Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares

Abstract

Valid linear inequalities are substantial in linear and convex mixed-integer programming. This article deals with the computation of valid linear inequalities for nonlinear programs. Given a point in the feasible set, we consider the task of computing a tight valid inequality. We reformulate this geometrically as the problem of finding a hyperplane which minimizes the distance to the given point. A characterization of the existence of optimal solutions is given. If the constraints are given by polynomial functions, we show that it is possible to approximate the minimal distance by solving a hierarchy of sum of squares programs. Furthermore, using a result from real algebraic geometry, we show that the hierarchy converges if the relaxed feasible set is bounded. We have implemented our approach, showing that our ideas work in practice.

1 Introduction

The problem we want to solve is the following: Given a subset S of ${\mathbb {R}}^n$ and a point q in S, find a valid linear inequality for S which is as close as possible to q (a formal definition is given in Sect. 2). Our motivation stems from the fact that valid linear inequalities play an important role in solving mixed-integer linear and mixed-integer convex programs. It is thus a natural task to study valid inequalities for the more general class of mixed-integer nonlinear programs (MINLP). More specifically, we search for valid inequalities for the feasible set $F_{\mathcal {I}}$ and its continuous relaxation F. We also consider the special case where we require the objective and constraint functions to be polynomials, which we refer to as mixed-integer polynomial programming (MIPP). To avoid unnecessary clutter, we state our results for the set S which can be thought of being equal to F or $F_{\mathcal {I}}$.

In mixed-integer linear and convex programming, one is interested in finding valid inequalities for $F_{\mathcal {I}}$. One reason for this interest is that the convex hull of $F_{\mathcal {I}}$ can be described by finitely many valid inequalities for rational data in the mixed-integer linear case [1]. This result does not generalize to the convex case; however, in mixed-integer convex programming, a common solution approach is the generation of cuts. A cut is a valid linear inequality for the feasible set that is violated at some point of the relaxed feasible set. A second motivation to find valid linear inequalities is “polyhedrification”, a special form of convexification [2], that is, outer approximation of the sets F and $F_{\mathcal {I}}$ by polyhedra. Note that the meaning of outer approximation is twofold in the literature: It is the name of a celebrated solution method for a special class of MINLP [3, 4], and also describes the process of relaxing a complicated set to a larger set that is easier to handle.

An early result on cuts in mixed-integer linear programming is the algorithmic generation of so-called Gomory cuts [5]. Later, it was shown that the repeated application of all Gomory-type cuts yields the convex hull of $F_{\mathcal {I}}$ for linear integer programming, see Theorem 1.1 in [6]. Nowadays, the underlying theory of cuts has become quite deep and the number of different types of cuts is—even though the underlying ideas of the cuts are often related—vast. The article [6] is a modern presentation of the most influential cuts, and the article [7] explores the relationships in the cut zoo. Recently, maximal lattice-free polyhedra have attracted attention, since it can be shown that the strongest cuts are derived from maximal lattice-free polyhedra, see [8].

Methods for generating cuts also exist in convex programming; for an early approach, see [9]. A modern introduction into the key ideas on cuts for continuous convex problems is given in [10]. For an overview on cuts for mixed-integer convex problems, we refer to Chapter 4 in [11].

There is work on the computation of valid linear inequalities in the non-convex setting: In [12], the authors compute outer approximations for separable non-convex mixed-integer nonlinear programs, and require the feasible set to be contained in a known polytope, i.e., a compact polyhedron. Another recent approach which has proven to be quite successful is via so-called S-free sets, generalizing the idea of lattice-free polyhedra. For example, in [13], an oracle-based cut generating algorithm is presented that computes an arbitrarily precise—as measured by the Hausdorff-distance—approximation of the convex hull of a closed set, if the latter is contained in a known polytope. Related is the work [14], and its extension [15], where the authors derive cuts for a pre-image of a closed set by a linear mapping via so-called cut generating functions and show that these functions are intimately related to S-free sets. There are also new results on cut generation for special cases. A framework is proposed in [16] that creates valid inequalities for a reverse convex set using a cut generating linear program. Also, maximal S-free sets for quadratically constrained problems are computed [17].

