# Constrained Optimization

For stochastic-gradient optimization with very many inequality constraints, an adaptive constraint-sampling method stops drawing each step's constraint uniformly from all constraints and instead maintains an "important set" of recently violated constraints, sampling from that set or from all constraints with fixed probabilities. On 100-dimensional synthetic convex problems it lowers the final Euclidean distance of the iterates to the constrained minimizer by about two orders of magnitude on average for ABSGD, Projection-SGD and Huber-penalty SGD, and by one to two orders for APDSG; in Dubins Car control-barrier-function synthesis it lowers maximum constraint violation by half an order to one order of magnitude.

For stochastic-gradient optimization with very many inequality constraints, an adaptive constraint-sampling method stops drawing each step's constraint uniformly from all constraints and instead maintains an "important set" of recently violated constraints, sampling from that set or from all constraints with fixed probabilities. On 100-dimensional synthetic convex problems it lowers the final Euclidean distance of the iterates to the constrained minimizer by about two orders of magnitude on average for ABSGD, Projection-SGD and Huber-penalty SGD, and by one to two orders for APDSG; in Dubins Car control-barrier-function synthesis it lowers maximum constraint violation by half an order to one order of magnitude.

Evidential deep learning represents classification uncertainty through Dirichlet parameters, but the standard objective shows evidence contraction at high confidence. This work analyzes the optimization landscape of the regularized Dirichlet likelihood on a box domain, proves that the second-order sufficient conditions reduce to a scalar self-consistency derivative being positive, and gives an enumeration algorithm with a finite-convergence guarantee. On logits from a ResNet-18 trained on CIFAR-10, total evidence rises monotonically with confidence yet never exceeds a theoretical threshold near 19.75, while the active set shrinks from about 2.05 classes to one.

Evidential deep learning represents classification uncertainty through Dirichlet parameters, but the standard objective shows evidence contraction at high confidence. This work analyzes the optimization landscape of the regularized Dirichlet likelihood on a box domain, proves that the second-order sufficient conditions reduce to a scalar self-consistency derivative being positive, and gives an enumeration algorithm with a finite-convergence guarantee. On logits from a ResNet-18 trained on CIFAR-10, total evidence rises monotonically with confidence yet never exceeds a theoretical threshold near 19.75, while the active set shrinks from about 2.05 classes to one.