By Satoru Iwata, Satoko Moriguchi, Kazuo Murota

This paper provides a swifter set of rules for the M-convex submodular How challenge, that's a generalization of the minimum-cost How challenge with an M-convex fee functionality for the How-boundary, the place an M-convex functionality is a nonlinear nonseparable cliserete convex functionality on integer issues. The set of rules extends the potential sealing strategy lor the submodular How challenge through Fleischer. Iwata and MeCormiek (2002) because of a singular means of altering the capability by means of fixing greatest submodular How difficulties.

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We call these functions self-concordant barriers. As we shall see later, these barriers play the central role in the interior point machinery. In this section, we introduce the notion of a self-concordant barrier and study the basic properties of these barriers. As already mentioned, a self-concordant barrier F should satisfy the following relations: SELF-CONCORDANT BARRIERS 33 Thus, generally speaking, there are two Lipschitz constants, consti and const2, responsible for the barrier. Note that a scaling F —> cF updates these constants as follows: consti —*• c"1'2 consti, const2 —> c1'2 const2.

Proof. Without loss of generality, we can restrict ourselves to the case when G does not contain any straight line. 1°. For x e intG, the polar G*(x) clearly is a bounded closed convex domain (note that int G* (x) ^ 0 since G does not contain lines and that the boundedness of G*(x) follows from the inclusion x € int G). Hence the function is well defined and positive on int G. If Xi G int G and Xi —>• x £ dG, then all of the sets G*(xi) contain certain fixed open nonempty set (since the sequence {xi} is bounded), and, at the same time, these sets are not uniformly bounded (since limxj 6 dG).

Let Fi e B#f(Ki), i = 1,2. Then F1(x) + F2(y) € B#1+dy(K! x K2), and, if Fi € NB^(Ki), i = 1,2, then also F1(x) + F2(y) € NB^1+^(Ki x K2). A logarithmically homogeneous barrier satisfies a number of useful identities. )-logarithmically homogeneous barrier for a cone K. Then for each x € int K, h 6 E and each t > 0, Proof. 13). 14). 15). 14)). 2 A ^-normal barrier for K is a '&-self-concordant barrier forK. 15). 5 Let K C E be a closed convex cone and let F be a •&logarithmically homogeneous barrier for K.

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