plq_proj - Piecewise linear quadratic (plq), projection of a (not necessarily convex) function on the set of plq (convex) functions

Calling Sequence

[plq,d] = plq_proj(f,x,p)

Parameters

• plq : matrix. A plq function.
• d : float. distance between the function f and its projection plq functions d=||f-plq||_p.
• f : vector of size n. The values of the function f on the grid x: f=f(x(i)).
• x : vector of size n. The grid on which the function f is evaluated: f=f(x(i)).
• p : integer. The norm for which the projection is computed. p=1, 2, or %inf for the norm ||.||_p.

Description

Computes the projection of the function f on the cone of plq convex functions. Return the result as a plq function.

The computation assumes the function f is sampled on the uniform grid x. The values of the projection are computed by solving the optimization problem: min ||f-u||_p such that u is convex For the L_2 norm, a quadratic program with linear constraints is solved.

Examples

n=300;
x=linspace(-2,2,n)';
f=abs(abs(x)-1);

[p,d]=plq_proj(f,x,2);
u=plq_eval(p,x);

scf(0);clf();
plot2d(x,f,style=1);
plot2d(x,u,style=2);
legends(["f","plq"],[1,2],opt="lr");
xtitle(sprintf("Projection of f onto the cone of convex plq functions. ||f-plq||_p=%1.2f",d));
  

See Also

plq_function,  

Author

Yves Lucet, University of British Columbia, BC, Canada