[Kaiyao-Ge]

NonsmoothNewton descent strategy

Adds NonsmoothNewton, a Newton variant for objectives that split as f = s + n, with s smooth and n a stiff nonsmooth term such as a contact barrier.

When the current iterate has active nonsmooth terms, the strategy minimizes a reduced model: a frozen quadratic of s plus the exact n. It does this with Levenberg–Marquardt steps in a growing subspace. Every segment of the resulting path is certified with max_step_size. The end point, or a retraction along the path, is accepted by its true decrease. On other iterates the Newton direction is line searched as usual. If the strategy fails, the solver falls back to the existing Newton chain.

Near convergence the true decrease of a step falls below what the energy can resolve in floating point, so the acceptance test can no longer verify the path. From then on the strategy hands the rest of the solve to the Newton chain, whose line search compares gradient norms instead of energies once the gradient is small (line_search/use_grad_norm_tol). On a pin-cushion contact scene (steps 11–20), the handoff and the energy resolution cut the solve time from 205 s to 131 s.

Changes

• Problem: optional nonsmooth-split callbacks (has_nonsmooth_term, smooth_*, nonsmooth_*, metric_diagonal) and HessianBlock. The defaults report that the problem has no split, so existing problems are unaffected.
• DescentStrategy::take_step: a hook that lets a strategy return a step it has already verified. Solver::minimize then applies x += step and skips the line search. The default returns false, so behaviour is unchanged for all existing strategies.
• Solver::minimize evaluated the objective at the start point twice (once only for a debug message). It now evaluates it once.
• Solver::create("NonsmoothNewton"): builds the chain NonsmoothNewton → Newton chain (Newton::create_solver(true, ...)) → GD.
• nonlinear-solver-spec.json: the NonsmoothNewton options (max_inner, relative_stop, fraction_to_boundary, rho_min, max_columns, handoff, resolution, guard/*) and internal debug/* switches.
  • handoff (off / first / any, default any): hand the solve to the Newton chain after an outer iteration whose inner loop stopped at the noise floor of the energy, or whose path is below the energy resolution (first: only if no inner step was accepted before). Except for off, also when the gradient norm is below line_search/use_grad_norm_tol.
  • resolution (default true): count 32 eps |E| as the resolution of the true energy E, in the inner noise floor and in the acceptance. A path whose model decrease does not exceed it is not verified by the true energy; the Newton step is line searched instead.
• New linear solver Eigen::SimplicialLLT (Eigen's simplicial Cholesky; in the factory, available_solvers and linear-solver-spec.json). Like Eigen::CholmodSupernodalLLT, it reports a non-positive pivot, so NonsmoothNewton also runs in builds without CHOLMOD.
• Tests: tests/test_nonsmooth_newton.cpp. They cover option parsing, a log-barrier problem (with and without the handoff, and the resolution on an energy with a large constant), and the no-split case, where the result is bitwise equal to Newton. The log-barrier test uses Eigen::CholmodSupernodalLLT when CHOLMOD is available and Eigen::SimplicialLLT otherwise, and also checks a solve with Eigen::SimplicialLLT.

When the problem provides the split, the linear solver must be an LLT that reports indefinite matrices: Eigen::CholmodSupernodalLLT or Eigen::SimplicialLLT. This is checked on first use.