git-svn-id: file:///home/svn/opt_utilities@129 ed2142bd-67ad-457f-ba7c-d818d4011675

1 files changed, 1276 insertions, 0 deletions

diff --git a/methods/lbfgs/lbfgs.cpp b/methods/lbfgs/lbfgs.cpp
new file mode 100644
index 0000000..d7b5494
--- /dev/null
+++ b/methods/lbfgs/lbfgs.cpp
@@ -0,0 +1,1276 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include "arithmetic_ansi.h"
+
+
+//#define min2(a, b)      ((a) <= (b) ? (a) : (b))
+template<typename T>
+static T min2(const T& a,const T& b)
+{
+  return (a) <= (b) ? (a) : (b);
+}
+//#define max2(a, b)      ((a) >= (b) ? (a) : (b))
+template<typename T>
+static T max2(const T& a,const T& b)
+{
+  return (a) >= (b) ? (a) : (b);
+}
+//#define max3(a, b, c)   max2(max2((a), (b)), (c));
+template<typename T>
+static T max3(const T& a,const T& b,const T& c)
+{
+  return max2(max2((a), (b)), (c));
+}
+
+
+struct tag_callback_data {
+  int n;
+  void *instance;
+  lbfgs_evaluate_t proc_evaluate;
+  lbfgs_progress_t proc_progress;
+};
+typedef struct tag_callback_data callback_data_t;
+
+struct tag_iteration_data {
+  lbfgsfloatval_t alpha;
+  lbfgsfloatval_t *s;     /* [n] */
+  lbfgsfloatval_t *y;     /* [n] */
+  lbfgsfloatval_t ys;     /* vecdot(y, s) */
+};
+typedef struct tag_iteration_data iteration_data_t;
+
+static const lbfgs_parameter_t _defparam = {
+  6, 1e-5, 0, 1e-5,
+  0, LBFGS_LINESEARCH_DEFAULT, 40,
+  1e-20, 1e20, 1e-4, 0.9, 0.9, 1.0e-16,
+  0.0, 0, -1,
+};
+
+/* Forward function declarations. */
+
+typedef int (*line_search_proc)(
+				int n,
+				lbfgsfloatval_t *x,
+				lbfgsfloatval_t *f,
+				lbfgsfloatval_t *g,
+				lbfgsfloatval_t *s,
+				lbfgsfloatval_t *stp,
+				const lbfgsfloatval_t* xp,
+				const lbfgsfloatval_t* gp,
+				lbfgsfloatval_t *wa,
+				callback_data_t *cd,
+				const lbfgs_parameter_t *param
+				);
+
+static int line_search_backtracking(
+				    int n,
+				    lbfgsfloatval_t *x,
+				    lbfgsfloatval_t *f,
+				    lbfgsfloatval_t *g,
+				    lbfgsfloatval_t *s,
+				    lbfgsfloatval_t *stp,
+				    const lbfgsfloatval_t* xp,
+				    const lbfgsfloatval_t* gp,
+				    lbfgsfloatval_t *wa,
+				    callback_data_t *cd,
+				    const lbfgs_parameter_t *param
+				    );
+
+static int line_search_backtracking_owlqn(
+					  int n,
+					  lbfgsfloatval_t *x,
+					  lbfgsfloatval_t *f,
+					  lbfgsfloatval_t *g,
+					  lbfgsfloatval_t *s,
+					  lbfgsfloatval_t *stp,
+					  const lbfgsfloatval_t* xp,
+					  const lbfgsfloatval_t* gp,
+					  lbfgsfloatval_t *wp,
+					  callback_data_t *cd,
+					  const lbfgs_parameter_t *param
+					  );
+
+static int line_search_morethuente(
+				   int n,
+				   lbfgsfloatval_t *x,
+				   lbfgsfloatval_t *f,
+				   lbfgsfloatval_t *g,
+				   lbfgsfloatval_t *s,
+				   lbfgsfloatval_t *stp,
+				   const lbfgsfloatval_t* xp,
+				   const lbfgsfloatval_t* gp,
+				   lbfgsfloatval_t *wa,
+				   callback_data_t *cd,
+				   const lbfgs_parameter_t *param
+				   );
+
+static int update_trial_interval(
+				 lbfgsfloatval_t *x,
+				 lbfgsfloatval_t *fx,
+				 lbfgsfloatval_t *dx,
+				 lbfgsfloatval_t *y,
+				 lbfgsfloatval_t *fy,
+				 lbfgsfloatval_t *dy,
+				 lbfgsfloatval_t *t,
+				 lbfgsfloatval_t *ft,
+				 lbfgsfloatval_t *dt,
+				 const lbfgsfloatval_t tmin,
+				 const lbfgsfloatval_t tmax,
+				 int *brackt
+				 );
+
+static lbfgsfloatval_t owlqn_x1norm(
+				    const lbfgsfloatval_t* x,
+				    const int start,
+				    const int n
+				    );
+
+static void owlqn_pseudo_gradient(
+				  lbfgsfloatval_t* pg,
+				  const lbfgsfloatval_t* x,
+				  const lbfgsfloatval_t* g,
+				  const int n,
+				  const lbfgsfloatval_t c,
+				  const int start,
+				  const int end
+				  );
+
+static void owlqn_project(
+			  lbfgsfloatval_t* d,
+			  const lbfgsfloatval_t* sign,
+			  const int start,
+			  const int end
+			  );
+
+
+static lbfgsfloatval_t* lbfgs_malloc(int n)
+{
+  return (lbfgsfloatval_t*)vecalloc(sizeof(lbfgsfloatval_t) * n);
+}
+
+static void lbfgs_free(lbfgsfloatval_t *x)
+{
+  vecfree(x);
+}
+
+static void lbfgs_parameter_init(lbfgs_parameter_t *param)
+{
+  memcpy(param, &_defparam, sizeof(*param));
+}
+
+static int lbfgs(
+	  int n,
+	  lbfgsfloatval_t *x,
+	  lbfgsfloatval_t *ptr_fx,
+	  lbfgs_evaluate_t proc_evaluate,
+	  lbfgs_progress_t proc_progress,
+	  void *instance,
+	  lbfgs_parameter_t *_param
+	  )
+{
+  int ret;
+  int i, j, k, ls, end, bound;
+  lbfgsfloatval_t step;
+  
+  /* Constant parameters and their default values. */
+  lbfgs_parameter_t param = (_param != NULL) ? (*_param) : _defparam;
+  const int m = param.m;
+  
+  lbfgsfloatval_t *xp = NULL;
+  lbfgsfloatval_t *g = NULL, *gp = NULL, *pg = NULL;
