Stabilized Optimization Via an NCL Algorithm

Abstract

For optimization problems involving many nonlinear inequality constraints, we extend the bound-constrained (BCL) and linearly constrained (LCL) augmented Lagrangian approaches of LANCELOT and MINOS to an algorithm that solves a sequence of nonlinearly constrained augmented Lagrangian subproblems whose nonlinear constraints satisfy the LICQ everywhere. The NCL algorithm is implemented in AMPL and tested on large instances of a tax policy model that could not be solved directly by the state-of-the-art solvers that we tested, because of singularity in the Jacobian of the active constraints. Algorithm NCL with IPOPT as subproblem solver proves to be effective, with IPOPT using second derivatives and successfully warm starting each subproblem.

Appendix A ampl Models, Data, and Scripts

Algorithm NCL has been implemented in the AMPL modeling language [6] and tested on problem TAX. The following sections list each relevant file. The files are available from [17].

1.1 A.1 Tax Model

File pTax5Dncl.mod codes subproblem ${\text {NC}}_k$ for problem TAX with five parameters w, $\eta$, $\alpha$, $\psi$, $\gamma$, using $\mu := 1/\eta$. Note that for U(c, y) in the objective and constraint functions, the first term ${(c - \alpha )^{1 - 1/\gamma }} / (1 - 1/\gamma )$ is replaced by a piecewise-smooth function that is defined for all values of c and $\alpha$ (see [13]).

Primal regularization ${\textstyle \frac{1}{2}}\delta \Vert (c,y)\Vert ^2$ with $\delta = 10^{-8}$ is added to the objective function to promote uniqueness of the minimizer. The vector r is called R to avoid a clash with subscript r.

1.2 A.2 Tax Model Data

File pTax5Dncl.dat provides data for a specific problem.

1.3 A.3 Initial Values

File pTax5Dinitial.run solves a simplified model to compute starting values for Algorithm NCL. The nonlinear inequality constraints are removed, and $y=c$ is enforced. This model solves easily with MINOS or SNOPT on all cases tried. Solution values are output to file p5Dinitial.dat.

1.4 A.4 NCL Implementation

File pTax5Dnclipopt.run uses files

to implement Algorithm NCL. Subproblems ${\text {NC}}_k$ are solved in a loop until $\Vert r^*_k\Vert _{\infty } \le \texttt {rtol = 1e-6}$, or $\eta _k$ has been reduced to parameter etamin = 1e-8, or $\rho _k$ has been increased to parameter rhomax = 1e+8. The loop variable k is called K to avoid a clash with subscript k in the model file. The definitions of etak and rhok inside the loop are simpler than (but similar to) the settings of $\eta _k$ and $\rho _k$ in Algorithm 2.

Optimality tolerance $\omega _k = \omega _* = 10^{-6}$ is used throughout to ensure that the solution of the final subproblem ${\text {NC}}_k$ will be close to a solution of the original problem if $\Vert r^*_k\Vert _{\infty }$ is small enough for the final k ($\Vert r^*_k\Vert _{\infty } \le \texttt {rtol = 1e-6}$).

IPOPT is used to solve each subproblem ${\text {NC}}_k$, with runtime options set to implement increasingly warm starts.