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A generic algorithm is developed to reduce the problem of obtaining linearand nonlinear entanglement witnesses of a given quantum system, to convexoptimization problem. This approach is completely general and can be appliedfor the entanglement detection of any N-partite quantum system. For thispurpose, a map from convex space of separable density matrices to a convexregion called feasible region is defined, where by using exact convexoptimization method, the linear entanglement witnesses can be obtained frompolygonal shape feasible regions, while for curved shape feasible regions,envelope of the family of linear entanglement witnesses can be considered asnonlinear entanglement witnesses. This method proposes a new methodologicalframework within which most of previous EWs can be studied. To conclude and inorder to demonstrate the capability of the proposed approach, besides providingsome nonlinear witnesses for entanglement detection of density matrices inunextendible product bases, W-states, and GHZ with W-states, some furtherexamples of three qubits systems and their classification and entanglementdetection are included. Also it is explained how one can manipulate most of thenon-decomposable linear and nonlinear three qubits entanglement witnessesappearing in some of the papers published by us and other authors, by themethod proposed in this paper. Keywords: non-linear and linear entanglementwitnesses, convex optimization. PACS number(s): 03.67.Mn, 03.65.Ud

A general algorithm for manipulating non-linear and linear entanglement witnesses by using exact convex optimization