/// <summary>
/// Note that we implement a hidden feature not in the standard nlopt_minimize_constrained interface: whenever the
/// constraint function returns NaN, that constraint becomes inactive.
/// Originally described at
/// </summary>
/// <param name="n"></param>
/// <param name="f"></param>
/// <param name="f_data"></param>
/// <param name="m"></param>
/// <param name="fc"></param>
/// <param name="lb">Lower bounds. Will not be modified.</param>
/// <param name="ub">Upper bounds. Will not be modified.</param>
/// <param name="x">On input: initial guess. On output: minimizer</param>
/// <param name="minf">
/// The minimum value of the objective function: <paramref name="minf"/> = <paramref name="f"/>(<paramref name="x"/>).
/// </param>
/// <param name="stop"></param>
/// <param name="dual_opt"></param>
internal static nlopt_result mma_minimize(int n, nlopt_func f, object[] f_data, int m, nlopt_constraint[] fc,
double[] lb, double[] ub, double[] x, ref double minf, nlopt_stopping stop, nlopt_opt dual_opt)
{
// Translated from https://github.com/stevengj/nlopt/blob/master/src/algs/mma/mma.c#L145
nlopt_result ret = nlopt_result.NLOPT_SUCCESS;
double rho, fcur;
double[] xcur, sigma, dfdx, dfdx_cur, xprev, xprevprev;
double[] dfcdx, dfcdx_cur;
double[] fcval, fcval_cur, rhoc, gcval, y, dual_lb, dual_ub;
int i, ifc, j, k = 0;
dual_data dd;
bool feasible;
double infeasibility;
int mfc;
m = NLoptStopping.nlopt_count_constraints(mfc = m, fc);
if (NLoptApi.nlopt_get_dimension(dual_opt) != m)
{
NLoptStopping.nlopt_stop_msg(stop,
string.Format("dual optimizer has wrong dimension %d != %d", NLoptApi.nlopt_get_dimension(dual_opt), m));
return(nlopt_result.NLOPT_INVALID_ARGS);
}
try
{
// The original source malloc'd a large array sigma and set the rest as offsets.
sigma = new double[n];
dfdx = new double[n];
dfdx_cur = new double[n];
xcur = new double[n];
xprev = new double[n];
xprevprev = new double[n];
fcval = new double[m];
fcval_cur = new double[m];
rhoc = new double[m];
gcval = new double[m];
dual_lb = new double[m];
dual_ub = new double[m];
y = new double[m];
dfcdx = new double[m * n];
dfcdx_cur = new double[m * n];
}
catch (OutOfMemoryException)
{
return(nlopt_result.NLOPT_OUT_OF_MEMORY);
}
dd = new dual_data();
dd.n = n;
dd.x = x;
dd.lb = lb;
dd.ub = ub;
dd.sigma = sigma;
dd.dfdx = dfdx;
dd.dfcdx = dfcdx;
dd.fcval = fcval;
dd.rhoc = rhoc;
dd.xcur = xcur;
dd.gcval = gcval;
for (j = 0; j < n; ++j)
{
if (NLoptStopping.nlopt_isinf(ub[j]) || NLoptStopping.nlopt_isinf(lb[j]))
{
sigma[j] = 1.0; // arbitrary default
}
else
{
sigma[j] = 0.5 * (ub[j] - lb[j]);
}
}
rho = 1.0;
for (i = 0; i < m; ++i)
{
rhoc[i] = 1.0;
dual_lb[i] = y[i] = 0.0;
dual_ub[i] = double.MaxValue; // Originally dual_ub[i] = HUGE_VAL;
}
dd.fval = fcur = minf = f(n, x, 0, dfdx, 0, f_data, 0);
++stop.nevals_p;
Array.Copy(x, xcur, n);
if (NLoptStopping.nlopt_stop_forced(stop))
{
ret = nlopt_result.NLOPT_FORCED_STOP;
return(ret);
}
feasible = true; infeasibility = 0;
for (i = ifc = 0; ifc < mfc; ++ifc)
{
NLoptStopping.nlopt_eval_constraint(fcval, i, dfcdx, i * n, fc, ifc, n, x, 0);
i += fc[ifc].m;
if (NLoptStopping.nlopt_stop_forced(stop))
{
ret = nlopt_result.NLOPT_FORCED_STOP;
return(ret);
}
}
for (i = 0; i < m; ++i)
{
feasible = feasible && (fcval[i] <= 0 || NLoptStopping.nlopt_isnan(fcval[i]));
if (fcval[i] > infeasibility)
{
infeasibility = fcval[i];
}
}
// For non-feasible initial points, set a finite (large) upper-bound on the dual variables. What this means is that,
// if no feasible solution is found from the dual problem, it will minimize the dual objective with the unfeasible
// constraint weighted by 1e40 -- basically, minimizing the unfeasible constraint until it becomes feasible or until
// we at least obtain a step towards a feasible point. Svanberg suggested a different approach in his 1987 paper,
// basically introducing additional penalty variables for unfeasible constraints, but this is easier to implement
// and at least as efficient.
