A safeguarded nonmonotone Barzilai–Borwein smoothing Newton method for weighted complementarity problems: convergence analysis and numerical validation

We developed a nonmonotone smoothing Newton method for weighted complementarity problems (WCPs) in $ \mathbb{R}^n $. After full-step rejection, a safeguarded Barzilai–Borwein (BB) factor with step-length memory initialized backtracking. Under the structural rank–monotonicity condition and a bounded initial level set, we proved that the residual tends to zero, and every accumulation point solves the WCP; uniqueness gives whole-sequence convergence. Nonsingularity of every matrix in the generalized Jacobian at an accumulation point, together with a locally Lipschitz derivative of the original equation block, yielded quadratic convergence and eventual full-step acceptance. For symmetric positive-definite linear problems, we verified level-set boundedness, uniqueness, and local regularity without strict complementarity. On 150 paired nonconvex instances, rejected-trial counts were significantly lower than those of a one-step comparator in five of six settings after Holm correction, whereas iteration-count differences were not significant. An ablation study supported the BB initializer's benefit without establishing a general timing or success-rate advantage. Further tests included an independent recent one-step comparator, monotone linear tests, sparse positive-weight problems with up to 102400 complementarity pairs, and three zero-weight partial differential equation (PDE) benchmarks with grid-refinement and independent reference checks. On nonconvex tests, success depends on the fixed complementarity parameter and initialization; failures at larger parameter values persist from the original signed starts. Some signed-start Bratu failures persisted under a larger budget. These results support reduced post-rejection line-search work, not uniform robustness across problem classes.