+  lbfgsfloatval_t *d = NULL, *w = NULL, *pf = NULL;
+  iteration_data_t *lm = NULL, *it = NULL;
+  lbfgsfloatval_t ys, yy;
+  lbfgsfloatval_t xnorm, gnorm, beta;
+  lbfgsfloatval_t fx = 0.;
+  lbfgsfloatval_t rate = 0.;
+  line_search_proc linesearch = line_search_morethuente;
+  
+  /* Construct a callback data. */
+  callback_data_t cd;
+  cd.n = n;
+  cd.instance = instance;
+  cd.proc_evaluate = proc_evaluate;
+  cd.proc_progress = proc_progress;
+  
+  /* Check the input parameters for errors. */
+  if (n <= 0) {
+    return LBFGSERR_INVALID_N;
+  }
+  
+  if (param.epsilon < 0.) {
+    return LBFGSERR_INVALID_EPSILON;
+  }
+  if (param.past < 0) {
+    return LBFGSERR_INVALID_TESTPERIOD;
+  }
+  if (param.delta < 0.) {
+    return LBFGSERR_INVALID_DELTA;
+  }
+  if (param.min_step < 0.) {
+    return LBFGSERR_INVALID_MINSTEP;
+  }
+  if (param.max_step < param.min_step) {
+    return LBFGSERR_INVALID_MAXSTEP;
+  }
+  if (param.ftol < 0.) {
+    return LBFGSERR_INVALID_FTOL;
+  }
+  if (param.linesearch == LBFGS_LINESEARCH_BACKTRACKING_WOLFE ||
+      param.linesearch == LBFGS_LINESEARCH_BACKTRACKING_STRONG_WOLFE) {
+    if (param.wolfe <= param.ftol || 1. <= param.wolfe) {
+      return LBFGSERR_INVALID_WOLFE;
+    }
+  }
+  if (param.gtol < 0.) {
+    return LBFGSERR_INVALID_GTOL;
+  }
+  if (param.xtol < 0.) {
+    return LBFGSERR_INVALID_XTOL;
+  }
+  if (param.max_linesearch <= 0) {
+    return LBFGSERR_INVALID_MAXLINESEARCH;
+  }
+  if (param.orthantwise_c < 0.) {
+    return LBFGSERR_INVALID_ORTHANTWISE;
+  }
+  if (param.orthantwise_start < 0 || n < param.orthantwise_start) {
+    return LBFGSERR_INVALID_ORTHANTWISE_START;
+  }
+  if (param.orthantwise_end < 0) {
+    param.orthantwise_end = n;
+  }
+  if (n < param.orthantwise_end) {
+    return LBFGSERR_INVALID_ORTHANTWISE_END;
+  }
+  if (param.orthantwise_c != 0.) {
+    switch (param.linesearch) {
+    case LBFGS_LINESEARCH_BACKTRACKING:
+      linesearch = line_search_backtracking_owlqn;
+      break;
+    default:
+      /* Only the backtracking method is available. */
+      return LBFGSERR_INVALID_LINESEARCH;
+    }
+  } else {
+    switch (param.linesearch) {
+    case LBFGS_LINESEARCH_MORETHUENTE:
+      linesearch = line_search_morethuente;
+      break;
+    case LBFGS_LINESEARCH_BACKTRACKING_ARMIJO:
+    case LBFGS_LINESEARCH_BACKTRACKING_WOLFE:
+    case LBFGS_LINESEARCH_BACKTRACKING_STRONG_WOLFE:
+      linesearch = line_search_backtracking;
+      break;
+    default:
+      return LBFGSERR_INVALID_LINESEARCH;
+    }
+  }
+  
+  /* Allocate working space. */
+  xp = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+  g = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+  gp = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+  d = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+  w = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+  if (xp == NULL || g == NULL || gp == NULL || d == NULL || w == NULL) {
+    ret = LBFGSERR_OUTOFMEMORY;
+    goto lbfgs_exit;
+  }
+  
+  if (param.orthantwise_c != 0.) {
+    /* Allocate working space for OW-LQN. */
+    pg = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+    if (pg == NULL) {
+      ret = LBFGSERR_OUTOFMEMORY;
+      goto lbfgs_exit;
+    }
+  }
+  
+  /* Allocate limited memory storage. */
+  lm = (iteration_data_t*)vecalloc(m * sizeof(iteration_data_t));
+  if (lm == NULL) {
+    ret = LBFGSERR_OUTOFMEMORY;
+    goto lbfgs_exit;
+  }
+  
+  /* Initialize the limited memory. */
+  for (i = 0;i < m;++i) {
+    it = &lm[i];
+    it->alpha = 0;
+    it->ys = 0;
+    it->s = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+    it->y = (lbfgsfloatval_t*)vecalloc(n * sizeof(lbfgsfloatval_t));
+    if (it->s == NULL || it->y == NULL) {
+      ret = LBFGSERR_OUTOFMEMORY;
+      goto lbfgs_exit;
+    }
+  }
+  
+  /* Allocate an array for storing previous values of the objective function. */
+  if (0 < param.past) {
+    pf = (lbfgsfloatval_t*)vecalloc(param.past * sizeof(lbfgsfloatval_t));
+  }
+  
+  /* Evaluate the function value and its gradient. */
+  fx = cd.proc_evaluate(cd.instance, x, g, cd.n, 0);
+  if (0. != param.orthantwise_c) {
+    /* Compute the L1 norm of the variable and add it to the object value. */
+    xnorm = owlqn_x1norm(x, param.orthantwise_start, param.orthantwise_end);
+    fx += xnorm * param.orthantwise_c;
+    owlqn_pseudo_gradient(
+			  pg, x, g, n,
+			  param.orthantwise_c, param.orthantwise_start, param.orthantwise_end
+			  );
+  }
+  
+  /* Store the initial value of the objective function. */
+  if (pf != NULL) {
+    pf[0] = fx;
+  }
+  
+  /*
+    Compute the direction;
+    we assume the initial hessian matrix H_0 as the identity matrix.