Finding a valid linear inequality can be considered as a hyperplane location problem. For an overview on a location theory approach, we refer to [18, 19].

In this article, sos programming plays a key role. This technique can be traced back to [20,21,22,23]. See [24] for a survey, and [25] for a focus on geometric aspects. An algebraic approach is [26]. For a concise treatment, let us mention [27].

Nonlinear mixed-integer programming itself is a large problem class, and the literature is extensive. For an overview and several pointers to key results, let us mention the survey [28], as well as [11, 29].

The remainder of the paper is structured as follows. Section 2 settles notation and prerequisites from sum of squares programming. Section 3 formulates the task of finding a valid linear inequality for a given subset S of ${\mathbb {R}}^n$ that is close to a point q in S. The distance is measured by a gauge. We give geometric characterizations that ensure the existence of feasible and optimal solutions and formulate the problem as a non-convex and semi-infinite optimization problem. In order to make the problem tractable, we first linearize the objective function using a result from [30] in Sect. 4. In Sect. 5, we give a convex reformulation if the distance is measured by a polyhedral gauge and furthermore, restricting ourselves to a semi-algebraic set S in Sect. 6 we receive a hierarchy of sos programs, which converge to the optimal value of the original program if S is bounded. Illustrating examples are provided in Sect. 7. Extensions are discussed in Sect. 8.

Our first main contribution—definitions deferred to Sect. 2—is Proposition 5.1: If valid linear inequalities exist, we can find one that is tight with respect to a polyhedral gauge by solving finitely many linear semi-infinite problems. This result holds without any structural assumptions on the set S, which is to the best of our knowledge a new contribution. The second main contribution is Theorem 6.1: If S is semi-algebraic and the gauge polyhedral, we can give a weakened formulation in terms of a hierarchy of sos programs. Feasibility provided, the hierarchy yields a sequence of hyperplanes with decreasing distances to q. If the corresponding quadratic module is Archimedean, we can guarantee convergence of the hierarchy towards a tight valid inequality. In contrast with the approach of [12], we do not require S to be contained in a polytope to produce feasible solutions (see Sect. 7 for an unbounded example). Similarly, for the method in [13], an oracle is needed, and in the case of polynomial constraints, an oracle is only provided if the feasible set is contained in a polytope. The approach in [14] is not algorithmic and is, without further research, not directly applicable in our setting.

Regarding the scope of the paper, let us note that, for MINLP, it is an important question how cuts can be generated from valid inequalities. One reason for this is that they can improve the strength of an optimization model by only using information inherent in the model, see, e.g., [5, 6, 8, 13,14,15,16,17]. Also, it is an interesting question how different choices of q affect the obtained valid inequalities. Both questions are not within the scope of this article and left as starting points for future research.

2 Preliminaries for Our Approach

In this section, we introduce basic concepts and notation that is used throughout this article. The natural, integer, and real numbers are denoted by ${\mathbb {N}}$, ${\mathbb {Z}}$ and ${\mathbb {R}}$. The natural numbers do not contain 0, and we denote ${\mathbb {N}}_0:={\mathbb {N}}\cup \{0\}$ and put $[k]:= \{1, \ldots , k \}$ for $k \in {\mathbb {N}}$.

2.1 Tight Valid Inequalities

An inequality $(a^Tx \le b)$ is given by some $a \in {\mathbb {R}}^n$, $a \ne 0$, $b \in {\mathbb {R}}$. We say that $(a^Tx \le b)$ is

• valid for $S \subset {\mathbb {R}}^n$ if it is satisfied for all $x \in S$, that is, $a^Tx \le b$ holds for all $x \in S$,

• violated by $x \in S$ if $a^T x > b$,

• tight for S if it is valid for S and for any $b'< b$, the inequality $(a^Tx \le b')$ is violated by some $x \in S$,

• tight for S at $q \in S$ if the inequality is tight for S and $a^Tq = b$.

