doc: fix typo in annotation

[isl.git] / isl_ilp.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 */

#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl/ilp.h>
#include <isl/union_set.h>
#include "isl_sample.h"
#include <isl_seq.h>
#include "isl_equalities.h"
#include <isl_aff_private.h>
#include <isl_local_space_private.h>
#include <isl_mat_private.h>
#include <isl_val_private.h>
#include <isl_vec_private.h>
#include <isl_lp_private.h>
#include <isl_ilp_private.h>

/* Given a basic set "bset", construct a basic set U such that for
 * each element x in U, the whole unit box positioned at x is inside
 * the given basic set.
 * Note that U may not contain all points that satisfy this property.
 *
 * We simply add the sum of all negative coefficients to the constant
 * term.  This ensures that if x satisfies the resulting constraints,
 * then x plus any sum of unit vectors satisfies the original constraints.
 */
static __isl_give isl_basic_set *unit_box_base_points(
        __isl_take isl_basic_set *bset)
{
        int i, j, k;
        struct isl_basic_set *unit_box = NULL;
        unsigned total;

        if (!bset)
                goto error;

        if (bset->n_eq != 0) {
                isl_space *space = isl_basic_set_get_space(bset);
                isl_basic_set_free(bset);
                return isl_basic_set_empty(space);
        }

        total = isl_basic_set_total_dim(bset);
        unit_box = isl_basic_set_alloc_space(isl_basic_set_get_space(bset),
                                        0, 0, bset->n_ineq);

        for (i = 0; i < bset->n_ineq; ++i) {
                k = isl_basic_set_alloc_inequality(unit_box);
                if (k < 0)
                        goto error;
                isl_seq_cpy(unit_box->ineq[k], bset->ineq[i], 1 + total);
                for (j = 0; j < total; ++j) {
                        if (isl_int_is_nonneg(unit_box->ineq[k][1 + j]))
                                continue;
                        isl_int_add(unit_box->ineq[k][0],
                                unit_box->ineq[k][0], unit_box->ineq[k][1 + j]);
                }
        }

        isl_basic_set_free(bset);
        return unit_box;
error:
        isl_basic_set_free(bset);
        isl_basic_set_free(unit_box);
        return NULL;
}

/* Find an integer point in "bset", preferably one that is
 * close to minimizing "f".
 *
 * We first check if we can easily put unit boxes inside bset.
 * If so, we take the best base point of any of the unit boxes we can find
 * and round it up to the nearest integer.
 * If not, we simply pick any integer point in "bset".
 */
static __isl_give isl_vec *initial_solution(__isl_keep isl_basic_set *bset,
        isl_int *f)
{
        enum isl_lp_result res;
        struct isl_basic_set *unit_box;
        struct isl_vec *sol;

        unit_box = unit_box_base_points(isl_basic_set_copy(bset));

        res = isl_basic_set_solve_lp(unit_box, 0, f, bset->ctx->one,
                                        NULL, NULL, &sol);
        if (res == isl_lp_ok) {
                isl_basic_set_free(unit_box);
                return isl_vec_ceil(sol);
        }

        isl_basic_set_free(unit_box);

        return isl_basic_set_sample_vec(isl_basic_set_copy(bset));
}

/* Restrict "bset" to those points with values for f in the interval [l, u].
 */
static __isl_give isl_basic_set *add_bounds(__isl_take isl_basic_set *bset,
        isl_int *f, isl_int l, isl_int u)
{
        int k;
        unsigned total;

        total = isl_basic_set_total_dim(bset);
        bset = isl_basic_set_extend_constraints(bset, 0, 2);

        k = isl_basic_set_alloc_inequality(bset);
        if (k < 0)
                goto error;
        isl_seq_cpy(bset->ineq[k], f, 1 + total);
        isl_int_sub(bset->ineq[k][0], bset->ineq[k][0], l);

        k = isl_basic_set_alloc_inequality(bset);
        if (k < 0)
                goto error;
        isl_seq_neg(bset->ineq[k], f, 1 + total);
        isl_int_add(bset->ineq[k][0], bset->ineq[k][0], u);