if (!feasible)
{
for (i = 0; i < m; ++i)
{
dual_ub[i] = 1e40;
}
}
NLoptOptions.nlopt_set_min_objective(dual_opt, dual_func, new object[] { dd }, 0);
NLoptOptions.nlopt_set_lower_bounds(dual_opt, dual_lb);
NLoptOptions.nlopt_set_upper_bounds(dual_opt, dual_ub);
NLoptApi.nlopt_set_stopval(dual_opt, double.MinValue);
NLoptOptions.nlopt_remove_inequality_constraints(dual_opt);
NLoptOptions.nlopt_remove_equality_constraints(dual_opt);
while (true) //outer iterations
{
double fprev = fcur;
if (NLoptStopping.nlopt_stop_forced(stop))
{
ret = nlopt_result.NLOPT_FORCED_STOP;
}
else if (NLoptStopping.nlopt_stop_evals(stop))
{
ret = nlopt_result.NLOPT_MAXEVAL_REACHED;
}
else if (NLoptStopping.nlopt_stop_time(stop))
{
ret = nlopt_result.NLOPT_MAXTIME_REACHED;
}
else if (feasible && (minf < stop.minf_max))
{
ret = nlopt_result.NLOPT_MINF_MAX_REACHED;
}
if (ret != nlopt_result.NLOPT_SUCCESS)
{
return(ret);
}
if (++k > 1)
{
Array.Copy(xprev, xprevprev, n);
}
Array.Copy(xcur, xprev, n);
while (true) //inner iterations
{
double min_dual = double.NaN;
double infeasibility_cur;
bool feasible_cur, inner_done;
int save_verbose;
bool new_infeasible_constraint;
nlopt_result reti;
// Solve dual problem
dd.rho = rho; dd.count = 0;
save_verbose = mma_verbose;
mma_verbose = 0; // no recursive verbosity
reti = NLoptOptimize.nlopt_optimize_limited(dual_opt, y, 0, ref min_dual, 0,
stop.maxtime - (stop.watch.Elapsed.TotalSeconds - stop.start));
mma_verbose = save_verbose;
if (reti < 0 || reti == nlopt_result.NLOPT_MAXTIME_REACHED)
{
ret = reti;
return(ret);
}
dual_func(m, y, 0, null, 0, new object[] { dd }, 0); // evaluate final xcur etc.
if (mma_verbose == 1)
{
Console.WriteLine("MMA dual converged in %d iterations to g=%g:", dd.count, dd.gval);
for (i = 0; i < Math.Min(mma_verbose, m); ++i)
{
Console.WriteLine(" MMA y[%u]=%g, gc[%u]=%g\n", i, y[i], i, dd.gcval[i]);
}
}
fcur = f(n, xcur, 0, dfdx_cur, 0, f_data, 0);
++stop.nevals_p;
if (NLoptStopping.nlopt_stop_forced(stop))
{
ret = nlopt_result.NLOPT_FORCED_STOP;
return(ret);
}
feasible_cur = true; infeasibility_cur = 0;
new_infeasible_constraint = false;
inner_done = dd.gval >= fcur;
for (i = ifc = 0; ifc < mfc; ++ifc)
{
NLoptStopping.nlopt_eval_constraint(fcval_cur, i, dfcdx_cur, i * n, fc, ifc, n, xcur, 0);
i += fc[ifc].m;
if (NLoptStopping.nlopt_stop_forced(stop))
{
ret = nlopt_result.NLOPT_FORCED_STOP;
return(ret);
}
}
for (i = ifc = 0; ifc < mfc; ++ifc)
{
int i0 = i, inext = i + fc[ifc].m;
for (; i < inext; ++i)
{
if (!NLoptStopping.nlopt_isnan(fcval_cur[i]))
{
feasible_cur = feasible_cur && (fcval_cur[i] <= fc[ifc].tol[i - i0]);
if (!NLoptStopping.nlopt_isnan(fcval[i]))
{
inner_done = inner_done && (dd.gcval[i] >= fcval_cur[i]);
}
else if (fcval_cur[i] > 0)
{
new_infeasible_constraint = true;
}
if (fcval_cur[i] > infeasibility_cur)
{
infeasibility_cur = fcval_cur[i];
}
}
}
}
if ((fcur < minf && (inner_done || feasible_cur || !feasible)) ||
(!feasible && infeasibility_cur < infeasibility))
{
if (mma_verbose == 1 && !feasible_cur)
{
Console.WriteLine("MMA - using infeasible point?");
}
dd.fval = minf = fcur;
infeasibility = infeasibility_cur;
Array.Copy(fcval_cur, fcval, m);
Array.Copy(xcur, x, n);
Array.Copy(dfdx_cur, dfdx, n);
Array.Copy(dfcdx_cur, dfcdx, n * m);
// once we have reached a feasible solution, the algorithm should never make the solution infeasible
// again (if inner_done), although the constraints may be violated slightly by rounding errors etc. so we
// must be a little careful about checking feasibility
if (infeasibility_cur == 0)
{
if (!feasible) // reset upper bounds to infin.
