Self-concordant function

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A self-concordant function is a function satisfying a certain differential inequality, which makes it particularly easy for optimization using Newton's method[1]: Sub.6.2.4.2  A self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set. Self-concordant barriers are important ingredients in interior point methods for optimization.

Self-concordant functions

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Multivariate self-concordant function

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Here is the general definition of a self-concordant function.[2]: Def.2.0.1 

Let C be a convex nonempty open set in Rn. Let f be a function that is three-times continuously differentiable defined on C. We say that f is self-concordant on C if it satisfies the following properties:

1. Barrier property: on any sequence of points in C that converges to a boundary point of C, f converges to ∞.

2. Differential inequality: for every point x in C, and any direction h in Rn, let gh be the function f restricted to the direction h, that is: gh(t) = f(x+t*h). Then the one-dimensional function gh should satisfy the following differential inequality:

 .

Equivalently:[3]

 

Univariate self-concordant function

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A function   is self-concordant on   if:

 

Equivalently: if wherever   it satisfies:

 

and satisfies   elsewhere.

Examples

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  • Linear and convex quadratic functions are self-concordant, since their third derivative is zero.
  • Any function   where   is defined and convex for all   and verifies  , is self concordant on its domain which is  . Some examples are
    •   for  
    •  
    •   for  
    •  
    • for any function   satisfying the conditions, the function   with   also satisfies the conditions.

Some functions that are not self-concordant:

  •  
  •  
  •  

Self-concordant barriers

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Here is the general definition of a self-concordant barrier (SCB).[2]: Def.3.1.1 

Let C be a convex closed set in Rn with a non-empty interior. Let f be a function from interior(C) to R. Let M>0 be a real parameter. We say that f is a M-self-concordant barrier for C if it satisfies the following:

1. f is a self-concordant function on interior(C).

2. For every point x in interior(C), and any direction h in Rn, let gh be the function f restricted to the direction h, that is: gh(t) = f(x+t*h). Then the one-dimensional function gh should satisfy the following differential inequality:

 .

Constructing SCBs

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Due to the importance of SCBs in interior-point methods, it is important to know how to construct SCBs for various domains.

In theory, it can be proved that every closed convex domain in Rn has a self-concordant barrier with parameter O(n). But this “universal barrier” is given by some multivariate integrals, and it is too complicated for actual computations. Hence, the main goal is to construct SCBs that are efficiently computable.[4]: Sec.9.2.3.3 

SCBs can be constructed from some basic SCBs, that are combined to produce SCBs for more complex domains, using several combination rules.

Basic SCBs

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Every constant is a self-concordant barrier for all Rn, with parameter M=0. It is the only self-concordant barrier for the entire space, and the only self-concordant barrier with M < 1.[2]: Example 3.1.1  [Note that linear and quadratic functions are self-concordant functions, but they are not self concordant barriers].

For the positive half-line  ( ),   is a self-concordant barrier with parameter  . This can be proved directly from the definition.

Substitution rule

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Let G be a closed convex domain in Rn, and g an M-SCB for G. Let x = Ay+b be an affine mapping from Rk to Rn with its image intersecting the interior of G. Let H be the inverse image of G under the mapping: H = {y in Rk | Ay+b in G}. Let h be the composite function h(y) := g(Ay+b). Then, h is an M-SCB for H.[2]: Prop.3.1.1 

For example, take n=1, G the positive half-line, and  . For any k, let a be a k-element vector and b a scalar. Let H = {y in Rk | aTy+b ≥ 0} = a k-dimensional half-space. By the substitution rule,   is a 1-SCB for H. A more common format is H = {x in Rk | aTx ≤ b}, for which the SCB is  .

The substitution rule can be extended from affine mappings to a certain class of "appropriate" mappings,[2]: Thm.9.1.1  and to quadratic mappings.[2]: Sub.9.3 

Cartesian product rule

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For all i in 1,...,m, let Gi be a closed convex domains in Rni, and let gi be an Mi-SCB for Gi. Let G be the cartesian product of all Gi. Let g(x1,...,xm) := sumi gi(xi). Then, g is a SCB for G, with parameter sumi Mi.[2]: Prop.3.1.1 

For example, take all Gi to be the positive half-line, so that G is the positive orthant  . Let   is an m-SCB for G.