+  */
+  if (param.orthantwise_c == 0.) {
+    vecncpy(d, g, n);
+  } else {
+    vecncpy(d, pg, n);
+  }
+  
+  /*
+    Make sure that the initial variables are not a minimizer.
+  */
+  vec2norm(&xnorm, x, n);
+  if (param.orthantwise_c == 0.) {
+    vec2norm(&gnorm, g, n);
+  } else {
+    vec2norm(&gnorm, pg, n);
+  }
+  if (xnorm < 1.0) xnorm = 1.0;
+  if (gnorm / xnorm <= param.epsilon) {
+    ret = LBFGS_ALREADY_MINIMIZED;
+    goto lbfgs_exit;
+  }
+  
+  /* Compute the initial step:
+     step = 1.0 / sqrt(vecdot(d, d, n))
+  */
+  vec2norminv(&step, d, n);
+  
+  k = 1;
+  end = 0;
+  for (;;) {
+    /* Store the current position and gradient vectors. */
+    veccpy(xp, x, n);
+    veccpy(gp, g, n);
+    
+    /* Search for an optimal step. */
+    if (param.orthantwise_c == 0.) {
+      ls = linesearch(n, x, &fx, g, d, &step, xp, gp, w, &cd, &param);
+    } else {
+      ls = linesearch(n, x, &fx, g, d, &step, xp, pg, w, &cd, &param);
+      owlqn_pseudo_gradient(
+			    pg, x, g, n,
+			    param.orthantwise_c, param.orthantwise_start, param.orthantwise_end
+			    );
+    }
+    if (ls < 0) {
+      /* Revert to the previous point. */
+      veccpy(x, xp, n);
+      veccpy(g, gp, n);
+      ret = ls;
+      goto lbfgs_exit;
+    }
+    
+    /* Compute x and g norms. */
+    vec2norm(&xnorm, x, n);
+    if (param.orthantwise_c == 0.) {
+      vec2norm(&gnorm, g, n);
+    } else {
+      vec2norm(&gnorm, pg, n);
+    }
+    
+    /* Report the progress. */
+    if (cd.proc_progress) {
+      if (ret = cd.proc_progress(cd.instance, x, g, fx, xnorm, gnorm, step, cd.n, k, ls)) {
+	goto lbfgs_exit;
+      }
+    }
+    
+    /*
+      Convergence test.
+      The criterion is given by the following formula:
+      |g(x)| / \max(1, |x|) < \epsilon
+    */
+    if (xnorm < 1.0) xnorm = 1.0;
+    if (gnorm / xnorm <= param.epsilon) {
+      /* Convergence. */
+      ret = LBFGS_SUCCESS;
+      break;
+    }
+    
+    /*
+      Test for stopping criterion.
+      The criterion is given by the following formula:
+      (f(past_x) - f(x)) / f(x) < \delta
+    */
+    if (pf != NULL) {
+      /* We don't test the stopping criterion while k < past. */
+      if (param.past <= k) {
+	/* Compute the relative improvement from the past. */
+	rate = (pf[k % param.past] - fx) / fx;
+	
+	/* The stopping criterion. */
+	if (rate < param.delta) {
+	  ret = LBFGS_STOP;
+	  break;
+	}
+      }
+      
+      /* Store the current value of the objective function. */
+      pf[k % param.past] = fx;
+    }
+    
+    if (param.max_iterations != 0 && param.max_iterations < k+1) {
+      /* Maximum number of iterations. */
+      ret = LBFGSERR_MAXIMUMITERATION;
+      break;
+    }
+    
+    /*
+      Update vectors s and y:
+      s_{k+1} = x_{k+1} - x_{k} = \step * d_{k}.
+      y_{k+1} = g_{k+1} - g_{k}.
+    */
+    it = &lm[end];
+    vecdiff(it->s, x, xp, n);
+    vecdiff(it->y, g, gp, n);
+    
+    /*
+      Compute scalars ys and yy:
+      ys = y^t \cdot s = 1 / \rho.
+      yy = y^t \cdot y.
+      Notice that yy is used for scaling the hessian matrix H_0 (Cholesky factor).
+    */
+    vecdot(&ys, it->y, it->s, n);
+    vecdot(&yy, it->y, it->y, n);
+    it->ys = ys;
+    
+    /*
+      Recursive formula to compute dir = -(H \cdot g).
+      This is described in page 779 of:
+      Jorge Nocedal.
+      Updating Quasi-Newton Matrices with Limited Storage.
+      Mathematics of Computation, Vol. 35, No. 151,
+      pp. 773--782, 1980.
