Central-Path Saddle Optimizer

(1) Integrate this at the optimizer level for a constrained robust-training problem. Let theta be network parameters, delta be an adversarial perturbation, and optionally u be a low-dimensional routing variable. Solve min_theta max_delta L(theta,delta), subject to equality constraints A(theta,delta)=b and nonnegative variables z. For signed variables, use q=q_plus-q_minus with q_plus,q_minus>=epsilon, or apply the primal-dual system only to delta and u while keeping theta in ordinary unconstrained coordinates. Maintain equality multipliers lambda and positive dual variables s.

(2) On each minibatch, construct R(v)=[grad_theta L; -grad_delta L; A(z)-b; s*z-mu*ones]. Compute the Jacobian J=dR/dv by autodiff. Solve (J+gamma I)d=-R using dense linear algebra for small adversaries or a Krylov method with Jacobian-vector products for larger systems. Backtrack over alpha in {1,1/2,1/4,...} until z+alpha*d_z>=epsilon, s+alpha*d_s>=epsilon, and ||R(v+alpha d)||<=(1-c*alpha)||R(v)||. Update v<-v+alpha d. After successful steps set mu<-max(mu_min,0.2*mu), warm-starting the next minibatch.

(3) The paper's mathematics determines the residual structure, complementarity barrier, positivity safeguards, and the nonsingular-Jacobian/local-attraction diagnostic. Estimate J with autodiff or Hessian-vector products; estimate conditioning using a few Lanczos iterations and increase gamma when the smallest-singular-value proxy is too small. Because minibatches are noisy, use an exponential moving average of R or damp the Newton direction.

(4) First test a two-layer MLP on MNIST or CIFAR-10 with a 10- to 32-dimensional l-infinity adversary. Compare against projected gradient descent/ascent, extragradient, and Adam alternating updates at equal backward-pass budgets. Record robust validation accuracy, constraint violation, residual norm, adversary iterations, wall-clock cost, and failed runs across seeds. Success means equal or higher robust accuracy with fewer inner iterations, fewer cycling or exploding runs, and maximum violation below 1e-3 without projection. A second test can use a four-expert MoE with per-batch load equality constraints and compare routing imbalance and loss against softmax routing plus an auxiliary balancing loss.