branch_distribution.h

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/**@file   branch_distribution.h
 * @ingroup BRANCHINGRULES
 * @brief  probability based branching rule based on an article by J. Pryor and J.W. Chinneck
 * @author Gregor Hendel
 *
 * The distribution branching rule selects a variable based on its impact on row activity distributions. More formally,
 * let \f$ a(x) = a_1 x_1 + \dots + a_n x_n \leq b \f$ be a row of the LP. Let further \f$ l_i, u_i \in R\f$ denote the
 * (finite) lower and upper bound, respectively, of the \f$ i \f$-th variable \f$x_i\f$.
 * Viewing every variable \f$x_i \f$ as (continuously) uniformly distributed within its bounds, we can approximately
 * understand the row activity \f$a(x)\f$ as a gaussian random variate with mean value \f$ \mu = E[a(x)] = \sum_i a_i\frac{l_i + u_i}{2}\f$
 * and variance \f$ \sigma^2 = \sum_i a_i^2 \sigma_i^2 \f$, with \f$ \sigma_i^2 = \frac{(u_i - l_i)^2}{12}\f$ for
 * continuous and \f$ \sigma_i^2 = \frac{(u_i - l_i + 1)^2 - 1}{12}\f$ for discrete variables.
 * With these two parameters, we can calculate the probability to satisfy the row in terms of the cumulative distribution
 * of the standard normal distribution: \f$ P(a(x) \leq b) = \Phi(\frac{b - \mu}{\sigma})\f$.
 *
 * The impact of a variable on the probability to satisfy a constraint after branching can be estimated by altering
 * the variable contribution to the sums described above. In order to keep the tree size small,
 * variables are preferred which decrease the probability
 * to satisfy a row because it is more likely to create infeasible subproblems.
 *
 * The selection of the branching variable is subject to the parameter @p scoreparam. For both branching directions,
 * an individual score is calculated. Available options for scoring methods are:
 * - @b d: select a branching variable with largest difference in satisfaction probability after branching
 * - @b l: lowest cumulative probability amongst all variables on all rows (after branching).
 * - @b h: highest cumulative probability amongst all variables on all rows (after branching).
 * - @b v: highest number of votes for lowering row probability for all rows a variable appears in.
 * - @b w: highest number of votes for increasing row probability for all rows a variable appears in.
 *
 * If the parameter @p usescipscore is set to @a TRUE, a single branching score is calculated from the respective
 * up and down scores as defined by the SCIP branching score function (see advanced branching parameter @p scorefunc),
 * otherwise, the variable with the single highest score is selected, and the maximizing direction is assigned
 * higher branching priority.
 *
 * The original idea of probability based branching schemes appeared in:
 *
 * J. Pryor and J.W. Chinneck:@n
 * Faster Integer-Feasibility in Mixed-Integer Linear Programs by Branching to Force Change@n
 * Computers and Operations Research, vol. 38, 2011, p. 1143–1152@n
 * (http://www.sce.carleton.ca/faculty/chinneck/docs/PryorChinneck.pdf)
 */

/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/

#ifndef __SCIP_BRANCH_DISTRIBUTION_H__
#define __SCIP_BRANCH_DISTRIBUTION_H__


#include "scip/def.h"
#include "scip/type_lp.h"
#include "scip/type_retcode.h"
#include "scip/type_scip.h"
#include "scip/type_var.h"

#ifdef __cplusplus
extern "C" {
#endif

/** creates the distribution branching rule and includes it in SCIP
 *
 *  @ingroup BranchingRuleIncludes
 */
SCIP_EXPORT
SCIP_RETCODE SCIPincludeBranchruleDistribution(
   SCIP*                 scip                /**< SCIP data structure */
   );

/**@addtogroup BRANCHINGRULES
 *
 * @{
 */

/** calculate the variable's distribution parameters (mean and variance) for the bounds specified in the arguments.
 *  special treatment of infinite bounds necessary */
SCIP_EXPORT
void SCIPvarCalcDistributionParameters(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_Real             varlb,              /**< variable lower bound */
   SCIP_Real             varub,              /**< variable upper bound */
   SCIP_VARTYPE          vartype,            /**< type of the variable */
   SCIP_Real*            mean,               /**< pointer to store mean value */
   SCIP_Real*            variance            /**< pointer to store the variance of the variable uniform distribution */
   );

/** calculates the cumulative distribution P(-infinity <= x <= value) that a normally distributed
 *  random variable x takes a value between -infinity and parameter \p value.
 *
 *  The distribution is given by the respective mean and deviation. This implementation
 *  uses the error function erf().
 */
SCIP_EXPORT
SCIP_Real SCIPcalcCumulativeDistribution(
   SCIP*                 scip,               /**< current SCIP */
   SCIP_Real             mean,               /**< the mean value of the distribution */
   SCIP_Real             variance,           /**< the square of the deviation of the distribution */
   SCIP_Real             value               /**< the upper limit of the calculated distribution integral */
   );

/** calculates the probability of satisfying an LP-row under the assumption
 *  of uniformly distributed variable values.
 *
 *  For inequalities, we use the cumulative distribution function of the standard normal
 *  distribution PHI(rhs - mu/sqrt(sigma2)) to calculate the probability
 *  for a right hand side row with mean activity mu and variance sigma2 to be satisfied.
 *  Similarly, 1 - PHI(lhs - mu/sqrt(sigma2)) is the probability to satisfy a left hand side row.
 *  For equations (lhs==rhs), we use the centeredness measure p = min(PHI(lhs'), 1-PHI(lhs'))/max(PHI(lhs'), 1 - PHI(lhs')),
 *  where lhs' = lhs - mu / sqrt(sigma2).
 */
SCIP_EXPORT
SCIP_Real SCIProwCalcProbability(
   SCIP*                 scip,               /**< current scip */
   SCIP_ROW*             row,                /**< the row */
   SCIP_Real             mu,                 /**< the mean value of the row distribution */
   SCIP_Real             sigma2,             /**< the variance of the row distribution */
   int                   rowinfinitiesdown,  /**< the number of variables with infinite bounds to DECREASE activity */
   int                   rowinfinitiesup     /**< the number of variables with infinite bounds to INCREASE activity */
   );

/** update the up- and downscore of a single variable after calculating the impact of branching on a
 *  particular row, depending on the chosen score parameter
 */
SCIP_EXPORT
SCIP_RETCODE SCIPupdateDistributionScore(
   SCIP*                 scip,               /**< current SCIP pointer */
   SCIP_Real             currentprob,        /**< the current probability */
   SCIP_Real             newprobup,          /**< the new probability if branched upwards */
   SCIP_Real             newprobdown,        /**< the new probability if branched downwards */
   SCIP_Real*            upscore,            /**< pointer to store the new score for branching up */
   SCIP_Real*            downscore,          /**< pointer to store the new score for branching down */
   char                  scoreparam          /**< parameter to determine the way the score is calculated */
   );

/** @} */

#ifdef __cplusplus
}
#endif

#endif