{
for (i = 0; i < m; ++i)
{
dual_ub[i] = double.MaxValue;
}
NLoptOptions.nlopt_set_upper_bounds(dual_opt, dual_ub);
}
feasible = true;
}
else if (new_infeasible_constraint)
{
feasible = false;
}
}
if (NLoptStopping.nlopt_stop_forced(stop))
{
ret = nlopt_result.NLOPT_FORCED_STOP;
}
else if (NLoptStopping.nlopt_stop_evals(stop))
{
ret = nlopt_result.NLOPT_MAXEVAL_REACHED;
}
else if (NLoptStopping.nlopt_stop_time(stop))
{
ret = nlopt_result.NLOPT_MAXTIME_REACHED;
}
else if (feasible && minf < stop.minf_max)
{
ret = nlopt_result.NLOPT_MINF_MAX_REACHED;
}
if (ret != nlopt_result.NLOPT_SUCCESS)
{
return(ret);
}
if (inner_done)
{
break;
}
if (fcur > dd.gval)
{
rho = Math.Min(10 * rho, 1.1 * (rho + (fcur - dd.gval) / dd.wval));
}
for (i = 0; i < m; ++i)
{
if (!NLoptStopping.nlopt_isnan(fcval_cur[i]) && fcval_cur[i] > dd.gcval[i])
{
rhoc[i] = Math.Min(10 * rhoc[i],
1.1 * (rhoc[i] + (fcval_cur[i] - dd.gcval[i]) / dd.wval));
}
}
if (mma_verbose == 1)
{
Console.WriteLine("MMA inner iteration: rho -> %g", rho);
}
for (i = 0; i < Math.Min(mma_verbose, m); ++i)
{
Console.WriteLine(" MMA rhoc[%u] -> %g", i, rhoc[i]);
}
}
if (NLoptStopping.nlopt_stop_ftol(stop, fcur, fprev))
{
ret = nlopt_result.NLOPT_FTOL_REACHED;
}
if (NLoptStopping.nlopt_stop_x(stop, xcur, xprev))
{
ret = nlopt_result.NLOPT_XTOL_REACHED;
}
if (ret != nlopt_result.NLOPT_SUCCESS)
{
return(ret);
}
/* update rho and sigma for iteration k+1 */
rho = Math.Max(0.1 * rho, MMA_RHOMIN);
if (mma_verbose == 1)
{
Console.WriteLine("MMA outer iteration: rho -> %g", rho);
}
for (i = 0; i < m; ++i)
{
rhoc[i] = Math.Max(0.1 * rhoc[i], MMA_RHOMIN);
}
for (i = 0; i < Math.Min(mma_verbose, m); ++i)
{
Console.WriteLine(" MMA rhoc[%u] -> %g", i, rhoc[i]);
}
if (k > 1)
{
for (j = 0; j < n; ++j)
{
double dx2 = (xcur[j] - xprev[j]) * (xprev[j] - xprevprev[j]);
double gam = dx2 < 0 ? 0.7 : (dx2 > 0 ? 1.2 : 1);
sigma[j] *= gam;
if (!NLoptStopping.nlopt_isinf(ub[j]) && !NLoptStopping.nlopt_isinf(lb[j]))
{
sigma[j] = Math.Min(sigma[j], 10 * (ub[j] - lb[j]));
sigma[j] = Math.Max(sigma[j], 0.01 * (ub[j] - lb[j]));
}
}
for (j = 0; j < Math.Min(mma_verbose, n); ++j)
{
Console.WriteLine(" MMA sigma[%u] -> %g\n", j, sigma[j]);
}
}
}
}
/// <summary>
/// Function for MMA's dual solution of the approximate problem.