We can now apply the substitution rule. We get that, for the polytope defined by the linear inequalities ajTxbj for j in 1,...,m, if it satisfies Slater's condition, then   is an m-SCB. The linear functions   can be replaced by quadratic functions.

Intersection rule

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Let G1,...,Gm be closed convex domains in Rn. For each i in 1,...,m, let gi be an Mi-SCB for Gi, and ri a real number. Let G be the intersection of all Gi, and suppose its interior is nonempty. Let g := sumi ri*gi. Then, g is a SCB for G, with parameter sumi ri*Mi.[2]: Prop.3.1.1 

Therefore, if G is defined by a list of constraints, we can find a SCB for each constraint separately, and then simply sum them to get a SCB for G.

For example, suppose the domain is defined by m linear constraints of the form ajTxbj, for j in 1,...,m. Then we can use the Intersection rule to construct the m-SCB   (the same one that we previously computed using the Cartesian product rule).

SCBs for epigraphs

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The epigraph of a function f(x) is the area above the graph of the function, that is,  . The epigraph of f is a convex set if and only if f is a convex function. The following theorems present some functions f for which the epigraph has an SCB.

Let g(t) be a 3-times continuously-differentiable concave function on t>0, such that   is bounded by a constant (denoted 3*b) for all t>0. Let G be the 2-dimensional convex domain:  Then, the function f(x,t) = -ln(f(t)-x) - max[1,b2]*ln(t) is a self-concordant barrier for G, with parameter (1+max[1,b2]).[2]: Prop.9.2.1 

Examples:

  • Let g(t) = t1/p, for some p≥1, and b=(2p-1)/(3p). Then   has a 2-SCB. Similarly,   has a 2-SCB. Using the Intersection rule, we get that   has a 4-SCB.
  • Let g(t)=ln(t) and b=2/3. Then   has a 2-SCB.

We can now construct a SCB for the problem of minimizing the p-norm:  , where vj are constant scalars, uj are constant vectors, and p>0 is a constant. We first convert it into minimization of a linear objective:  , with the constraints:  for all j in [m]. For each constraint, we have a 4-SCB by the affine substitution rule. Using the Intersection rule, we get a (4n)-SCB for the entire feasible domain.

Similarly, let g be a 3-times continuously-differentiable convex function on the ray x>0, such that:   for all x>0. Let G be the 2-dimensional convex domain: closure({ (t,x) in R2: x>0, tg(x) }). Then, the function f(x,t) = -ln(t-f(x)) - max[1,b2]*ln(x) is a self-concordant barrier for G, with parameter (1+max[1,b2]).[2]: Prop.9.2.2 

Examples:

  • Let g(x) = x-p, for some p>0, and b=(2+p)/3. Then   has a 2-SCB.
  • Let g(x)=x ln(x) and b=1/3. Then   has a 2-SCB.

SCBs for cones

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  • For the second order cone  , the function   is a self-concordant barrier.
  • For the cone of positive semidefinite of m*m symmetric matrices, the function   is a self-concordant barrier.
  • For the quadratic region defined by   where   where   is a positive semi-definite symmetric matrix, the logarithmic barrier   is self-concordant with  
  • For the exponential cone  , the function   is a self-concordant barrier.
  • For the power cone  , the function   is a self-concordant barrier.

History

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As mentioned in the "Bibliography Comments"[5] of their 1994 book,[6] self-concordant functions were introduced in 1988 by Yurii Nesterov[7][8] and further developed with Arkadi Nemirovski.[9] As explained in[10] their basic observation was that the Newton method is affine invariant, in the sense that if for a function   we have Newton steps   then for a function   where   is a non-degenerate linear transformation, starting from   we have the Newton steps   which can be shown recursively

 .

However, the standard analysis of the Newton method supposes that the Hessian of   is Lipschitz continuous, that is   for some constant  . If we suppose that   is 3 times continuously differentiable, then this is equivalent to

 for all  

where   . Then the left hand side of the above inequality is invariant under the affine transformation  , however the right hand side is not.

The authors note that the right hand side can be made also invariant if we replace the Euclidean metric by the scalar product defined by the Hessian of   defined as   for  . They then arrive at the definition of a self concordant function as

 .

Properties

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Linear combination

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If   and   are self-concordant with constants   and   and  , then   is self-concordant with constant  .

Affine transformation

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If   is self-concordant with constant   and   is an affine transformation of  , then   is also self-concordant with parameter  .