        return bset;
error:
        isl_basic_set_free(bset);
        return NULL;
}

/* Find an integer point in "bset" that minimizes f (in any) such that
 * the value of f lies inside the interval [l, u].
 * Return this integer point if it can be found.
 * Otherwise, return sol.
 *
 * We perform a number of steps until l > u.
 * In each step, we look for an integer point with value in either
 * the whole interval [l, u] or half of the interval [l, l+floor(u-l-1/2)].
 * The choice depends on whether we have found an integer point in the
 * previous step.  If so, we look for the next point in half of the remaining
 * interval.
 * If we find a point, the current solution is updated and u is set
 * to its value minus 1.
 * If no point can be found, we update l to the upper bound of the interval
 * we checked (u or l+floor(u-l-1/2)) plus 1.
 */
static __isl_give isl_vec *solve_ilp_search(__isl_keep isl_basic_set *bset,
        isl_int *f, isl_int *opt, __isl_take isl_vec *sol, isl_int l, isl_int u)
{
        isl_int tmp;
        int divide = 1;

        isl_int_init(tmp);

        while (isl_int_le(l, u)) {
                struct isl_basic_set *slice;
                struct isl_vec *sample;

                if (!divide)
                        isl_int_set(tmp, u);
                else {
                        isl_int_sub(tmp, u, l);
                        isl_int_fdiv_q_ui(tmp, tmp, 2);
                        isl_int_add(tmp, tmp, l);
                }
                slice = add_bounds(isl_basic_set_copy(bset), f, l, tmp);
                sample = isl_basic_set_sample_vec(slice);
                if (!sample) {
                        isl_vec_free(sol);
                        sol = NULL;
                        break;
                }
                if (sample->size > 0) {
                        isl_vec_free(sol);
                        sol = sample;
                        isl_seq_inner_product(f, sol->el, sol->size, opt);
                        isl_int_sub_ui(u, *opt, 1);
                        divide = 1;
                } else {
                        isl_vec_free(sample);
                        if (!divide)
                                break;
                        isl_int_add_ui(l, tmp, 1);
                        divide = 0;
                }
        }

        isl_int_clear(tmp);

        return sol;
}

/* Find an integer point in "bset" that minimizes f (if any).
 * If sol_p is not NULL then the integer point is returned in *sol_p.
 * The optimal value of f is returned in *opt.
 *
 * The algorithm maintains a currently best solution and an interval [l, u]
 * of values of f for which integer solutions could potentially still be found.
 * The initial value of the best solution so far is any solution.
 * The initial value of l is minimal value of f over the rationals
 * (rounded up to the nearest integer).
 * The initial value of u is the value of f at the initial solution minus 1.
 *
 * We then call solve_ilp_search to perform a binary search on the interval.
 */
static enum isl_lp_result solve_ilp(__isl_keep isl_basic_set *bset,
        isl_int *f, isl_int *opt, __isl_give isl_vec **sol_p)
{
        enum isl_lp_result res;
        isl_int l, u;
        struct isl_vec *sol;

        res = isl_basic_set_solve_lp(bset, 0, f, bset->ctx->one,
                                        opt, NULL, &sol);
        if (res == isl_lp_ok && isl_int_is_one(sol->el[0])) {
                if (sol_p)
                        *sol_p = sol;
                else
                        isl_vec_free(sol);
                return isl_lp_ok;
        }
        isl_vec_free(sol);
        if (res == isl_lp_error || res == isl_lp_empty)
                return res;

        sol = initial_solution(bset, f);
        if (!sol)
                return isl_lp_error;
        if (sol->size == 0) {
                isl_vec_free(sol);
                return isl_lp_empty;
        }
        if (res == isl_lp_unbounded) {
                isl_vec_free(sol);
                return isl_lp_unbounded;
        }

        isl_int_init(l);
        isl_int_init(u);

        isl_int_set(l, *opt);

