Polyhedral Functions

LP and QP > Polyhedra | Standard forms | Polyhedral Functions | Min-Cardinality | Applications

Polyhedral functions: definition and examples
Minimization via LP
CVX implementation

Definition and examples

Definition

We say that a function $f : mathbf{R}^n rightarrow mathbf{R}$ is polyhedral if its epigraph is a polyhedron.

That is, a function $f : mathbf{R}^n rightarrow mathbf{R}$ is polyhedral if and only if the epigraph

$mbox{bf epi} f := left{ (x,t) in mathbf{R}^{n+1} ~:~ t ge f(x) right}$

can be expressed as a polyhedron: there exist a matrix $C in mathbf{R}^{m times (n+1)}$ and a vector $d in mathbf{R}^m$ such that

$mbox{bf epi} f = left{ (x,t) in mathbf{R}^{n+1} ~:~ C left( begin{array}{c} x t end{array} right) le d right} .$

Maxima of affine functions

Polyhedral functions include in particular, functions that can be expressed as a maximum of a finite number of affine functions:

$f(x) = max_{1 le i le m} : (a_i^Tx+b_i),$

where $a_i in mathbf{R}^n$ , $b_i in mathbf{R}$ , i=1,ldots,m . Indeed, the epigraph of :

$mbox{bf epi} f = left{ (x,t) in mathbf{R}^{n+1} ~:~ t ge max_{1 le i le m} : (a_i^Tx+b_i) right}$

can be expressed as the polyhedron

$mbox{bf epi} f = left{ (x,t) in mathbf{R}^{n+1} ~:~ t ge a_i^Tx+b_i, ;; i=1,ldots,m right} .$

Example: The l_infty -norm function, with values f(x) = |x|_infty , is polyhedral, as it can be written as the maximum of affine functions:

$f(x) = max_{1 le i le n} : max : (x_i,-x_i).$

Sums of maxima of affine functions

Polyhedral functions include more general functions. For example, a function that can be expressed as a sum of functions themselves expressed as maxima of affine functions:

$f(x) = sum_{j=1}^p max_{1 le i le m} (a_{ij}^Tx+b_{ij})$

for appropriate vectors $a_{ij} in mathbf{R}^n$ and scalars $b_{ij}$ , is polyhedral.

Indeed, the condition (x,t) in mbox{bf epi}f is equivalent to the existence of a vector $u in mathbf{R}^p$ such that

$t ge sum_{j=1}^p u_j ~:~ begin{array}[t]{l} u_j ge a_{ij}^Tx+b_{ij}, ;; i=1,ldots,m, ;; j=1,ldots,p, Cx le d . end{array}$

Hence, mbox{bf epi}f is the projection (on the space of (x,t) -variables) of a polyhedron, which is itself a polyhedron. Note however that representing this polyhedron in terms of a set of affine inequalities involving (x,t) only, is complicated.

Example: The l_1 -norm function, with values f(x) = |x|_1 , is polyhedral, as it can be written as the sum of maxima of affine functions:

$f(x) = sum_{i=1}^n : max(x_i,-x_i).$

Minimization of Polyhedral Functions

Using LPs, we can minimize polyhedral functions, under polyhedral constraints.

Indeed, consider the problem

with polyhedral. We can solve the problem as the LP

$min_{x,t} : t ~:~(x,t) in mbox{bf epi} f, ;; Ax le b.$

Minimization of maxima of affine functions

For example, assume that is defined as the maximum of linear functions as above. The problem

$min_x : max_{1 le i le m} : (a_i^Tx+b_i) ~:~ Cx le d,$

can be expressed as the LP (in variables $x in mathbf{R}^n$ and t in mathbf{R})

$min_{x,t} : t ~:~ Cx le d, ;; t ge a_i^Tx+b_i, ;; i=1,ldots,m.$

The above problem is indeed an LP:

The objective of the problem is linear in the variables ;
the constraints are ordinary inequalities involving affine functions.

Minimization of a sum of maxima of affine functions

We can formulate the problem of minimizing the function with values

$f(x) = sum_{j=1}^p max_{1 le i le m} (a_{ij}^Tx+b_{ij})$

under polytopic constraints, as an LP.

We use the same trick as before, introducing a new variable for each max-linear function that appears in the function . We obtain the LP representation

$min_{x,t} : sum_{j=1}^p t_j ~:~ begin{array}[t]{l} t_j ge a_{ij}^Tx+b_{ij}, ;; i=1,ldots,m, ;; j=1,ldots,p, Cx le d . end{array}$

Examples:

- and -norm regression problems.

CVX implementation

In CVX, there is no need to introduce new variables in order to transform a polyhedral function minimization. We can directly minimize the function expressed as a maximum of affine functions.

The snippet below implements the problem

$min_x : max_{1 le i le m} : c_i^Tx ~:~ Ax le b,$

where c_1,ldots,c_m are given vectors in $mathbf{R}^n$ . We assume that C = [c_1,ldots,c_m] exists in the workspace.

CVX implementation
cvx_begin
   variable x(n,1)
   minimize( max(C'*x) )
   subject to
        A*x <= b;
cvx_end

For minimizing l_1 - or l_infty -norms, we can simply rely on the corresponding matlab functions, as they are overloaded in CVX. The following snippet solves

where $A in mathbf{R}^{m times n}$ , $b in mathbf{R}^m$ exist in the workspace.

CVX implementation
cvx_begin
   variable x(n,1)
   minimize( norm(A*x-b,inf) + norm(x,1) )
cvx_end