It seems like most solvers that can tackle nonlinear nonconvex optimization problems (e.g. IPOPT) operate on ultimately solving for the first-order optimality conditions. Can it therefore be assumed that these solvers do not guarantee convergence to a local minimzer, since saddle points and maximizers also satisfy the optimality conditions? If this is the case, what techniques do solvers use to try and avoid the points that are not minimizers (if at all)?

Second order KKT conditions are as follows: A point satisfying 1st order KKT conditions and for which objective function and constraints are twice continuously differentiable is (sufficient for) a local minimum if the the Hessian of the Lagrangian projected into the nullspace of the Jacobian of active constraints is positive semidefinite. Letting $$Z$$ be a basis for the nullspace of the Jacobian of active constraints, second order KKT condition is that $$Z^THZ$$ is positive semi-definite, where $$H$$ is the Hessian of the Lagrangian. If $$Z^THZ$$ is indefinite, i.e., has at least one positive eigenvalue and at least one negative eigenvalue, then the point is a saddle point. Active constraints consist of all equality constraints plus all inequality constraints which are satisfied with equality at the point under consideration. If no constraints are active at the 1st order KKT point under consideration, the identity matrix is a nullspace basis $$Z$$, and all Lagrange multipliers must be zero, therefore, the second order necessary condition for a local minimum reduces to the familiar condition from unconstrained optimization that the Hessian of the objective function is positive semidefinite. If all constraints are linear, the Hessian of the Lagrangian = Hessian of objective function because the 2nd derivative of a linear function = 0.