+    */
+    bound = (m <= k) ? m : k;
+    ++k;
+    end = (end + 1) % m;
+    
+    /* Compute the steepest direction. */
+    if (param.orthantwise_c == 0.) {
+      /* Compute the negative of gradients. */
+      vecncpy(d, g, n);
+    } else {
+      vecncpy(d, pg, n);
+    }
+    
+    j = end;
+    for (i = 0;i < bound;++i) {
+      j = (j + m - 1) % m;    /* if (--j == -1) j = m-1; */
+      it = &lm[j];
+      /* \alpha_{j} = \rho_{j} s^{t}_{j} \cdot q_{k+1}. */
+      vecdot(&it->alpha, it->s, d, n);
+      it->alpha /= it->ys;
+      /* q_{i} = q_{i+1} - \alpha_{i} y_{i}. */
+      vecadd(d, it->y, -it->alpha, n);
+    }
+    
+    vecscale(d, ys / yy, n);
+    
+    for (i = 0;i < bound;++i) {
+      it = &lm[j];
+      /* \beta_{j} = \rho_{j} y^t_{j} \cdot \gamma_{i}. */
+      vecdot(&beta, it->y, d, n);
+      beta /= it->ys;
+      /* \gamma_{i+1} = \gamma_{i} + (\alpha_{j} - \beta_{j}) s_{j}. */
+      vecadd(d, it->s, it->alpha - beta, n);
+      j = (j + 1) % m;        /* if (++j == m) j = 0; */
+    }
+    
+    /*
+      Constrain the search direction for orthant-wise updates.
+    */
+    if (param.orthantwise_c != 0.) {
+      for (i = param.orthantwise_start;i < param.orthantwise_end;++i) {
+	if (d[i] * pg[i] >= 0) {
+	  d[i] = 0;
+	}
+      }
+    }
+    
+    /*
+      Now the search direction d is ready. We try step = 1 first.
+    */
+    step = 1.0;
+  }
+  
+ lbfgs_exit:
+  /* Return the final value of the objective function. */
+  if (ptr_fx != NULL) {
+    *ptr_fx = fx;
+  }
+  
+  vecfree(pf);
+  
+  /* Free memory blocks used by this function. */
+  if (lm != NULL) {
+    for (i = 0;i < m;++i) {
+      vecfree(lm[i].s);
+      vecfree(lm[i].y);
+    }
+    vecfree(lm);
+  }
+  vecfree(pg);
+  vecfree(w);
+  vecfree(d);
+  vecfree(gp);
+  vecfree(g);
+  vecfree(xp);
+  
+  return ret;
+}
+
+
+
+static int line_search_backtracking(
+				    int n,
+				    lbfgsfloatval_t *x,
+				    lbfgsfloatval_t *f,
+				    lbfgsfloatval_t *g,
+				    lbfgsfloatval_t *s,
+				    lbfgsfloatval_t *stp,
+				    const lbfgsfloatval_t* xp,
+				    const lbfgsfloatval_t* gp,
+				    lbfgsfloatval_t *wp,
+				    callback_data_t *cd,
+				    const lbfgs_parameter_t *param
+				    )
+{
+  int ret = 0, count = 0;
+  lbfgsfloatval_t width, dg, norm = 0.;
+  lbfgsfloatval_t finit, dginit = 0., dgtest;
+  const lbfgsfloatval_t dec = 0.5, inc = 2.1;
+  
+  /* Check the input parameters for errors. */
+  if (*stp <= 0.) {
+    return LBFGSERR_INVALIDPARAMETERS;
+  }
+  
+  /* Compute the initial gradient in the search direction. */
+  vecdot(&dginit, g, s, n);
+  
+  /* Make sure that s points to a descent direction. */
+  if (0 < dginit) {
+    return LBFGSERR_INCREASEGRADIENT;
+  }
+  
+  /* The initial value of the objective function. */
+  finit = *f;
+  dgtest = param->ftol * dginit;
+  
+  for (;;) {
+    veccpy(x, xp, n);
+    vecadd(x, s, *stp, n);
+    
+    /* Evaluate the function and gradient values. */
+    *f = cd->proc_evaluate(cd->instance, x, g, cd->n, *stp);
+    
+    ++count;
+    
+    if (*f > finit + *stp * dgtest) {
+      width = dec;
+    } else {
+      /* The sufficient decrease condition (Armijo condition). */
+      if (param->linesearch == LBFGS_LINESEARCH_BACKTRACKING_ARMIJO) {
+	/* Exit with the Armijo condition. */
+	return count;
+      }
+      
+      /* Check the Wolfe condition. */
+      vecdot(&dg, g, s, n);
+      if (dg < param->wolfe * dginit) {
+	width = inc;
+      } else {
+	if(param->linesearch == LBFGS_LINESEARCH_BACKTRACKING_WOLFE) {
+	  /* Exit with the regular Wolfe condition. */
+	  return count;
+	}
+	
+	/* Check the strong Wolfe condition. */
+	if(dg > -param->wolfe * dginit) {
+	  width = dec;
+	} else {
+	  /* Exit with the strong Wolfe condition. */
+	  return count;
+	}
+      }
+    }
+    
+    if (*stp < param->min_step) {
+      /* The step is the minimum value. */
+      return LBFGSERR_MINIMUMSTEP;
+    }
+    if (*stp > param->max_step) {
+      /* The step is the maximum value. */
+      return LBFGSERR_MAXIMUMSTEP;
+    }
+    if (param->max_linesearch <= count) {
+      /* Maximum number of iteration. */
+      return LBFGSERR_MAXIMUMLINESEARCH;
+    }
+    
+    (*stp) *= width;
+  }
+}
+
+
+
+static int line_search_backtracking_owlqn(
+					  int n,
+					  lbfgsfloatval_t *x,
+					  lbfgsfloatval_t *f,
+					  lbfgsfloatval_t *g,
