Solving Constraint Integer Programs

prop_nlobbt.h File Reference

Detailed Description

nonlinear OBBT propagator

Benjamin Mueller

In Nonlinear Optimization-Based Bound Tightening (NLOBBT), we solve auxiliary NLPs of the form

\[ \min / \max \, x_i \\ \]

\[ s.t. \; g_j(x) \le 0 \, \forall j=1,\ldots,m \\ \]

\[ c'x \le \mathcal{U} \]

\[ x \in [\ell,u] \]

where each \( g_j \) is a convex function and \( \mathcal{U} \) the solution value of the current incumbent. Clearly, the optimal objective value of this nonlinear program provides a valid lower/upper bound on variable \( x_i \).

The propagator sorts all variables w.r.t. their occurrences in convex nonlinear constraints and solves sequentially all convex NLPs. Variables which could be successfully tightened by the propagator will be prioritized in the next call of a new node in the branch-and-bound tree. By default, the propagator requires at least one nonconvex constraints to be executed. For purely convex problems, the benefit of having tighter bounds is negligible.

By default, NLOBBT is only applied for non-binary variables. A reason for this can be found here . Variables which do not appear non-linearly in the nonlinear constraints will not be considered even though they might lead to additional tightenings.

After solving the NLP to optimize \( x_i \) we try to exploit the dual information to generate a globally valid inequality, called Generalized Variable Bound (see prop_genvbounds.h). Let \( \lambda_j \), \( \mu \), \( \alpha \), and \( \beta \) be the dual multipliers for the constraints of the NLP where \( \alpha \) and \( \beta \) correspond to the variable bound constraints. Because of the convexity of \( g_j \) we know that

\[ g_j(x) \ge g_j(x^*) + \nabla g_j(x^*)(x-x^*) \]

holds for every \( x^* \in [\ell,u] \). Let \( x^* \) be the optimal solution after solving the NLP for the case of minimizing \( x_i \) (similiar for the case of maximizing \( x_i \)). Since the NLP is convex we know that the KKT conditions

\[ e_i + \lambda' \nabla g(x^*) + \mu' c + \alpha - \beta = 0 \]

\[ \lambda_j g_j(x^*) = 0 \]

hold. Aggregating the inequalities \( x_i \ge x_i \) and \( g_j(x) \le 0 \) leads to the inequality

\[ x_i \ge x_i + \sum_{j} g_j(x_i) \]

Instead of calling the (expensive) propagator during the tree search we can use this inequality to derive further reductions on \( x_i \). Multiplying the first KKT condition by \( (x - x^*) \) and using the fact that each \( g_j \) is convex we can rewrite the previous inequality to

\[ x_i \ge (\beta - \alpha)'x + (e_i + \alpha - \beta) x^* + \mu \mathcal{U}. \]

which is passed to the genvbounds propagator. Note that if \( \alpha_i \neq \beta_i \) we know that the bound of \( x_i \) is the proof for optimality and thus no useful genvbound can be found.

Definition in file prop_nlobbt.h.

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

Go to the source code of this file.


SCIP_RETCODE SCIPincludePropNlobbt (SCIP *scip)