        isl_seq_inner_product(f, sol->el, sol->size, opt);
        isl_int_sub_ui(u, *opt, 1);

        sol = solve_ilp_search(bset, f, opt, sol, l, u);
        if (!sol)
                res = isl_lp_error;

        isl_int_clear(l);
        isl_int_clear(u);

        if (sol_p)
                *sol_p = sol;
        else
                isl_vec_free(sol);

        return res;
}

static enum isl_lp_result solve_ilp_with_eq(__isl_keep isl_basic_set *bset,
        int max, isl_int *f, isl_int *opt, __isl_give isl_vec **sol_p)
{
        unsigned dim;
        enum isl_lp_result res;
        struct isl_mat *T = NULL;
        struct isl_vec *v;

        bset = isl_basic_set_copy(bset);
        dim = isl_basic_set_total_dim(bset);
        v = isl_vec_alloc(bset->ctx, 1 + dim);
        if (!v)
                goto error;
        isl_seq_cpy(v->el, f, 1 + dim);
        bset = isl_basic_set_remove_equalities(bset, &T, NULL);
        v = isl_vec_mat_product(v, isl_mat_copy(T));
        if (!v)
                goto error;
        res = isl_basic_set_solve_ilp(bset, max, v->el, opt, sol_p);
        isl_vec_free(v);
        if (res == isl_lp_ok && sol_p) {
                *sol_p = isl_mat_vec_product(T, *sol_p);
                if (!*sol_p)
                        res = isl_lp_error;
        } else
                isl_mat_free(T);
        isl_basic_set_free(bset);
        return res;
error:
        isl_mat_free(T);
        isl_basic_set_free(bset);
        return isl_lp_error;
}

/* Find an integer point in "bset" that minimizes (or maximizes if max is set)
 * f (if any).
 * If sol_p is not NULL then the integer point is returned in *sol_p.
 * The optimal value of f is returned in *opt.
 *
 * If there is any equality among the points in "bset", then we first
 * project it out.  Otherwise, we continue with solve_ilp above.
 */
enum isl_lp_result isl_basic_set_solve_ilp(__isl_keep isl_basic_set *bset,
        int max, isl_int *f, isl_int *opt, __isl_give isl_vec **sol_p)
{
        unsigned dim;
        enum isl_lp_result res;

        if (!bset)
                return isl_lp_error;
        if (sol_p)
                *sol_p = NULL;

        isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0,
                return isl_lp_error);

        if (isl_basic_set_plain_is_empty(bset))
                return isl_lp_empty;

        if (bset->n_eq)
                return solve_ilp_with_eq(bset, max, f, opt, sol_p);

        dim = isl_basic_set_total_dim(bset);

        if (max)
                isl_seq_neg(f, f, 1 + dim);

        res = solve_ilp(bset, f, opt, sol_p);

        if (max) {
                isl_seq_neg(f, f, 1 + dim);
                isl_int_neg(*opt, *opt);
        }

        return res;
}

static enum isl_lp_result basic_set_opt(__isl_keep isl_basic_set *bset, int max,
        __isl_keep isl_aff *obj, isl_int *opt)
{
        enum isl_lp_result res;

        if (!obj)
                return isl_lp_error;
        bset = isl_basic_set_copy(bset);
        bset = isl_basic_set_underlying_set(bset);
        res = isl_basic_set_solve_ilp(bset, max, obj->v->el + 1, opt, NULL);
        isl_basic_set_free(bset);
        return res;
}

static __isl_give isl_mat *extract_divs(__isl_keep isl_basic_set *bset)
{
        int i;
        isl_ctx *ctx = isl_basic_set_get_ctx(bset);
        isl_mat *div;

        div = isl_mat_alloc(ctx, bset->n_div,
                            1 + 1 + isl_basic_set_total_dim(bset));
        if (!div)
                return NULL;

        for (i = 0; i < bset->n_div; ++i)
                isl_seq_cpy(div->row[i], bset->div[i], div->n_col);

