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/*++
Copyright (c) 2017 Microsoft Corporation
Author:
Lev Nachmanson (levnach)
Nikolaj Bjorner (nbjorner)
--*/
#include "math/lp/nla_tangent_lemmas.h"
#include "math/lp/nla_core.h"
namespace nla {
class tangent_imp {
point m_a;
point m_b;
point m_xy;
rational m_correct_v;
// "below" means that the incorrect value is less than the correct one, that is m_v < m_correct_v
bool m_below;
// pl is in the strict interior of the bound box (model-driven points
// get_initial_points + push_point); McCormick at the box corners
// requires non-strict inequality because the tangent meets the surface
// along the box's edges (xy = pl.y*x + pl.x*y - pl.x*pl.y at x = pl.x
// or y = pl.y).
bool m_pl_strict_interior = true;
rational m_v; // the monomial value
lpvar m_j; // the monic variable
const monic& m_m;
const factor& m_x;
const factor& m_y;
lpvar m_jx;
lpvar m_jy;
tangents& m_tang;
bool m_is_mon;
public:
tangent_imp(point xy,
const rational& v,
const monic& m,
const factorization& f,
tangents& tang) : m_xy(xy),
m_correct_v(xy.x * xy.y),
m_below(v < m_correct_v),
m_v(v),
m_j(m.var()),
m_m(m),
m_x(f[0]),
m_y(f[1]),
m_jx(m_x.var()),
m_jy(m_y.var()),
m_tang(tang),
m_is_mon(f.is_mon()) {
SASSERT(f.size() == 2);
}
void operator()() {
get_points();
TRACE(nla_solver, print_tangent_domain(tout << "tang domain = ") << std::endl;);
generate_line1();
generate_line2();
generate_plane(m_a);
generate_plane(m_b);
}
private:
core & c() { return m_tang.c(); }
void explain(lemma_builder& lemma) {
if (!m_is_mon) {
lemma &= m_m;
lemma &= m_x;
lemma &= m_y;
}
}
void generate_plane(const point & pl) {
if (c().throttle().insert_new(nla_throttle::TANGENT_LEMMA, m_j, m_jx, m_jy, m_below))
return;
lemma_builder lemma(c(), "generate tangent plane");
c().negate_relation(lemma, m_jx, m_x.rat_sign()*pl.x);
c().negate_relation(lemma, m_jy, m_y.rat_sign()*pl.y);
#if Z3DEBUG
SASSERT(c().val(m_x) == m_xy.x && c().val(m_y) == m_xy.y);
// int mult_sign = nla::rat_sign(pl.x - m_xy.x)*nla::rat_sign(pl.y - m_xy.y);
SASSERT((nla::rat_sign(pl.x - m_xy.x)*nla::rat_sign(pl.y - m_xy.y) == 1) == m_below);
// If "mult_sign is 1" then (a - x)(b-y) > 0 and ab - bx - ay + xy > 0
// or -ab + bx + ay < xy or -ay - bx + xy > -ab
// val(j) stands for xy. So, finally we have -ay - bx + j > - ab
#endif
lp::lar_term t;
t.add_monomial(- m_y.rat_sign()*pl.x, m_jy);
t.add_monomial(- m_x.rat_sign()*pl.y, m_jx);
t.add_var(m_j);
llc cmp = m_below
? (m_pl_strict_interior ? llc::GT : llc::GE)
: (m_pl_strict_interior ? llc::LT : llc::LE);
lemma |= ineq(t, cmp, - pl.x*pl.y);
explain(lemma);
}
void generate_line1() {
// Use plane_type 1 to distinguish line1 from other tangent lemmas
if (c().throttle().insert_new(nla_throttle::TANGENT_LEMMA, m_j, m_jx, m_jy, m_below, 1))
return;
lemma_builder lemma(c(), "tangent line 1");
// Should be v = val(m_x)*val(m_y), and val(factor) = factor.rat_sign()*var(factor.var())
lemma |= ineq(m_jx, llc::NE, c().val(m_jx));
lemma |= ineq(lp::lar_term(m_j, - m_y.rat_sign() * m_xy.x, m_jy), llc::EQ, 0);
explain(lemma);
}
void generate_line2() {
// Use plane_type 2 to distinguish line2 from other tangent lemmas
if (c().throttle().insert_new(nla_throttle::TANGENT_LEMMA, m_j, m_jx, m_jy, m_below, 2))
return;
lemma_builder lemma(c(), "tangent line 2");
lemma |= ineq(m_jy, llc::NE, c().val(m_jy));
lemma |= ineq(lp::lar_term(m_j, - m_x.rat_sign() * m_xy.y, m_jx), llc::EQ, 0);
explain(lemma);
}
// Get two planes tangent to surface z = xy, one at point a, and another at point b, creating a cut
void get_initial_points() {
const rational& x = m_xy.x;
const rational& y = m_xy.y;
bool all_ints = m_v.is_int() && x.is_int() && y.is_int();
rational delta = rational(1);
if (!all_ints )
delta = std::min(delta, abs(m_correct_v - m_v));
TRACE(nla_solver, tout << "delta = " << delta << "\n";);
if (!m_below){
m_a = point(x - delta, y + delta);
m_b = point(x + delta, y - delta);
}
else {
// denote x = xy.x and y = xy.y, and vx, vy - the values of x and y.
