WSNet: Learning to Warm-Start Interior-Point Methods for Nonlinear Optimization

Interior-point methods (IPMs) are widely used to solve constrained nonlinear programs, yet warm-starting an IPM is particularly challenging because an initial point must not only be close to a solution, but also preserve the primal-dual interiority and centrality required for reliable numerical progress. We propose the Warm-Start Network (WSNet), a learning-based framework that reuses the primal-dual trajectory of a reference problem to warm-start a perturbed instance. WSNet first employs an attention-based selection module to identify an informative trajectory iterate. It then adjusts the selected point by predicting a dual-slack correction, recovering the remaining primal-dual updates through finite-step residual minimization, and enforcing centrality with neighborhood-aware training losses. Under explicit optimization and regularity assumptions, we show that mini-batch training produces strictly interior primal-dual points with high probability. Combined with finite-step recovery, this result guarantees membership in a prescribed central-path neighborhood. Experiments on convex and nonconvex quadratic programs, quadratically constrained quadratic programs, and second-order cone programs demonstrate that WSNet effectively accelerates IPOPT, reducing its iteration count by up to 59.9% and total solution time by up to 60.3%.