        return div;
}

enum isl_lp_result isl_basic_set_opt(__isl_keep isl_basic_set *bset, int max,
        __isl_keep isl_aff *obj, isl_int *opt)
{
        int *exp1 = NULL;
        int *exp2 = NULL;
        isl_ctx *ctx;
        isl_mat *bset_div = NULL;
        isl_mat *div = NULL;
        enum isl_lp_result res;
        int bset_n_div, obj_n_div;

        if (!bset || !obj)
                return isl_lp_error;

        ctx = isl_aff_get_ctx(obj);
        if (!isl_space_is_equal(bset->dim, obj->ls->dim))
                isl_die(ctx, isl_error_invalid,
                        "spaces don't match", return isl_lp_error);
        if (!isl_int_is_one(obj->v->el[0]))
                isl_die(ctx, isl_error_unsupported,
                        "expecting integer affine expression",
                        return isl_lp_error);

        bset_n_div = isl_basic_set_dim(bset, isl_dim_div);
        obj_n_div = isl_aff_dim(obj, isl_dim_div);
        if (bset_n_div == 0 && obj_n_div == 0)
                return basic_set_opt(bset, max, obj, opt);

        bset = isl_basic_set_copy(bset);
        obj = isl_aff_copy(obj);

        bset_div = extract_divs(bset);
        exp1 = isl_alloc_array(ctx, int, bset_n_div);
        exp2 = isl_alloc_array(ctx, int, obj_n_div);
        if (!bset_div || (bset_n_div && !exp1) || (obj_n_div && !exp2))
                goto error;

        div = isl_merge_divs(bset_div, obj->ls->div, exp1, exp2);

        bset = isl_basic_set_expand_divs(bset, isl_mat_copy(div), exp1);
        obj = isl_aff_expand_divs(obj, isl_mat_copy(div), exp2);

        res = basic_set_opt(bset, max, obj, opt);

        isl_mat_free(bset_div);
        isl_mat_free(div);
        free(exp1);
        free(exp2);
        isl_basic_set_free(bset);
        isl_aff_free(obj);

        return res;
error:
        isl_mat_free(div);
        isl_mat_free(bset_div);
        free(exp1);
        free(exp2);
        isl_basic_set_free(bset);
        isl_aff_free(obj);
        return isl_lp_error;
}

/* Compute the minimum (maximum if max is set) of the integer affine
 * expression obj over the points in set and put the result in *opt.
 *
 * The parameters are assumed to have been aligned.
 */
static enum isl_lp_result isl_set_opt_aligned(__isl_keep isl_set *set, int max,
        __isl_keep isl_aff *obj, isl_int *opt)
{
        int i;
        enum isl_lp_result res;
        int empty = 1;
        isl_int opt_i;

        if (!set || !obj)
                return isl_lp_error;
        if (set->n == 0)
                return isl_lp_empty;

        res = isl_basic_set_opt(set->p[0], max, obj, opt);
        if (res == isl_lp_error || res == isl_lp_unbounded)
                return res;
        if (set->n == 1)
                return res;
        if (res == isl_lp_ok)
                empty = 0;

        isl_int_init(opt_i);
        for (i = 1; i < set->n; ++i) {
                res = isl_basic_set_opt(set->p[i], max, obj, &opt_i);
                if (res == isl_lp_error || res == isl_lp_unbounded) {
                        isl_int_clear(opt_i);
                        return res;
                }
                if (res == isl_lp_empty)
                        continue;
                empty = 0;
                if (max ? isl_int_gt(opt_i, *opt) : isl_int_lt(opt_i, *opt))
                        isl_int_set(*opt, opt_i);
        }
        isl_int_clear(opt_i);

        return empty ? isl_lp_empty : isl_lp_ok;
}

/* Compute the minimum (maximum if max is set) of the integer affine
 * expression obj over the points in set and put the result in *opt.
 */
enum isl_lp_result isl_set_opt(__isl_keep isl_set *set, int max,
        __isl_keep isl_aff *obj, isl_int *opt)
{
        enum isl_lp_result res;
        isl_bool aligned;

