The optimality conditions give a characterization for locally optimal solutions of a nonlinear optimization problem. The goal of an optimization process is to find a point that satisfies those conditions. During the optimization process (information written to the log by the solver), the optimality error gives an estimate of how close the NLP solver is to a locally optimal solution. The NLP solver terminates, declaring convergence to a local solution, only if the optimality error (which is the norm of the optimality conditions) is less than the user-defined or default tolerance (usually around 1e-6).
The optimization process is an iterative process that starts from an initial estimate of the solution (starting point) and stops eventually at a locally optimal solution. During this process, the NLP solver generates a sequence of iterates. At every iteration, the NLP solver prints the value of the objective function at the current iterate, the infeasibility error and the optimality error. These information provide an idea about the progress made by the NLP solver towards a locally optimally solution. The infeasibility indicates the maximum value out of all constraint violations (for the original problem before introducing slack variables). It is always equal to zero for unconstrained problems.
When monitoring the information written to the log, the ideal case is that the infeasibility (given by the third column) and the optimality error (given by the fourth column) decrease every iteration. However, given the nonlinearity and nonconvexity of the problem, it is difficult to guarantee that those quantities decrease every iteration. Those quantities can oscillate up and down during the optimization process but converge to zero asymptotically (under some assumptions).
For unconstrained problems, the optimality conditions are defined based on the gradient of the objective function and/or the positive (semi-)definiteness of the Hessian. For general constrained optimization problems, the optimality conditions are more complicated and involve the primal and dual variables and the Lagrangian function. I would recommend reading Chapter 12 of the book “Numerical Optimization” by Jorge Nocedal and Stephen Wright. It gives a detailed description of the basic theory associated with nonlinear optimization. This is a great book to learn about numerical optimization.
In the SAS NLP solver’s documentation, we give more details about nonlinear optimization and the algorithms that are implemented. Here is a link https://support.sas.com/documentation/onlinedoc/or/151/nlpsolver.pdf