+					  lbfgsfloatval_t *s,
+					  lbfgsfloatval_t *stp,
+					  const lbfgsfloatval_t* xp,
+					  const lbfgsfloatval_t* gp,
+					  lbfgsfloatval_t *wp,
+					  callback_data_t *cd,
+					  const lbfgs_parameter_t *param
+					  )
+{
+  int i, ret = 0, count = 0;
+  lbfgsfloatval_t width = 0.5, norm = 0.;
+  lbfgsfloatval_t finit = *f, dgtest;
+  
+  /* Check the input parameters for errors. */
+  if (*stp <= 0.) {
+    return LBFGSERR_INVALIDPARAMETERS;
+  }
+  
+  /* Choose the orthant for the new point. */
+  for (i = 0;i < n;++i) {
+    wp[i] = (xp[i] == 0.) ? -gp[i] : xp[i];
+  }
+  
+  for (;;) {
+    /* Update the current point. */
+    veccpy(x, xp, n);
+    vecadd(x, s, *stp, n);
+    
+    /* The current point is projected onto the orthant. */
+    owlqn_project(x, wp, param->orthantwise_start, param->orthantwise_end);
+    
+    /* Evaluate the function and gradient values. */
+    *f = cd->proc_evaluate(cd->instance, x, g, cd->n, *stp);
+    
+    /* Compute the L1 norm of the variables and add it to the object value. */
+    norm = owlqn_x1norm(x, param->orthantwise_start, param->orthantwise_end);
+    *f += norm * param->orthantwise_c;
+    
+    ++count;
+    
+    dgtest = 0.;
+    for (i = 0;i < n;++i) {
+      dgtest += (x[i] - xp[i]) * gp[i];
+    }
+    
+    if (*f <= finit + param->ftol * dgtest) {
+      /* The sufficient decrease condition. */
+      return count;
+    }
+    
+    if (*stp < param->min_step) {
+      /* The step is the minimum value. */
+      return LBFGSERR_MINIMUMSTEP;
+    }
+    if (*stp > param->max_step) {
+      /* The step is the maximum value. */
+      return LBFGSERR_MAXIMUMSTEP;
+    }
+    if (param->max_linesearch <= count) {
+      /* Maximum number of iteration. */
+      return LBFGSERR_MAXIMUMLINESEARCH;
+    }
+    
+    (*stp) *= width;
+  }
+}
+
+
+
+static int line_search_morethuente(
+				   int n,
+				   lbfgsfloatval_t *x,
+				   lbfgsfloatval_t *f,
+				   lbfgsfloatval_t *g,
+				   lbfgsfloatval_t *s,
+				   lbfgsfloatval_t *stp,
+				   const lbfgsfloatval_t* xp,
+				   const lbfgsfloatval_t* gp,
+				   lbfgsfloatval_t *wa,
+				   callback_data_t *cd,
+				   const lbfgs_parameter_t *param
+				   )
+{
+  int count = 0;
+  int brackt, stage1, uinfo = 0;
+  lbfgsfloatval_t dg;
+  lbfgsfloatval_t stx, fx, dgx;
+  lbfgsfloatval_t sty, fy, dgy;
+  lbfgsfloatval_t fxm, dgxm, fym, dgym, fm, dgm;
+  lbfgsfloatval_t finit, ftest1, dginit, dgtest;
+  lbfgsfloatval_t width, prev_width;
+  lbfgsfloatval_t stmin, stmax;
+  
+  /* Check the input parameters for errors. */
+  if (*stp <= 0.) {
+    return LBFGSERR_INVALIDPARAMETERS;
+  }
+  
+  /* Compute the initial gradient in the search direction. */
+  vecdot(&dginit, g, s, n);
+  
+  /* Make sure that s points to a descent direction. */
+  if (0 < dginit) {
+    return LBFGSERR_INCREASEGRADIENT;
+  }
+  
+  /* Initialize local variables. */
+  brackt = 0;
+  stage1 = 1;
+  finit = *f;
+  dgtest = param->ftol * dginit;
+  width = param->max_step - param->min_step;
+  prev_width = 2.0 * width;
+  
+  /*
+    The variables stx, fx, dgx contain the values of the step,
+    function, and directional derivative at the best step.
+    The variables sty, fy, dgy contain the value of the step,
+    function, and derivative at the other endpoint of
+    the interval of uncertainty.
+    The variables stp, f, dg contain the values of the step,
+    function, and derivative at the current step.
+  */
+  stx = sty = 0.;
+  fx = fy = finit;
+  dgx = dgy = dginit;
+  
+  for (;;) {
+    /*
+      Set the minimum and maximum steps to correspond to the
+      present interval of uncertainty.
+    */
+    if (brackt) {
+      stmin = min2(stx, sty);
+      stmax = max2(stx, sty);
+    } else {
+      stmin = stx;
+      stmax = *stp + 4.0 * (*stp - stx);
+    }
+    
+    /* Clip the step in the range of [stpmin, stpmax]. */
+    if (*stp < param->min_step) *stp = param->min_step;
+    if (param->max_step < *stp) *stp = param->max_step;
+    
+    /*
+      If an unusual termination is to occur then let
+      stp be the lowest point obtained so far.
+    */
+    if ((brackt && ((*stp <= stmin || stmax <= *stp) || param->max_linesearch <= count + 1 || uinfo != 0)) || (brackt && (stmax - stmin <= param->xtol * stmax))) {
+      *stp = stx;
+    }
+	
+    /*
+      Compute the current value of x:
+      x <- x + (*stp) * s.