        if (!set || !obj)
                return isl_lp_error;

        aligned = isl_set_space_has_equal_params(set, obj->ls->dim);
        if (aligned < 0)
                return isl_lp_error;
        if (aligned)
                return isl_set_opt_aligned(set, max, obj, opt);

        set = isl_set_copy(set);
        obj = isl_aff_copy(obj);
        set = isl_set_align_params(set, isl_aff_get_domain_space(obj));
        obj = isl_aff_align_params(obj, isl_set_get_space(set));

        res = isl_set_opt_aligned(set, max, obj, opt);

        isl_set_free(set);
        isl_aff_free(obj);

        return res;
}

/* Convert the result of a function that returns an isl_lp_result
 * to an isl_val.  The numerator of "v" is set to the optimal value
 * if lp_res is isl_lp_ok.  "max" is set if a maximum was computed.
 *
 * Return "v" with denominator set to 1 if lp_res is isl_lp_ok.
 * Return NULL on error.
 * Return a NaN if lp_res is isl_lp_empty.
 * Return infinity or negative infinity if lp_res is isl_lp_unbounded,
 * depending on "max".
 */
static __isl_give isl_val *convert_lp_result(enum isl_lp_result lp_res,
        __isl_take isl_val *v, int max)
{
        isl_ctx *ctx;

        if (lp_res == isl_lp_ok) {
                isl_int_set_si(v->d, 1);
                return isl_val_normalize(v);
        }
        ctx = isl_val_get_ctx(v);
        isl_val_free(v);
        if (lp_res == isl_lp_error)
                return NULL;
        if (lp_res == isl_lp_empty)
                return isl_val_nan(ctx);
        if (max)
                return isl_val_infty(ctx);
        else
                return isl_val_neginfty(ctx);
}

/* Return the minimum (maximum if max is set) of the integer affine
 * expression "obj" over the points in "bset".
 *
 * Return infinity or negative infinity if the optimal value is unbounded and
 * NaN if "bset" is empty.
 *
 * Call isl_basic_set_opt and translate the results.
 */
__isl_give isl_val *isl_basic_set_opt_val(__isl_keep isl_basic_set *bset,
        int max, __isl_keep isl_aff *obj)
{
        isl_ctx *ctx;
        isl_val *res;
        enum isl_lp_result lp_res;

        if (!bset || !obj)
                return NULL;

        ctx = isl_aff_get_ctx(obj);
        res = isl_val_alloc(ctx);
        if (!res)
                return NULL;
        lp_res = isl_basic_set_opt(bset, max, obj, &res->n);
        return convert_lp_result(lp_res, res, max);
}

/* Return the maximum of the integer affine
 * expression "obj" over the points in "bset".
 *
 * Return infinity or negative infinity if the optimal value is unbounded and
 * NaN if "bset" is empty.
 */
__isl_give isl_val *isl_basic_set_max_val(__isl_keep isl_basic_set *bset,
        __isl_keep isl_aff *obj)
{
        return isl_basic_set_opt_val(bset, 1, obj);
}

/* Return the minimum (maximum if max is set) of the integer affine
 * expression "obj" over the points in "set".
 *
 * Return infinity or negative infinity if the optimal value is unbounded and
 * NaN if "set" is empty.
 *
 * Call isl_set_opt and translate the results.
 */
__isl_give isl_val *isl_set_opt_val(__isl_keep isl_set *set, int max,
        __isl_keep isl_aff *obj)
{
        isl_ctx *ctx;
        isl_val *res;
        enum isl_lp_result lp_res;

        if (!set || !obj)
                return NULL;

        ctx = isl_aff_get_ctx(obj);
        res = isl_val_alloc(ctx);
        if (!res)
                return NULL;
        lp_res = isl_set_opt(set, max, obj, &res->n);
        return convert_lp_result(lp_res, res, max);
}