/// Originally described at
/// </summary>
/// <param name="y">Will not be modified.</param>
private static double dual_func(int m, double[] y, int offset_y, double[] grad, int grad_offset,
object[] d, int offset_d)
{
var dd = (dual_data)d[offset_d];
// Unpack dual_data DTO. TODO: copy comments for these
int n = dd.n;
double[] x = dd.x, lb = dd.lb, ub = dd.ub, sigma = dd.sigma, dfdx = dd.dfdx;
double[] dfcdx = dd.dfcdx;
double rho = dd.rho, fval = dd.fval;
double[] rhoc = dd.rhoc, fcval = dd.fcval;
double[] xcur = dd.xcur;
double[] gcval = dd.gcval;
double val;
dd.count++;
val = dd.gval = fval;
dd.wval = 0;
for (int i = 0; i < m; ++i)
{
val += y[offset_y + i] * (gcval[i] = NLoptStopping.nlopt_isnan(fcval[i]) ? 0 : fcval[i]);
}
for (int j = 0; j < n; ++j)
{
double u, v, dx, denominv, c, sigma2, dx2;
// First, compute xcur[j] for y. Because this objective is separable, we can minimize over x analytically,
// and the minimum dx is given by the solution of a quadratic equation: u dx^2 + 2 v sigma^2 dx + u sigma^2 = 0
// where u and v are defined by the sums below. Because of the definitions, it is guaranteed that
// |u/v| <= sigma, and it follows that the only dx solution with |dx| <= sigma is given by:
// (v/u) sigma^2 (-1 + sqrt(1 - (u / v sigma)^2)) = (u/v) / (-1 - sqrt(1 - (u / v sigma)^2))
// (which goes to zero as u -> 0). The latter expression is less susceptible to roundoff error.
if (sigma[j] == 0) // special case for lb[i] == ub[i] dims, dx=0
{
xcur[j] = x[j];
continue;
}
u = dfdx[j];
v = Math.Abs(dfdx[j]) * sigma[j] + 0.5 * rho;
for (int i = 0; i < m; ++i)
{
if (!NLoptStopping.nlopt_isnan(fcval[i]))
{
u += dfcdx[i * n + j] * y[offset_y + i];
v += (Math.Abs(dfcdx[i * n + j]) * sigma[j] + 0.5 * rhoc[i]) * y[offset_y + i];
}
}
u *= (sigma2 = sigma[j] * sigma[j]);
dx = (u / v) / (-1 - Math.Sqrt(Math.Abs(1 - Math.Pow(u / (v * sigma[j]), 2))));
xcur[j] = x[j] + dx;
if (xcur[j] > ub[j])
{
xcur[j] = ub[j];
}
else if (xcur[j] < lb[j])
{
xcur[j] = lb[j];
}
if (xcur[j] > x[j] + 0.9 * sigma[j])
{
xcur[j] = x[j] + 0.9 * sigma[j];
}
else if (xcur[j] < x[j] - 0.9 * sigma[j])
{
xcur[j] = x[j] - 0.9 * sigma[j];
}
dx = xcur[j] - x[j];
// Function value:
dx2 = dx * dx;
denominv = 1.0 / (sigma2 - dx2);
val += (u * dx + v * dx2) * denominv;
// Update gval, wval, gcval (approximant functions)
c = sigma2 * dx;
dd.gval += (dfdx[j] * c + (Math.Abs(dfdx[j]) * sigma[j] + 0.5 * rho) * dx2) * denominv;
dd.wval += 0.5 * dx2 * denominv;
for (int i = 0; i < m; ++i)
{
if (!NLoptStopping.nlopt_isnan(fcval[i]))
{
gcval[i] += (dfcdx[i * n + j] * c + (Math.Abs(dfcdx[i * n + j]) * sigma[j]
+ 0.5 * rhoc[i]) * dx2) * denominv;
}
}
}
// Gradient is easy to compute: since we are at a minimum x (dval/dx=0), we only need the partial derivative
// with respect to y, and we negate because we are maximizing:
if (grad != null)
{
for (int i = 0; i < m; ++i)
{
grad[grad_offset + i] = -gcval[i];
}
}
return(-val);
}