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The Branch and Bound Algorithm Note that, since ! is not allowed to be a vertex of S , the number of indices i such that i 6= 0 is at least 2. 1. If the point ! is taken to be the midpoint of the longest edge of S , then the partitioning method is called bisection. If ! is not the midpoint but an arbitrary point on the longest edge, then one speaks of generalized bisection. More on this topic can be found in Horst 42 . Some partitioning procedures have a special quality: Every nested sequence of sets generated by the procedure eventually shrinks to a singleton.
On C . Then the function 'C;f : C ! IR, 'C;f x := supfh x : h : C ! IR convex, h f on C g is said to be the convex envelope of f over C . Notice that it is often convenient to eliminate formally the set C by setting 8 f x = : f x ; x 2 C +1; x 2= C and replacing 'C;f accordingly by its extension 'C;f : IRn ! IR f1g. c. on C , and hence is representable as the pointwise supremum of the a ne minorants of f . Geometrically 'C;f is the function whose closed epigraph coincides with the convex hull of the epigraph of f .
Nevertheless, at least for polynomials some decompositions can be given. First, consider polynomials of one variable. Obviously, there is no problem if the exponent is even. For polynomials of the form xn with n odd, we give a decomposition in the next theorem. c. decomposition of a function f is equivalent to nding a convex function ' such that f + ' is convex. Also recall that a twice di erentiable function is convex if and only if its Hessian is a positive semide nite matrix. 1 Let f : IR ! IR; f x = xn, with n 3 odd.