Convex conjugate

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If   is self-concordant, then its convex conjugate   is also self-concordant.[6][11]

Non-singular Hessian

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If   is self-concordant and the domain of   contains no straight line (infinite in both directions), then   is non-singular.

Conversely, if for some   in the domain of   and   we have  , then   for all   for which   is in the domain of   and then   is linear and cannot have a maximum so all of   is in the domain of  . We note also that   cannot have a minimum inside its domain.

Applications

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Among other things, self-concordant functions are useful in the analysis of Newton's method. Self-concordant barrier functions are used to develop the barrier functions used in interior point methods for convex and nonlinear optimization. The usual analysis of the Newton method would not work for barrier functions as their second derivative cannot be Lipschitz continuous, otherwise they would be bounded on any compact subset of  .

Self-concordant barrier functions

  • are a class of functions that can be used as barriers in constrained optimization methods
  • can be minimized using the Newton algorithm with provable convergence properties analogous to the usual case (but these results are somewhat more difficult to derive)
  • to have both of the above, the usual constant bound on the third derivative of the function (required to get the usual convergence results for the Newton method) is replaced by a bound relative to the Hessian

Minimizing a self-concordant function

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A self-concordant function may be minimized with a modified Newton method where we have a bound on the number of steps required for convergence. We suppose here that   is a standard self-concordant function, that is it is self-concordant with parameter  .

We define the Newton decrement   of   at   as the size of the Newton step   in the local norm defined by the Hessian of   at  

 

Then for   in the domain of  , if   then it is possible to prove that the Newton iterate

 

will be also in the domain of  . This is because, based on the self-concordance of  , it is possible to give some finite bounds on the value of  . We further have

 

Then if we have

 

then it is also guaranteed that  , so that we can continue to use the Newton method until convergence. Note that for   for some   we have quadratic convergence of   to 0 as  . This then gives quadratic convergence of   to   and of   to  , where  , by the following theorem. If   then

 
 

with the following definitions

 
 
 

If we start the Newton method from some   with   then we have to start by using a damped Newton method defined by

 

For this it can be shown that   with   as defined previously. Note that   is an increasing function for   so that   for any  , so the value of   is guaranteed to decrease by a certain amount in each iteration, which also proves that   is in the domain of  .

References

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  1. ^ Nemirovsky and Ben-Tal (2023). "Optimization III: Convex Optimization" (PDF).
  2. ^ a b c d e f g h i j Arkadi Nemirovsky (2004). "Interior point polynomial time methods in convex programming".
  3. ^ Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
  4. ^ Nemirovsky and Ben-Tal (2023). "Optimization III: Convex Optimization" (PDF).
  5. ^ Nesterov, Yurii; Nemirovskii, Arkadii (January 1994). Interior-Point Polynomial Algorithms in Convex Programming (Bibliography Comments). Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611970791.bm. ISBN 978-0-89871-319-0.
  6. ^ a b Nesterov, Yurii; Nemirovskii, Arkadii (1994). Interior-Point Polynomial Algorithms in Convex Programming. Studies in Applied and Numerical Mathematics. Vol. 13. doi:10.1137/1.9781611970791. ISBN 978-0-89871-319-0. OCLC 29310677.[page needed]
  7. ^ Yu. E. NESTEROV, Polynomial time methods in linear and quadratic programming, Izvestija AN SSSR, Tekhnitcheskaya kibernetika, No. 3, 1988, pp. 324-326. (In Russian.)
  8. ^ Yu. E. NESTEROV, Polynomial time iterative methods in linear and quadratic programming, Voprosy kibernetiki, Moscow,1988, pp. 102-125. (In Russian.)
  9. ^ Y.E. Nesterov and A.S. Nemirovski, Self–concordant functions and polynomial–time methods in convex programming, Technical report, Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, USSR, 1989.
  10. ^ Nesterov, I︠U︡. E. (December 2013). Introductory lectures on convex optimization : a basic course. Boston. ISBN 978-1-4419-8853-9. OCLC 883391994.{{cite book}}: CS1 maint: location missing publisher (link)
  11. ^ Sun, Tianxiao; Tran-Dinh, Quoc (2018). "Generalized Self-Concordant Functions: A Recipe for Newton-Type Methods". Mathematical Programming: Proposition 6. arXiv:1703.04599.