Norm bounds and underestimators for unconstrained polynomial integer minimization

Abstract

We consider the problem of minimizing a polynomial function over the integer lattice. Though impossible in general, we use a known sufficient condition for the existence of continuous minimizers to guarantee the existence of integer minimizers as well. In case this condition holds, we use sos programming to compute the radius of a p-norm ball which contains all integer minimizers. We prove that this radius is smaller than the radius known from the literature. Our numerical results show that the number of potentially optimal solutions is reduced by several orders of magnitude. Furthermore, we derive a new class of underestimators of the polynomial function. Using a Stellensatz from real algebraic geometry and again sos programming, we optimize over this class to get a strong lower bound on the integer minimum. Also our lower bounds are evaluated experimentally. They show a good performance, in particular within a branch and bound framework.

Computing the norm bounds

In Remark 7 we saw that we get a tighter norm bound R on the minimizers the closer the $c_j$ get to their optimal value $c_j^*$. In the following, we present two means that improve on the Approach 1. in Sect. 4.1 that do not rely on sos programming. The second method we present is a refinement of the first. For both, we improve the norm bound R by replacing the estimate $|x^\alpha | \le 1$ on $\mathbb {S}^{n-1}_p$ with $|x^\alpha | \le \hat{x}^\alpha$, where $\hat{x}$ is a continuous maximizer of the function $\mathbb {S}^{n-1}_p \rightarrow \mathbb {R}$, $x \mapsto x^\alpha$.

1.1 A direct improvement

One has the following closed form for the continuous minimizer $\hat{x}$ with nonnegative coordinates:

Lemma 20

Let $0 \ne \alpha \in \mathbb {N}_0^n$ and $p \in [1, \infty )$. Then, the monomial $X^\alpha$ attains its maximum on $\mathbb {S}^{n-1}_p$ at $\hat{x}$ with coordinates

(17)

Proof

By a simple analysis, the proof can be reduced to $\alpha _i \ge 1$ for $i=1, \ldots , n$ and then to maximization of $X^\alpha$ on $\{ x \in \mathbb {S}^{n-1}_p \ | \ x_1> 0, \ldots , x_n > 0 \}$. Using the method of Lagrange multipliers, the claim follows from a short calculation. $\square$

Observation 21

Denote by $\hat{x}_{(\alpha )}$ the maximizer of $X^\alpha$ on $\mathbb {S}^{n-1}_p$ as in (17). Hence for $x \in \mathbb {S}^{n-1}_p$ we have

$$\begin{aligned} f_j(x) = \sum _{|\alpha | = j} a_\alpha x^{\alpha } \ge \sum _{|\alpha | = j} - |a_\alpha | \cdot (\hat{x}_{(\alpha )})^\alpha =: c_j. \end{aligned}$$

(18)

This $c_j$ is as least as large as the $c_j$ from Proposition 8, since, for $0 \ne \alpha$, $(\hat{x}_{(\alpha )})^\alpha < 1$—unless $X^\alpha \in \mathbb {R}[X_i]$ for some i, in which case $\hat{x}_{(\alpha )} = e_i$, the ith unit vector, and thus $(\hat{x}_{(\alpha )})^\alpha = 1$.

1.2 A different approach

This last approach on computing bounds $c_j$ is different to the ones before, as we actually compute $2^n$ norm bounds: we restrict f to each of the $2^n$ orthants

$$\begin{aligned} H_\tau = \{x \in \mathbb {R}^n \ | \ \tau _i x_i \ge 0\} \text { for }\tau \in \{-1,1\}^n \end{aligned}$$

(19)

and compute a norm bound on integer minimizers of every $f|_{H_\tau }$. This has the advantage that, roughly speaking, we may neglect half of the terms of $f = \sum a_\alpha X^\alpha$. Also, minimization on $H_\tau$ can be reduced to minimization on $H_{(1, \ldots , 1)}$, i.e., the set of those $x \in \mathbb {R}^n$ with $x \ge 0$, as we shall see in a moment.

Introducing the notation $|a|^- = |\min (a, 0)|$ for $a \in \mathbb {R}$ and with $\hat{x}$ from (17), we have for every term $a_\alpha x^\alpha \ge - |a_\alpha |^- x^\alpha \ge -| a_\alpha |^- \hat{x}^\alpha$ as $x \ge 0$, thus

$$\begin{aligned} f_j(x) = \sum _{|\alpha | = j } a_\alpha x^\alpha \ge \underbrace{\sum _{|\alpha | = j} -|a_\alpha |^- \hat{x}^\alpha }_{=:c_j^{(1, \ldots , 1)}}, \quad x \in \mathbb {S}^{n-1}_p \text { and } x \ge 0, \end{aligned}$$

which means about half of the coefficients are neglected in comparison to (18), if signs are distributed equally among the $a_\alpha$. Now let $R^{(1, \ldots , 1)}$ be the largest real root of

$$\begin{aligned} q^{(1, \ldots , 1)}(\lambda ) := c_d \lambda ^d + \sum _{j=1}^{d-1} c_j^{(1, \ldots , 1)} \lambda ^j. \end{aligned}$$

The verbatim argument of Theorem 6 shows that $f(x) > f(0)$ for $\Vert x \Vert _p > R^{(1, \ldots , 1)}$ and $x \ge 0$. This bounds integer and continuous minimizers on $H_{(1, \ldots , 1)}$. Bounding the norm of minimizers of f on $H_\tau$, $\tau \in \{-1,1\}^{n}$, can be reduced to bounding the norm of minimizers on $H_{(1, \ldots , 1)}$ by a simple change of coordinates. To this end, let $\tau (x) = (\tau _1 x_1, \ldots , \tau _n x_n)$, $x \in \mathbb {R}^n$, and $f^\tau$ be the polynomial

$$\begin{aligned} f^{\tau }(x) := f(\tau (x)) = \sum _{\alpha }a_\alpha \tau ^\alpha x^\alpha , \quad \tau \in \{-1, 1 \}^n. \end{aligned}$$

As $\tau ^\alpha \in \{-1, 1\}$, f and $f^\tau$ merely differ in the sign of their coefficients, and $f^\tau _d(x) \ge c_d$ still holds for $x \in \mathbb {S}^{n-1}_p$ as the sphere is $\tau$-invariant, that is $\tau (\mathbb {S}^{n-1}_p) = \mathbb {S}^{n-1}_p$. Similarly to before, denote by $R^{\tau }$ the largest real root of

$$\begin{aligned} q^{\tau }(\lambda ) = c_d \lambda ^d + \sum _{j=1}^{d-1} c_j^{\tau } \lambda ^{j}, \end{aligned}$$

with $c_j^\tau = -|a_\alpha \tau ^{\alpha }|^- \hat{x}^\alpha$. It is now clear that $f^{\tau }(x) > f(0)$ for $\Vert x\Vert _p > R^\tau$ and $x \ge 0$, equivalently, $f(x) > f(0)$ for $\Vert x\Vert _p > R^\tau$ and $x \in H_\tau$.

This results in more effort in the preprocessing, but reduces the number of feasible solutions.