+    */
+    veccpy(x, xp, n);
+    vecadd(x, s, *stp, n);
+	
+    /* Evaluate the function and gradient values. */
+    *f = cd->proc_evaluate(cd->instance, x, g, cd->n, *stp);
+    vecdot(&dg, g, s, n);
+	
+    ftest1 = finit + *stp * dgtest;
+    ++count;
+	
+    /* Test for errors and convergence. */
+    if (brackt && ((*stp <= stmin || stmax <= *stp) || uinfo != 0)) {
+      /* Rounding errors prevent further progress. */
+      return LBFGSERR_ROUNDING_ERROR;
+    }
+    if (*stp == param->max_step && *f <= ftest1 && dg <= dgtest) {
+      /* The step is the maximum value. */
+      return LBFGSERR_MAXIMUMSTEP;
+    }
+    if (*stp == param->min_step && (ftest1 < *f || dgtest <= dg)) {
+      /* The step is the minimum value. */
+      return LBFGSERR_MINIMUMSTEP;
+    }
+    if (brackt && (stmax - stmin) <= param->xtol * stmax) {
+      /* Relative width of the interval of uncertainty is at most xtol. */
+      return LBFGSERR_WIDTHTOOSMALL;
+    }
+    if (param->max_linesearch <= count) {
+      /* Maximum number of iteration. */
+      return LBFGSERR_MAXIMUMLINESEARCH;
+    }
+    if (*f <= ftest1 && fabs(dg) <= param->gtol * (-dginit)) {
+      /* The sufficient decrease condition and the directional derivative condition hold. */
+      return count;
+    }
+	
+    /*
+      In the first stage we seek a step for which the modified
+      function has a nonpositive value and nonnegative derivative.
+    */
+    if (stage1 && *f <= ftest1 && min2(param->ftol, param->gtol) * dginit <= dg) {
+      stage1 = 0;
+    }
+
+    /*
+      A modified function is used to predict the step only if
+      we have not obtained a step for which the modified
+      function has a nonpositive function value and nonnegative
+      derivative, and if a lower function value has been
+      obtained but the decrease is not sufficient.
+    */
+    if (stage1 && ftest1 < *f && *f <= fx) {
+      /* Define the modified function and derivative values. */
+      fm = *f - *stp * dgtest;
+      fxm = fx - stx * dgtest;
+      fym = fy - sty * dgtest;
+      dgm = dg - dgtest;
+      dgxm = dgx - dgtest;
+      dgym = dgy - dgtest;
+
+      /*
+	Call update_trial_interval() to update the interval of
+	uncertainty and to compute the new step.
+      */
+      uinfo = update_trial_interval(
+				    &stx, &fxm, &dgxm,
+				    &sty, &fym, &dgym,
+				    stp, &fm, &dgm,
+				    stmin, stmax, &brackt
+				    );
+
+      /* Reset the function and gradient values for f. */
+      fx = fxm + stx * dgtest;
+      fy = fym + sty * dgtest;
+      dgx = dgxm + dgtest;
+      dgy = dgym + dgtest;
+    } else {
+      /*
+	Call update_trial_interval() to update the interval of
+	uncertainty and to compute the new step.
+      */
+      uinfo = update_trial_interval(
+				    &stx, &fx, &dgx,
+				    &sty, &fy, &dgy,
+				    stp, f, &dg,
+				    stmin, stmax, &brackt
+				    );
+    }
+
+    /*
+      Force a sufficient decrease in the interval of uncertainty.
+    */
+    if (brackt) {
+      if (0.66 * prev_width <= fabs(sty - stx)) {
+	*stp = stx + 0.5 * (sty - stx);
+      }
+      prev_width = width;
+      width = fabs(sty - stx);
+    }
+  }
+
+  return LBFGSERR_LOGICERROR;
+}
+
+
+
+/**
+ * Define the local variables for computing minimizers.
+ */
+#define USES_MINIMIZER					\
+  lbfgsfloatval_t a, d, gamma, theta, p, q, r, s;
+
+/**
+ * Find a minimizer of an interpolated cubic function.
+ *  @param  cm      The minimizer of the interpolated cubic.
+ *  @param  u       The value of one point, u.
+ *  @param  fu      The value of f(u).
+ *  @param  du      The value of f'(u).
+ *  @param  v       The value of another point, v.
+ *  @param  fv      The value of f(v).
+ *  @param  du      The value of f'(v).
+ */
+#define CUBIC_MINIMIZER(cm, u, fu, du, v, fv, dv)	\
+  d = (v) - (u);					\
+  theta = ((fu) - (fv)) * 3 / d + (du) + (dv);		\
+  p = fabs(theta);					\
+  q = fabs(du);						\
+  r = fabs(dv);						\
+  s = max3(p, q, r);					\
+  /* gamma = s*sqrt((theta/s)**2 - (du/s) * (dv/s)) */	\
+  a = theta / s;					\
+  gamma = s * sqrt(a * a - ((du) / s) * ((dv) / s));	\
+  if ((v) < (u)) gamma = -gamma;			\
+  p = gamma - (du) + theta;				\
+  q = gamma - (du) + gamma + (dv);			\
+  r = p / q;						\
+  (cm) = (u) + r * d;
+
+/**
+ * Find a minimizer of an interpolated cubic function.
+ *  @param  cm      The minimizer of the interpolated cubic.
+ *  @param  u       The value of one point, u.
+ *  @param  fu      The value of f(u).
+ *  @param  du      The value of f'(u).
+ *  @param  v       The value of another point, v.
+ *  @param  fv      The value of f(v).
+ *  @param  du      The value of f'(v).
+ *  @param  xmin    The maximum value.
+ *  @param  xmin    The minimum value.
