (MPCC for short): min f (x, y) s.t. g(x, y) ≤ 0,. A detailed overview of MPCC applications can be found in [1] and the monographs [2,3,4]. Since MPCC (1) is a nonconvex optimization problem and the standard Mangasarian– Fromovitz constraint qualification (MFCQ) is violated at any feasible point, the welldeveloped algorithms for the standard nonlinear programs (for example, [5,6,7,8,9,10,11,12,13,14,15,16,17,18]) typically have severe difficulties if they are directly used to solve the MPCC (1).
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