Knowledge about the characteristics of a discrete optimization problem can often be leveraged for improved algorithm performance. Having an understanding of the relationship between knowledge and performance can provide us useful hints and better prediction of algorithm behavior when we choose or design algorithms. Previously, we developed a Directional Tree (DT) model [9] which concurrently describes algorithm behavior and represents knowledge explicitly, and thus provided us with a tool to analyze the impact of knowledge on algorithm performance. By using DT, we analyzed the performance of an algorithm when that algorithm is used for solving problems from a problem space P. In our previous analysis, we only considered the situation where there is only one optimal solution in the solution space. Furthermore, our DT model assumes that knowledge about the problem is obtained prior to the execution of any probabilistic or deterministic algorithm. However, such prior knowledge may not be readily or immediately available. In this paper, we extend our DT results to cover multiple optimal solutions and then generalize our DT model to take into account acquisition of problem knowledge during execution. Our Generalized Directional Tree (GDT) model provides a framework for incorporating knowledge obtained online into algorithms, and thus enables us to develop online optimization algorithms that dynamically updates their search decisions in response to obtained knowledge so far.