+ */
+#define CUBIC_MINIMIZER2(cm, u, fu, du, v, fv, dv, xmin, xmax)	\
+  d = (v) - (u);						\
+  theta = ((fu) - (fv)) * 3 / d + (du) + (dv);			\
+  p = fabs(theta);						\
+  q = fabs(du);							\
+  r = fabs(dv);							\
+  s = max3(p, q, r);						\
+  /* gamma = s*sqrt((theta/s)**2 - (du/s) * (dv/s)) */		\
+  a = theta / s;						\
+  gamma = s * sqrt(max2(0., a * a - ((du) / s) * ((dv) / s)));	\
+  if ((u) < (v)) gamma = -gamma;				\
+  p = gamma - (dv) + theta;					\
+  q = gamma - (dv) + gamma + (du);				\
+  r = p / q;							\
+  if (r < 0. && gamma != 0.) {					\
+    (cm) = (v) - r * d;						\
+  } else if (a < 0) {						\
+    (cm) = (xmax);						\
+  } else {							\
+    (cm) = (xmin);						\
+  }
+
+/**
+ * Find a minimizer of an interpolated quadratic function.
+ *  @param  qm      The minimizer of the interpolated quadratic.
+ *  @param  u       The value of one point, u.
+ *  @param  fu      The value of f(u).
+ *  @param  du      The value of f'(u).
+ *  @param  v       The value of another point, v.
+ *  @param  fv      The value of f(v).
+ */
+#define QUARD_MINIMIZER(qm, u, fu, du, v, fv)			\
+  a = (v) - (u);						\
+  (qm) = (u) + (du) / (((fu) - (fv)) / a + (du)) / 2 * a;
+
+/**
+ * Find a minimizer of an interpolated quadratic function.
+ *  @param  qm      The minimizer of the interpolated quadratic.
+ *  @param  u       The value of one point, u.
+ *  @param  du      The value of f'(u).
+ *  @param  v       The value of another point, v.
+ *  @param  dv      The value of f'(v).
+ */
+#define QUARD_MINIMIZER2(qm, u, du, v, dv)	\
+  a = (u) - (v);				\
+  (qm) = (v) + (dv) / ((dv) - (du)) * a;
+
+/**
+ * Update a safeguarded trial value and interval for line search.
+ *
+ *  The parameter x represents the step with the least function value.
+ *  The parameter t represents the current step. This function assumes
+ *  that the derivative at the point of x in the direction of the step.
+ *  If the bracket is set to true, the minimizer has been bracketed in
+ *  an interval of uncertainty with endpoints between x and y.
+ *
+ *  @param  x       The pointer to the value of one endpoint.
+ *  @param  fx      The pointer to the value of f(x).
+ *  @param  dx      The pointer to the value of f'(x).
+ *  @param  y       The pointer to the value of another endpoint.
+ *  @param  fy      The pointer to the value of f(y).
+ *  @param  dy      The pointer to the value of f'(y).
+ *  @param  t       The pointer to the value of the trial value, t.
+ *  @param  ft      The pointer to the value of f(t).
+ *  @param  dt      The pointer to the value of f'(t).
+ *  @param  tmin    The minimum value for the trial value, t.
+ *  @param  tmax    The maximum value for the trial value, t.
+ *  @param  brackt  The pointer to the predicate if the trial value is
+ *                  bracketed.
+ *  @retval int     Status value. Zero indicates a normal termination.
+ *  
+ *  @see
+ *      Jorge J. More and David J. Thuente. Line search algorithm with
+ *      guaranteed sufficient decrease. ACM Transactions on Mathematical
+ *      Software (TOMS), Vol 20, No 3, pp. 286-307, 1994.
+ */
+static int update_trial_interval(
+				 lbfgsfloatval_t *x,
+				 lbfgsfloatval_t *fx,
+				 lbfgsfloatval_t *dx,
+				 lbfgsfloatval_t *y,
+				 lbfgsfloatval_t *fy,
+				 lbfgsfloatval_t *dy,
+				 lbfgsfloatval_t *t,
+				 lbfgsfloatval_t *ft,
+				 lbfgsfloatval_t *dt,
+				 const lbfgsfloatval_t tmin,
+				 const lbfgsfloatval_t tmax,
+				 int *brackt
+				 )
+{
+  int bound;
+  int dsign = fsigndiff(dt, dx);
+  lbfgsfloatval_t mc; /* minimizer of an interpolated cubic. */
+  lbfgsfloatval_t mq; /* minimizer of an interpolated quadratic. */
+  lbfgsfloatval_t newt;   /* new trial value. */
+  USES_MINIMIZER;     /* for CUBIC_MINIMIZER and QUARD_MINIMIZER. */
+
+  /* Check the input parameters for errors. */
+  if (*brackt) {
+    if (*t <= min2(*x, *y) || max2(*x, *y) <= *t) {
+      /* The trival value t is out of the interval. */
+      return LBFGSERR_OUTOFINTERVAL;
+    }
+    if (0. <= *dx * (*t - *x)) {
+      /* The function must decrease from x. */
+      return LBFGSERR_INCREASEGRADIENT;
+    }
+    if (tmax < tmin) {
+      /* Incorrect tmin and tmax specified. */
+      return LBFGSERR_INCORRECT_TMINMAX;
+    }
+  }
+
+  /*
+    Trial value selection.
+  */
+  if (*fx < *ft) {
+    /*
+      Case 1: a higher function value.
+      The minimum is brackt. If the cubic minimizer is closer
+      to x than the quadratic one, the cubic one is taken, else
+      the average of the minimizers is taken.
+    */
+    *brackt = 1;
+    bound = 1;
+    CUBIC_MINIMIZER(mc, *x, *fx, *dx, *t, *ft, *dt);
+    QUARD_MINIMIZER(mq, *x, *fx, *dx, *t, *ft);
+    if (fabs(mc - *x) < fabs(mq - *x)) {
+      newt = mc;
+    } else {
+      newt = mc + 0.5 * (mq - mc);
+    }
+  } else if (dsign) {
+    /*
+      Case 2: a lower function value and derivatives of
+      opposite sign. The minimum is brackt. If the cubic
+      minimizer is closer to x than the quadratic (secant) one,
+      the cubic one is taken, else the quadratic one is taken.
