Copyright (c) 2020 Microsoft Corporation
Author:
Nikolaj Bjorner (nbjorner)
Lev Nachmanson (levnach)
--*/
#include "math/lp/monomial_bounds.h"
#include "math/lp/nla_core.h"
#include "math/lp/nla_intervals.h"
namespace nla {
monomial_bounds::monomial_bounds(core* c):
common(c),
dep(c->m_intervals.get_dep_intervals()) {}
void monomial_bounds::operator()() {
for (lpvar v : c().m_to_refine) {
monic const& m = c().emons()[v];
propagate(m);
}
}
* Accumulate product of variables in monomial starting at position 'start'
*/
void monomial_bounds::compute_product(unsigned start, monic const& m, scoped_dep_interval& product) {
scoped_dep_interval vi(dep);
unsigned power = 1;
for (unsigned i = start; i < m.size(); ) {
lpvar v = m.vars()[i];
var2interval(v, vi);
++i;
for (power = 1; i < m.size() && m.vars()[i] == v; ++i, ++power);
dep.power<dep_intervals::with_deps>(vi, power, vi);
dep.mul<dep_intervals::with_deps>(product, vi, product);
}
}
* Monomial definition implies that a variable v is within 'range'
* If the current value of v is outside of the range, we add
* a bounds axiom.
*/
bool monomial_bounds::propagate_value(dep_interval& range, lpvar v) {
auto val = c().val(v);
if (dep.is_below(range, val)) {
lp::explanation ex;
dep.get_upper_dep(range, ex);
auto const& upper = dep.upper(range);
auto cmp = dep.upper_is_open(range) ? llc::LT : llc::LE;
new_lemma lemma(c(), "propagate value - upper bound of range is below value");
lemma &= ex;
lemma |= ineq(v, cmp, upper);
TRACE("nla_solver", dep.display(tout << val << " > ", range) << "\n" << lemma << "\n";);
return true;
}
else if (dep.is_above(range, val)) {
lp::explanation ex;
dep.get_lower_dep(range, ex);
auto const& lower = dep.lower(range);
auto cmp = dep.lower_is_open(range) ? llc::GT : llc::GE;
new_lemma lemma(c(), "propagate value - lower bound of range is above value");
lemma &= ex;
lemma |= ineq(v, cmp, lower);
TRACE("nla_solver", dep.display(tout << val << " < ", range) << "\n" << lemma << "\n";);
return true;
}
else {
return false;
}
}
* val(v)^p should be in range.
* if val(v)^p > upper(range) add
* v <= root(p, upper(range)) and v >= -root(p, upper(range)) if p is even
* v <= root(p, upper(range)) if p is odd
* if val(v)^p < lower(range) add
* v >= root(p, lower(range)) or v <= -root(p, lower(range)) if p is even
* v >= root(p, lower(range)) if p is odd
*/
bool monomial_bounds::propagate_value(dep_interval& range, lpvar v, unsigned p) {
SASSERT(p > 0);
if (p == 1)
return propagate_value(range, v);
auto val_v = c().val(v);
auto val = power(val_v, p);
rational r;
if (dep.is_below(range, val)) {
lp::explanation ex;
dep.get_upper_dep(range, ex);
if (p % 2 == 0 && rational(dep.upper(range)).is_neg()) {
new_lemma lemma(c(), "range requires a non-negative upper bound");
lemma &= ex;
return true;
}
else if (rational(dep.upper(range)).root(p, r)) {
if ((p % 2 == 1) || val_v.is_pos()) {
auto le = dep.upper_is_open(range) ? llc::LT : llc::LE;
new_lemma lemma(c(), "propagate value - root case - upper bound of range is below value");
lemma &= ex;
lemma |= ineq(v, le, r);
return true;
}
if (p % 2 == 0 && val_v.is_neg()) {
SASSERT(!r.is_neg());
auto ge = dep.upper_is_open(range) ? llc::GT : llc::GE;
new_lemma lemma(c(), "propagate value - root case - upper bound of range is below negative value");
lemma &= ex;
lemma |= ineq(v, ge, -r);
return true;
}
}
}
else if (dep.is_above(range, val)) {
if (rational(dep.lower(range)).root(p, r)) {
lp::explanation ex;
dep.get_lower_dep(range, ex);