/* Return the minimum of the integer affine
 * expression "obj" over the points in "set".
 *
 * Return infinity or negative infinity if the optimal value is unbounded and
 * NaN if "set" is empty.
 */
__isl_give isl_val *isl_set_min_val(__isl_keep isl_set *set,
        __isl_keep isl_aff *obj)
{
        return isl_set_opt_val(set, 0, obj);
}

/* Return the maximum of the integer affine
 * expression "obj" over the points in "set".
 *
 * Return infinity or negative infinity if the optimal value is unbounded and
 * NaN if "set" is empty.
 */
__isl_give isl_val *isl_set_max_val(__isl_keep isl_set *set,
        __isl_keep isl_aff *obj)
{
        return isl_set_opt_val(set, 1, obj);
}

/* Return the optimum (min or max depending on "max") of "v1" and "v2",
 * where either may be NaN, signifying an uninitialized value.
 * That is, if either is NaN, then return the other one.
 */
static __isl_give isl_val *val_opt(__isl_take isl_val *v1,
        __isl_take isl_val *v2, int max)
{
        if (!v1 || !v2)
                goto error;
        if (isl_val_is_nan(v1)) {
                isl_val_free(v1);
                return v2;
        }
        if (isl_val_is_nan(v2)) {
                isl_val_free(v2);
                return v1;
        }
        if (max)
                return isl_val_max(v1, v2);
        else
                return isl_val_min(v1, v2);
error:
        isl_val_free(v1);
        isl_val_free(v2);
        return NULL;
}

/* Internal data structure for isl_set_opt_pw_aff.
 *
 * "max" is set if the maximum should be computed.
 * "set" is the set over which the optimum should be computed.
 * "res" contains the current optimum and is initialized to NaN.
 */
struct isl_set_opt_data {
        int max;
        isl_set *set;

        isl_val *res;
};

/* Update the optimum in data->res with respect to the affine function
 * "aff" defined over "set".
 */
static isl_stat piece_opt(__isl_take isl_set *set, __isl_take isl_aff *aff,
        void *user)
{
        struct isl_set_opt_data *data = user;
        isl_val *opt;

        set = isl_set_intersect(set, isl_set_copy(data->set));
        opt = isl_set_opt_val(set, data->max, aff);
        isl_set_free(set);
        isl_aff_free(aff);

        data->res = val_opt(data->res, opt, data->max);
        if (!data->res)
                return isl_stat_error;

        return isl_stat_ok;
}

/* Return the minimum (maximum if "max" is set) of the integer piecewise affine
 * expression "obj" over the points in "set".
 *
 * Return infinity or negative infinity if the optimal value is unbounded and
 * NaN if the intersection of "set" with the domain of "obj" is empty.
 *
 * Initialize the result to NaN and then update it for each of the pieces
 * in "obj".
 */
static __isl_give isl_val *isl_set_opt_pw_aff(__isl_keep isl_set *set, int max,
        __isl_keep isl_pw_aff *obj)
{
        struct isl_set_opt_data data = { max, set };

        data.res = isl_val_nan(isl_set_get_ctx(set));
        if (isl_pw_aff_foreach_piece(obj, &piece_opt, &data) < 0)
                return isl_val_free(data.res);

        return data.res;
}

/* Internal data structure for isl_union_set_opt_union_pw_aff.
 *
 * "max" is set if the maximum should be computed.
 * "obj" is the objective function that needs to be optimized.
 * "res" contains the current optimum and is initialized to NaN.
 */
struct isl_union_set_opt_data {
        int max;
        isl_union_pw_aff *obj;

        isl_val *res;
};

