Supplement: Universal Self-Concordant Barrier Functions

Transcription

1 IE Supplement: Universal Self-Concordant Barrier Functions

2 IE Recall that a self-concordant barrier function for K is a barrier function satisfying 3 F (x)[h, h, h] 2( 2 F (x)[h, h]) 3/2, F (x)[h] θ( 2 F (x)[h, h]) 1/2, for any x int K and any direction h R n. Furthermore, we call a barrier function F (x) to be ν-logarithmically homogeneous if for all x int K and t > 0. F (tx) = F (x) ν ln t

3 IE A fundamentally important result due to Nesterov and Nemirovski is the following: Theorem 1 Any closed convex cone admits a self-concordant, logarithmically homogeneous barrier function where θ = O( n). In this note we shall show why this is the case. First, let us note some basic facts. Proposition 1 Suppose that F (x) is a smooth ν-logarithmically homogeneous barrier function for K. Then the following identities hold where x int K and t > 0: F (tx) = 1 F (x); t 2 F (tx) = 1 t 2 2 F (x); 2 F (x)x = F (x); ( F (x)) T x = ν.

4 IE For an ν-logarithmically homogeneous function, the second property of the self-concordance always holds F (x)[h] = h T F (x) = h T 2 F (x)x = [( 2 F (x)) 1/2 h] T [( 2 F (x)) 1/2 x] h T 2 F (x)h x T 2 F (x)x = ν( 2 F (x)[h, h]) 1/2, where we used the Cauchy-Schwartz inequality.

5 IE There are intimate relationships between a convex cone and its barrier function; the dual cone and the conjugate of the barrier funciton. Recall that the conjugate of a function f is: f (s) = sup{( s) T x f(x) x dom f}, where dom f stands for the domain of the function f. Suppose that f is a barrier function of K. Then f is a barrier function of K. Moreover, for x int dom f and s int dom f the following three statements are equivalent s = f(x); x = f (s); x T s = f(x) + f (s).

6 IE For a convex cone K, the following function is known as its characteristic function ϕ(x) = e x,y dy. K Let F (x) = ln ϕ(x). One can show the following properties of F : F (x) is a barrier function for K; F (x) is strictly convex; F (x) is n-logarithmically homogeneous.

7 IE Let us consider the last property. Consider a linear transformation A that keeps K invariant, i.e. AK = K; in that case, A T K = K. Then, ϕ(ax) = e Ax,y dy K = e x,a T y dy K }{{} = dy / det A y :=A T y K e x,y = ϕ(x)/ det A. Now, for t > 0, the transformation A = ti keeps K invariant. Therefore, ϕ(tx) = ϕ(x)/t n, and so F (tx) = F (x) nt.

8 IE Moreover, A keeps K invariant, because F (Ax) = F (x)/ det A we have F (Ax) = F (x) ln det A. Differentiating the above identity yields for any k 1. k F (Ax)[Ah, Ah,..., Ah] = k F (x)[h, h,..., h] }{{}}{{} k k In order to show that the first property of the self-concordance is also satisfied, we fix a given point ι int K. For any x int K, there is a mapping A x such that ι = A x x. Let h = A x h. It will be sufficient to prove that there exists a constant C > 0 such that 3 F (ι)[h, h, h ] C( 2 F (ι)[h, h ]) 3/2 h R n which is true because 2 F (ι) is positive definite.

9 IE The characteristic function exists for any convex cone. The barrier function defined that way is also known as the universal barrier function. As we have just seen, it is also a self-concordant barrier. One problem, however, is that the universal barrier function is not easily computable. For special cases, one may be able to compute the associated universal barrier function. For instance, it is easy to see that the characteristic function for R n + is: R n + e x 1y 1 x n y n dy = 1/ n x i, and so the universal barrier function is simply the logarithmic barrier function n i=1 ln x i. i=1

10 IE Interestingly, there is also a very nice geometric interpretation of the universal barrier function. Let S be a convex set. Introduce the set function S (x) := ( x + S). Let u(x) := ln(vol(s (x))). Nesterov and Nemirovski showed that u(x) is a self-concordant barrier function for S with complexity parameter θ = O( n). Güler proved that if S is a convex cone (S = K) then the following identity holds F (x) = u(x) + ln n!.

11 IE Key References: Yu. Nesterov and A. Nemirovski, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, 13, O. Güler, Barrier Functions in Interior Point Methods, Mathematics of Operations Research, 21, pp , 1996.