A new technique to derive tight convex underestimators (sometimes envelopes)
提出一种通过松弛内层最小化问题和对偶性来推导凸下界函数的新方法，在二次函数等情形下可得到凸包络，适用于多项式、多项式比及可分离函数。
Abstract The convex envelope value for a given function f over a region X at some point $$\textbf{x}\in X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> can be derived by searching for the largest value at that point among affine underestimators of f over X . This can be computed by solving a maximin problem, whose exact computation, however, may be a hard task. In this paper we show that by relaxation of the inner minimization problem, duality, and, in particular, by an enlargement of the class of underestimators (thus, not only affine ones) an easier derivation of good convex understimating functions, which can also be proved to be convex envelopes in some cases, is possible. The proposed approach is mainly applied to the derivation of convex underestimators (in fact, in some cases, convex envelopes) in the quadratic case. However, some results are also presented for polynomial, ratio of polynomials, and some other separable functions over regions defined by similarly defined separable functions.