A technique performs an efficient constrained shortest path first (CSPF) optimization of Traffic Engineering (TE) Label Switched Paths (LSPs) in a computer network. The novel CSPF technique is triggered upon the detection of an event in the computer network that could create a more optimal path, such as, e.g., a new or restored network element or increased path resources. Once the novel CSPF technique is triggered, the computing node (e.g., a head-end node of the TE-LSP or a Path Computation Element, PCE) determines the set of nodes adjacent to the event, and further determines which of those adjacent nodes are within the TE-LSP (“attached nodes”). The computing node performs a CSPF computation rooted at the closest attached node to determine whether a new computed path cost is less than a current path cost (e.g., by a configurable amount), and if so, triggers optimization of the TE-LSP along the new path.