heur_multistart.h File Reference

Detailed Description

multistart heuristic for convex and nonconvex MINLPs

Author

Benjamin Mueller

The heuristic applies multiple NLP local searches to a mixed-integer nonlinear program with, probably nonconvex, constraints of the form $g_j(x) \le 0$. The algorithm tries to identify clusters which approximate the boundary of the feasible set of the continuous relaxation by sampling and improving randomly generated points. For each cluster we use a local search heuristic to find feasible solutions. The algorithm consists of the following four steps:

1. sample points

   Sample random points $x^1, \ldots, x^n$ in the box $[\ell,u]$. For an unbounded variable $x_i$ we consider $[\ell_i,\ell_i + \alpha], [u_i - \alpha,u_i],$ or $[-\alpha / 2, \alpha / 2]$ for an $\alpha > 0$ depending on which bound is infinite.

2. reduce infeasibility

   For each point $x^i$ we use a gradient descent method to reduce the maximum infeasibility. We first compute

   $$d_j = -\frac{g_j(x^i)}{||\nabla g_j(x^i)||^2} \nabla g_j(x^i)$$

   and update the current point $x^i$ with

   $$x^i := x^i + \frac{1}{n_j} \sum_{j} d_j$$

   where $n_j$ is the number of strictly positive $d_j$. The algorithm is called Constraint Consensus Method and has been introduced by here .

3. cluster points

   We use a greedy algorithm to all of the resulting points of step 3. to find clusters which (hopefully) approximate the boundary of the feasible set locally. Points with a too large violations will be filtered.

4. solve sub-problems

   Depending on the current setting, we solve a sub-problem for each identified cluster. The default strategy is to compute a starting point for the sub-NLP heuristic (see heur_subnlp.h) by using a linear combination of the points in a cluster $C$, i.e.,

   $$s := \sum_{x \in C} \lambda_x x$$

   Since the sub-NLP heuristic requires a starting point which is integer feasible we round each fractional value $s_i$ to its closest integer.

Definition in file heur_multistart.h.

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

Go to the source code of this file.

Functions
SCIP_RETCODE	SCIPincludeHeurMultistart (SCIP *scip)