Network survivability is a crucial requirement in high-speed optical networks. Most research to date has been focused on the failure of a single component such as a link or a node. A double-link failures model in which any two links in the network may fail in an arbitrary order was proposed recently in literature. Three loop-back methods of recovering from double-link failures were also presented. The basic idea behind these methods is to pre-compute two backup paths for each link on the primary paths and reserve resources on these paths. Compared to protection methods for single-link failure model, the protection methods for double-link failure model require much more spare capacity. Reserving dedicated resources on every backup path at the time of establishing primary path itself would reserve excessive resources. In this thesis, we capture the surviving double link failures in WDM optical networks as a single Integer Linear Programming (ILP) based optimization problem. We use the double-link failures recovery method available in literature, develop rules to identify the scenarios where the backup capacity among intersecting demand sets can be shared. We employ the backup multiplexing technique and use ILP to optimize the capacity requirement while providing 100% protection for double-link failures. The numerical results indicate that, for the given example network and randomly picked demand matrix, the shared-link protection scheme that uses backup multiplexing provides 10-15% saving in capacity utilization over the dedicated-link protection scheme that reserves dedicated capacity on two backup paths for each link. The main contribution of this thesis is that we provide a way of adapting the heuristic based double-link failure recovery method into a mathematical framework, and use technique to improve wavelength utilization for optimal capacity usage.