proved that this optimization problem , which we term Max-Trans-Graph, is NP-hard, and so described it as an Integer Linear Program (ILP).

Consequently, a natural variant of the Max-Trans-Graph problem is to restrict the required output graph of the optimization problem (1) to an FRG.

(2010) formulated the problem of learning entailment rules as a graph optimization problem , where nodes are predicates and edges represent entailment rules that respect transitivity.

Substituting back into (4) and dropping constant terms, we get the following optimization problem : minimize

This optimization problem is non-convex, and we do not know of a closed-form solution.

Gradient projection methods are attractive solutions to constrained optimization problems , particularly when the constraints on the parameters are simple (Bert-sekas, 1999).

In this section, we briefly introduce two methods previously studied by (Juola, 1997) and (Teahan, 2000) as representative of the two types, and we further explain a modification that we integrate into the final optimization problem .

The segmentation task was modeled as an optimization problem of finding the best segment and language sequences to minimize the description length of a given text.

This article presents one way to formulate the segmentation and identification problem as a combinatorial optimization problem ; specifically, to find the set of segments and their languages that minimizes the description length of a given multilingual text.