auto ge = dep.lower_is_open(range) ? llc::GT : llc::GE;
auto le = dep.lower_is_open(range) ? llc::LT : llc::LE;
new_lemma lemma(c(), "propagate value - root case - lower bound of range is above value");
lemma &= ex;
lemma |= ineq(v, ge, r);
if (p % 2 == 0) {
lemma |= ineq(v, le, -r);
}
return true;
}
}
return false;
}
void monomial_bounds::var2interval(lpvar v, scoped_dep_interval& i) {
lp::constraint_index ci;
rational bound;
bool is_strict;
if (c().has_lower_bound(v, ci, bound, is_strict)) {
dep.set_lower_is_open(i, is_strict);
dep.set_lower(i, bound);
dep.set_lower_dep(i, dep.mk_leaf(ci));
dep.set_lower_is_inf(i, false);
}
else {
dep.set_lower_is_inf(i, true);
}
if (c().has_upper_bound(v, ci, bound, is_strict)) {
dep.set_upper_is_open(i, is_strict);
dep.set_upper(i, bound);
dep.set_upper_dep(i, dep.mk_leaf(ci));
dep.set_upper_is_inf(i, false);
}
else {
dep.set_upper_is_inf(i, true);
}
}
* Propagate bounds for monomial 'm'.
* For each variable v in m, compute the intervals of the remaining variables in m.
* Compute also the interval for m.var() as mi
* If the value of v is outside of mi / product_of_other, add a bounds lemma.
* If the value of m.var() is outside of product_of_all_vars, add a bounds lemma.
*/
bool monomial_bounds::propagate(monic const& m) {
unsigned num_free, power;
lpvar free_var;
analyze_monomial(m, num_free, free_var, power);
bool do_propagate_up = num_free == 0;
bool do_propagate_down = !is_free(m.var()) && num_free <= 1;
if (!do_propagate_up && !do_propagate_down)
return false;
scoped_dep_interval product(dep);
scoped_dep_interval vi(dep), mi(dep);
scoped_dep_interval other_product(dep);
var2interval(m.var(), mi);
dep.set_value(product, rational::one());
for (unsigned i = 0; i < m.size(); ) {
lpvar v = m.vars()[i];
++i;
for (power = 1; i < m.size() && v == m.vars()[i]; ++i, ++power);
var2interval(v, vi);
dep.power<dep_intervals::with_deps>(vi, power, vi);
if (do_propagate_down && (num_free == 0 || free_var == v)) {
dep.set<dep_intervals::with_deps>(other_product, product);
compute_product(i, m, other_product);
if (propagate_down(m, mi, v, power, other_product))
return true;
}
dep.mul<dep_intervals::with_deps>(product, vi, product);
}
return do_propagate_up && propagate_value(product, m.var());
}
bool monomial_bounds::propagate_down(monic const& m, dep_interval& mi, lpvar v, unsigned power, dep_interval& product) {
if (!dep.separated_from_zero(product))
return false;
scoped_dep_interval range(dep);
dep.div<dep_intervals::with_deps>(mi, product, range);
return propagate_value(range, v, power);
}
bool monomial_bounds::is_free(lpvar v) const {
return !c().has_lower_bound(v) && !c().has_upper_bound(v);
}
bool monomial_bounds::is_zero(lpvar v) const {
return
c().has_lower_bound(v) &&
c().has_upper_bound(v) &&
c().get_lower_bound(v).is_zero() &&
c().get_upper_bound(v).is_zero();
}
* Count the number of unbound (free) variables.
* Variables with no lower and no upper bound multiplied
* to an odd degree have unbound ranges when it comes to
* bounds propagation.
*/
void monomial_bounds::analyze_monomial(monic const& m, unsigned& num_free, lpvar& fv, unsigned& fv_power) const {
unsigned power = 1;
num_free = 0;
fv = null_lpvar;
fv_power = 0;
for (unsigned i = 0; i < m.vars().size(); ) {
lpvar v = m.vars()[i];
++i;
for (power = 1; i < m.vars().size() && m.vars()[i] == v; ++i, ++power);
if (is_zero(v)) {
num_free = 0;
return;
}
if (power % 2 == 1 && is_free(v)) {
++num_free;
fv_power = power;
fv = v;
}
}
}
}