// we have val(xy) < vx*y + vy*x - vx*vy = pl(x, y);
// The plane with delta (1, 1) is (vx + 1)y + (vy + 1)x - (vx + 1)(vy + 1) =
// vx*y + vy*x - vx*vy + y + x - xv*vy - vx - vy - 1 = pl(x, y) - 1
// For integers the last expression is greater than or equal to val(xy) when x = vx and y = vy.
// If x <= vx+1 and y <= vy+1 then (vx+1-x)*(vy+1-y) > 0, that creates a cut
// - (vx + 1)y - (vy + 1)x + xy > - (vx+1)*(vx+1).
// If all_ints is false then we use the fact that
// tang_plane() will not change more than on delta*delta
m_a = point(x - delta, y - delta);
m_b = point(x + delta, y + delta);
}
}
void push_point(point & a) {
SASSERT(plane_is_correct_cut(a));
int steps = 10;
point del = a - m_xy;
while (steps-- && !c().done()) {
del *= rational(2);
point na = m_xy + del;
TRACE(nla_solver_tp, tout << "del = " << del << std::endl;);
if (!plane_is_correct_cut(na)) {
TRACE(nla_solver_tp, tout << "exit\n";);
return;
}
a = na;
}
}
rational tang_plane(const point& a) const {
return a.x * m_xy.y + a.y * m_xy.x - a.x * a.y;
}
// McCormick at box corners: choose m_a, m_b at the corners of
// [x_lo, x_hi] x [y_lo, y_hi] that bound xy from the side dictated by
// m_below. Returns false if either factor has an unbounded side, the
// box is degenerate, or the current LP value of a factor coincides with
// a chosen corner — generate_plane's negate_relation requires
// val(j) != corner_coord (SASSERT in debug; trivially-true literal in
// release). The caller falls back to the model-driven point selection in
// these cases.
bool set_box_corners() {
if (!c().has_lower_bound(m_jx) || !c().has_upper_bound(m_jx))
return false;
if (!c().has_lower_bound(m_jy) || !c().has_upper_bound(m_jy))
return false;
rational const& x_lo = c().get_lower_bound(m_jx);
rational const& x_hi = c().get_upper_bound(m_jx);
rational const& y_lo = c().get_lower_bound(m_jy);
rational const& y_hi = c().get_upper_bound(m_jy);
if (x_lo == x_hi || y_lo == y_hi)
return false;
// negate_relation requires the model value to be strictly separated
// from the corner coordinate it's compared to. If LP currently sits
// exactly at a box edge, fall back.
rational const& vx = c().val(m_jx);
rational const& vy = c().val(m_jy);
if (vx == x_lo || vx == x_hi || vy == y_lo || vy == y_hi)
return false;
if (m_below) {
// Under-approximation: tangents at (x_lo, y_lo) and (x_hi, y_hi)
// bound xy from below across the box.
m_a = point(x_lo, y_lo);
m_b = point(x_hi, y_hi);
} else {
// Over-approximation: anti-diagonal corners.
m_a = point(x_lo, y_hi);
m_b = point(x_hi, y_lo);
}
m_pl_strict_interior = false;
return true;
}
void get_points() {
if (c().params().arith_nl_tangents_box_corners() && set_box_corners()) {
// Box corners are extremes; pushing further moves out of the box
// and would invalidate the McCormick property.
TRACE(nla_solver, tout << "xy = " << m_xy << ", box-corner points: ";
print_tangent_domain(tout) << std::endl;);
return;
}
get_initial_points();
TRACE(nla_solver, tout << "xy = " << m_xy << ", correct val = " << m_correct_v;
print_tangent_domain(tout << "\ntang points:") << std::endl;);
push_point(m_a);
push_point(m_b);
TRACE(nla_solver,
tout << "pushed a = " << m_a << std::endl
<< "pushed b = " << m_b << std::endl
<< "tang_plane(a) = " << tang_plane(m_a) << " , val = " << m_a << ", "
<< "tang_plane(b) = " << tang_plane(m_b) << " , val = " << m_b << std::endl;);
}
std::ostream& print_tangent_domain(std::ostream& out) {
return out << "(" << m_a << ", " << m_b << ")";
}
bool plane_is_correct_cut(const point& plane) const {
TRACE(nla_solver, tout << "plane = " << plane << "\n";
tout << "tang_plane() = " << tang_plane(plane) << ", v = " << m_v << ", correct_v = " << m_correct_v << "\n";);
SASSERT((m_below && m_v < m_correct_v) ||
((!m_below) && m_v > m_correct_v));
rational sign = rational(m_below ? 1 : -1);
rational px = tang_plane(plane);
return ((m_correct_v - px)*sign).is_pos() && !((px - m_v)*sign).is_neg();
}
};
tangents::tangents(core * c) : common(c) {}
void tangents::tangent_lemma() {
factorization bf(nullptr);
const monic* m = nullptr;
if (c().params().arith_nl_tangents() && c().find_bfc_to_refine(m, bf)) {
lpvar j = m->var();
tangent_imp tangent(point(val(bf[0]), val(bf[1])), c().val(j), *m, bf, *this);
tangent();
}
}
}