Continuation

Continuation, also known as homotopy, is a technique that solves a sequence of problems that are continuously homotopic that is one problem can be continuously deformed into the other. Optimal control problems can be quite hard to solve and efficient technique to ease the convergence is to start from a simpler problem and then continously deformed it into others that are solved using the solution of the previous problem as initial guess until the final desired problem is solved.

In PINS it is possible to create a continuation sequence for almost every parameter defined in the problem and for the parameters of the penalties/barrier functions used to enforce the inequality constraints. Continuation sequence for the parameters of the penalties/barrier functions is particularly useful to achieve the convergence on complex or highly non linear problems.

It is also possible to create continuation sequence on multiple parameters simultaneously or as blocks of continuation sequences.

Continuation sequence for parameters

  # definition of a continuation sequence on two parameters simultaneously
  CONTINUATION = [ [ M      = p_epsi0*(1-s)+p_epsi1*s,
                     mu__f  = mu__f0*(1-s)+mu__f0*s
                   ]
                ];

Continuation sequence for barrier/penalty parameters

  # definition of a continuation sequence on two penalty/barrier functions
  CONTINUATION = [ [p_epsi = p_epsi0*(1-s)+p_epsi1*s,
                    p_tol  = p_tol0*(1-s)+p_tol1*s,
                    [["barrier1", "penalty2"], "epsilon"]   = p_epsi,
                    [["barrier1", "penalty2"], "tolerance"] = p_tol
                   ]
                ];