/* Update the optimum in data->res with the optimum over "set".
 * Do so by first extracting the matching objective function
 * from data->obj.
 */
static isl_stat set_opt(__isl_take isl_set *set, void *user)
{
        struct isl_union_set_opt_data *data = user;
        isl_space *space;
        isl_pw_aff *pa;
        isl_val *opt;

        space = isl_set_get_space(set);
        space = isl_space_from_domain(space);
        space = isl_space_add_dims(space, isl_dim_out, 1);
        pa = isl_union_pw_aff_extract_pw_aff(data->obj, space);
        opt = isl_set_opt_pw_aff(set, data->max, pa);
        isl_pw_aff_free(pa);
        isl_set_free(set);

        data->res = val_opt(data->res, opt, data->max);
        if (!data->res)
                return isl_stat_error;

        return isl_stat_ok;
}

/* Return the minimum (maximum if "max" is set) of the integer piecewise affine
 * expression "obj" over the points in "uset".
 *
 * Return infinity or negative infinity if the optimal value is unbounded and
 * NaN if the intersection of "uset" with the domain of "obj" is empty.
 *
 * Initialize the result to NaN and then update it for each of the sets
 * in "uset".
 */
static __isl_give isl_val *isl_union_set_opt_union_pw_aff(
        __isl_keep isl_union_set *uset, int max,
        __isl_keep isl_union_pw_aff *obj)
{
        struct isl_union_set_opt_data data = { max, obj };

        data.res = isl_val_nan(isl_union_set_get_ctx(uset));
        if (isl_union_set_foreach_set(uset, &set_opt, &data) < 0)
                return isl_val_free(data.res);

        return data.res;
}

/* Return a list of minima (maxima if "max" is set) over the points in "uset"
 * for each of the expressions in "obj".
 *
 * An element in the list is infinity or negative infinity if the optimal
 * value of the corresponding expression is unbounded and
 * NaN if the intersection of "uset" with the domain of the expression
 * is empty.
 *
 * Iterate over all the expressions in "obj" and collect the results.
 */
static __isl_give isl_multi_val *isl_union_set_opt_multi_union_pw_aff(
        __isl_keep isl_union_set *uset, int max,
        __isl_keep isl_multi_union_pw_aff *obj)
{
        int i, n;
        isl_multi_val *mv;

        if (!uset || !obj)
                return NULL;

        n = isl_multi_union_pw_aff_dim(obj, isl_dim_set);
        mv = isl_multi_val_zero(isl_multi_union_pw_aff_get_space(obj));

        for (i = 0; i < n; ++i) {
                isl_val *v;
                isl_union_pw_aff *upa;

                upa = isl_multi_union_pw_aff_get_union_pw_aff(obj, i);
                v = isl_union_set_opt_union_pw_aff(uset, max, upa);
                isl_union_pw_aff_free(upa);
                mv = isl_multi_val_set_val(mv, i, v);
        }

        return mv;
}

/* Return a list of minima over the points in "uset"
 * for each of the expressions in "obj".
 *
 * An element in the list is infinity or negative infinity if the optimal
 * value of the corresponding expression is unbounded and
 * NaN if the intersection of "uset" with the domain of the expression
 * is empty.
 */
__isl_give isl_multi_val *isl_union_set_min_multi_union_pw_aff(
        __isl_keep isl_union_set *uset, __isl_keep isl_multi_union_pw_aff *obj)
{
        return isl_union_set_opt_multi_union_pw_aff(uset, 0, obj);
}

/* Return the maximal value attained by the given set dimension,
 * independently of the parameter values and of any other dimensions.
 *
 * Return infinity if the optimal value is unbounded and
 * NaN if "bset" is empty.
 */
__isl_give isl_val *isl_basic_set_dim_max_val(__isl_take isl_basic_set *bset,
        int pos)
{
        isl_local_space *ls;
        isl_aff *obj;
        isl_val *v;

        if (!bset)
                return NULL;
        if (pos < 0 || pos >= isl_basic_set_dim(bset, isl_dim_set))
                isl_die(isl_basic_set_get_ctx(bset), isl_error_invalid,
                        "position out of bounds", goto error);
        ls = isl_local_space_from_space(isl_basic_set_get_space(bset));
        obj = isl_aff_var_on_domain(ls, isl_dim_set, pos);
        v = isl_basic_set_max_val(bset, obj);
        isl_aff_free(obj);
        isl_basic_set_free(bset);

        return v;
error:
        isl_basic_set_free(bset);
        return NULL;
}