Smooth Formulations of Nonsmooth Functions - MATLAB & Simulink (original) (raw)

To smooth an otherwise nonsmooth problem, you can sometimes add auxiliary variables. For example,

f(x) = max(g(x),h(x))

can be a nonsmooth function even when g(x) and_h_(x) are smooth, as illustrated by the following functions.

f(x) is nonsmooth at the points x = π/4 and x = 5_π_/4.

Code for Creating the Figure

x = linspace(0,2pi); g = @sin; h = @cos; f = @(x)max(g(x),h(x)); plot(x,g(x),'b.-',x,h(x),'r.-') hold on plot(x,f(x),'k--',"LineWidth",2) t = [pi,5pi]/4; plot(t,sin(t),'mo',"MarkerSize",15) legend("sin","cos","max","nonsmooth","Location","north") hold off

This lack of smoothness can cause problems for Optimization Toolbox™ solvers, all of which assume that objective functions and nonlinear constraint functions are continuously differentiable. So, if you try to solve

x = min_t_(f(t)) starting from the point x0 = 1,

you do not get an exit flag of 1, because the solution is not differentiable at the locally minimizing point x = π/4.

fun1 = @sin; fun2 = @cos; fun = @(x)max(fun1(x),fun2(x)); [x1,fval1,eflag1] = fminunc(fun,1)

Local minimum possible.

fminunc stopped because it cannot decrease the objective function along the current search direction.

x1 =

0.7854

fval1 =

0.7071

eflag1 =

 5

Sometimes, you can use an auxiliary variable to turn a nonsmooth problem into a smooth problem. For the previous example, consider the auxiliary variable y with the smooth constraints

Consider the optimization problem, subject to these constraints,

The resulting solution x, y is the solution to the original problem

This formulation uses the problem-based approach.

myvar = optimvar("myvar"); auxvar = optimvar("auxvar"); smprob = optimproblem("Objective",auxvar); smprob.Constraints.cons1 = auxvar >= sin(myvar); smprob.Constraints.cons2 = auxvar >= cos(myvar); x0.myvar = 1; x0.auxvar = 1; [sol2,fval2,eflag2] = solve(smprob,x0)

Solving problem using fmincon.

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.

sol2 =

struct with fields:

auxvar: 0.7071
 myvar: 0.7854

fval2 =

0.7071

eflag2 =

OptimalSolution

This same concept underlies the formulation of the fminimax function; see Goal Attainment Method.

See Also

fgoalattain | fminimax

Topics

• Goal Attainment Method