A branch-and-bound algorithm for discrete multi-factor by Niu Shu-fen, Wang Guo-xin, Sun Xiao-ling

By Niu Shu-fen, Wang Guo-xin, Sun Xiao-ling

During this paper, a brand new branch-and-bound set of rules in keeping with the Lagrangian twin rest and non-stop leisure is proposed for discrete multi-factor portfolio choice version with roundlot limit in monetary optimization. This discrete portfolio version is of integer quadratic programming difficulties. The separable constitution of the version is investigated through the use of Lagrangian leisure and twin seek. Computational effects exhibit that the set of rules is able to fixing real-world portfolio issues of information from US inventory industry and randomly generated try issues of as much as a hundred and twenty securities.

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Then there exists an edge e such that G/e is also 3-connected. Proof: Suppose there is no such edge. e. has a connected component C with |V (C)| < |V (G)| − 3. Choose e, x and C such that |V (C)| is minimum. x has a neighbour y in C, because otherwise C is a connected component of G − {v, w} (but G is 3-connected). e. there exists a vertex z such that G − {x, y, z} is disconnected. Since {v, w} ∈ E(G), there exists a connected component D of G − {x, y, z} which contains neither v nor w. But D contains a neighbour d of y, since otherwise D is a connected component of G − {x, z} (again contradicting the fact that G is 3-connected).

Y form an undirected circuit with the properties (a). Case 2: x has not been labelled. Then let R consist of all labelled vertices. Obviously, the undirected cut δ + (R) ∪ δ − (R) has the properties (b). Suppose that an undirected circuit C as in (a) and an undirected cut δ + (X ) ∪ − δ (X ) as in (b) both exist. All edges in their (nonempty) intersection are black, they all have the same orientation with respect to C, and they all leave X or all enter X . This is a contradiction. A digraph is called strongly connected if there is a path from s to t and a path from t to s for all s, t ∈ V (G).

On the other hand, because of the minimality of F, F \ {{v, w}} is not an odd cover (for any w with {v, w} ∈ F), so |δG (V (Ci ))| is odd for i = 1, . . , k. Since k |δG (V (Ci ))| = |δG (X )| + |δG (v)| i=1 − 2|E G ({v}, V (G) \ X )| + 2 |E G (Ci , C j )|, 1≤i< j≤k we conclude that k has the same parity as |δG (v)|. 2. A bipartition of an undirected graph G is a partition of the vertex set V (G) = . A ∪ B such that the subgraphs induced by A and B are both empty. A graph is called . bipartite if it has a bipartition.

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