+    */
+    *brackt = 1;
+    bound = 0;
+    CUBIC_MINIMIZER(mc, *x, *fx, *dx, *t, *ft, *dt);
+    QUARD_MINIMIZER2(mq, *x, *dx, *t, *dt);
+    if (fabs(mc - *t) > fabs(mq - *t)) {
+      newt = mc;
+    } else {
+      newt = mq;
+    }
+  } else if (fabs(*dt) < fabs(*dx)) {
+    /*
+      Case 3: a lower function value, derivatives of the
+      same sign, and the magnitude of the derivative decreases.
+      The cubic minimizer is only used if the cubic tends to
+      infinity in the direction of the minimizer or if the minimum
+      of the cubic is beyond t. Otherwise the cubic minimizer is
+      defined to be either tmin or tmax. The quadratic (secant)
+      minimizer is also computed and if the minimum is brackt
+      then the the minimizer closest to x is taken, else the one
+      farthest away is taken.
+    */
+    bound = 1;
+    CUBIC_MINIMIZER2(mc, *x, *fx, *dx, *t, *ft, *dt, tmin, tmax);
+    QUARD_MINIMIZER2(mq, *x, *dx, *t, *dt);
+    if (*brackt) {
+      if (fabs(*t - mc) < fabs(*t - mq)) {
+	newt = mc;
+      } else {
+	newt = mq;
+      }
+    } else {
+      if (fabs(*t - mc) > fabs(*t - mq)) {
+	newt = mc;
+      } else {
+	newt = mq;
+      }
+    }
+  } else {
+    /*
+      Case 4: a lower function value, derivatives of the
+      same sign, and the magnitude of the derivative does
+      not decrease. If the minimum is not brackt, the step
+      is either tmin or tmax, else the cubic minimizer is taken.
+    */
+    bound = 0;
+    if (*brackt) {
+      CUBIC_MINIMIZER(newt, *t, *ft, *dt, *y, *fy, *dy);
+    } else if (*x < *t) {
+      newt = tmax;
+    } else {
+      newt = tmin;
+    }
+  }
+
+  /*
+    Update the interval of uncertainty. This update does not
+    depend on the new step or the case analysis above.
+
+    - Case a: if f(x) < f(t),
+    x <- x, y <- t.
+    - Case b: if f(t) <= f(x) && f'(t)*f'(x) > 0,
+    x <- t, y <- y.
+    - Case c: if f(t) <= f(x) && f'(t)*f'(x) < 0, 
+    x <- t, y <- x.
+  */
+  if (*fx < *ft) {
+    /* Case a */
+    *y = *t;
+    *fy = *ft;
+    *dy = *dt;
+  } else {
+    /* Case c */
+    if (dsign) {
+      *y = *x;
+      *fy = *fx;
+      *dy = *dx;
+    }
+    /* Cases b and c */
+    *x = *t;
+    *fx = *ft;
+    *dx = *dt;
+  }
+
+  /* Clip the new trial value in [tmin, tmax]. */
+  if (tmax < newt) newt = tmax;
+  if (newt < tmin) newt = tmin;
+
+  /*
+    Redefine the new trial value if it is close to the upper bound
+    of the interval.
+  */
+  if (*brackt && bound) {
+    mq = *x + 0.66 * (*y - *x);
+    if (*x < *y) {
+      if (mq < newt) newt = mq;
+    } else {
+      if (newt < mq) newt = mq;
+    }
+  }
+
+  /* Return the new trial value. */
+  *t = newt;
+  return 0;
+}
+
+
+
+
+
+static lbfgsfloatval_t owlqn_x1norm(
+				    const lbfgsfloatval_t* x,
+				    const int start,
+				    const int n
+				    )
+{
+  int i;
+  lbfgsfloatval_t norm = 0.;
+
+  for (i = start;i < n;++i) {
+    norm += fabs(x[i]);
+  }
+
+  return norm;
+}
+
+static void owlqn_pseudo_gradient(
+				  lbfgsfloatval_t* pg,
+				  const lbfgsfloatval_t* x,
+				  const lbfgsfloatval_t* g,
+				  const int n,
+				  const lbfgsfloatval_t c,
+				  const int start,
+				  const int end
+				  )
+{
+  int i;
+
+  /* Compute the negative of gradients. */
+  for (i = 0;i < start;++i) {
+    pg[i] = g[i];
+  }
+
+  /* Compute the psuedo-gradients. */
+  for (i = start;i < end;++i) {
+    if (x[i] < 0.) {
+      /* Differentiable. */
+      pg[i] = g[i] - c;
+    } else if (0. < x[i]) {
+      /* Differentiable. */
+      pg[i] = g[i] + c;
+    } else {
+      if (g[i] < -c) {
+	/* Take the right partial derivative. */
+	pg[i] = g[i] + c;
+      } else if (c < g[i]) {
+	/* Take the left partial derivative. */
+	pg[i] = g[i] - c;
+      } else {
+	pg[i] = 0.;
+      }
+    }
+  }
+
+  for (i = end;i < n;++i) {
+    pg[i] = g[i];
+  }
+}
+
+static void owlqn_project(
+			  lbfgsfloatval_t* d,
+			  const lbfgsfloatval_t* sign,
+			  const int start,
+			  const int end
+			  )
+{
+  int i;
+
+  for (i = start;i < end;++i) {
+    if (d[i] * sign[i] <= 0) {
+      d[i] = 0;
+    }
+  }
+}