A linear inequality $(a^T x \le b)$ corresponds to the half-space $\{ x \in {\mathbb {R}}^n : a^T x \le b \}$. The associated hyperplane is denoted by

$$\begin{aligned} H(a,b) := \{ x \in {\mathbb {R}}^n : a^T x = b \}. \end{aligned}$$

2.2 Polynomials, Sum of Squares, and Quadratic Modules

We denote the ring of polynomials in n unknowns $X_1, \ldots , X_n$ and coefficients in ${\mathbb {R}}$ by ${\mathbb {R}}[X_1,\ldots ,X_n]$. A polynomial is a sum of squares or sos for short if it has a representation as a sum of squared polynomials. Formally, we have $p \in {\mathbb {R}}[X_1, \ldots , X_n]$ is sos if there are $q_1, \ldots , q_l \in {\mathbb {R}}[X_1, \ldots , X_n]$ with

$$\begin{aligned} p = q_1^2 + \cdots + q_l^2. \end{aligned}$$

(1)

We denote the set of all sos polynomials by

$$\begin{aligned} \varSigma _n := \left\{ p \in {\mathbb {R}}[X_1, \ldots , X_n] : \exists q_1, \ldots , q_l \in {\mathbb {R}}[X_1, \ldots , X_n], \ p = \sum _{j=1}^l q_j^2 \right\} . \end{aligned}$$

What makes this notion useful is that an immediate consequence of a representation of p as in (1) is that p is nonnegative on all of ${\mathbb {R}}^n$, and the $q_i$ certify nonnegativity. It turns out that deciding if a polynomial is a sum of squares can be reformulated as a semidefinite program (SDP), and SDPs in turn are well-understood and can be solved efficiently, see, e.g., [31, 32]. The set $\varSigma _n$ of all sos polynomials is a convex cone in ${\mathbb {R}}[X_1, \ldots , X_n]$.

We also need the notion of semi-algebraic sets: Given a finite collection of multivariate polynomials $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$, consider the subset of ${\mathbb {R}}^n$ where all polynomials $h_i$ attain nonnegative values

$$\begin{aligned} K(h_1, \ldots , h_s) := \left\{ x \in {\mathbb {R}}^n : h_1(x) \ge 0, \ldots , h_s(x) \ge 0 \right\} . \end{aligned}$$

(2)

A subset of ${\mathbb {R}}^n$ is called basic closed semi-algebraic, or in this article semi-algebraic for short, if it is of the form (2) for some polynomials $h_1, \ldots , h_s$. For example, we note that for a MIPP with constraint polynomials $h_1, \ldots , h_s$, we have $F = K(h_1, \ldots , h_s)$.

Given the constraints $h_i(x) \ge 0$ to a MIPP, a way to infer further valid inequalities is to scale the $h_i$ by sos (and thus nonnegative) polynomials and add them up. This is formalized in the algebraic definition of a quadratic module generated by the $h_i$:

$$\begin{aligned} M(h_1, \ldots , h_s) := \left\{ \sum _{i=0}^s \sigma _i h_i : \sigma _0, \ldots , \sigma _s \in \varSigma _n \right\} \end{aligned}$$

(3)

where $h_0:=1$.

In sos programming, the coefficients of the $\sigma _i$ appearing in (3) are unknowns that we optimize. As there is no degree bound on the $\sigma _i$, this is impractical. Hence, we instead use the truncated quadratic module of order $k \in \{- \infty \} \cup {\mathbb {N}}$, given by

$$\begin{aligned} M(h_1, \ldots , h_s)[k] := \left\{ \sum _{i=0}^s \sigma _i h_i : \sigma _i \in \varSigma _n, \ \deg (\sigma _i h_i) \le k, \ i = 0, \ldots , s \right\} , \end{aligned}$$

(4)

where, again, $h_0 := 1$.

We address now the question how polynomials in $M(h_1, \ldots , h_s)$ and polynomials nonnegative on $K(h_1, \ldots , h_s)$ are related. The following observation which derives a geometric statement from an algebraic one, follows directly from the definitions of $K(h_1,\ldots ,h_s)$ and $M(h_1,\ldots ,h_s)$.

Observation 2.1

Let $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$ and $p \in M(h_1, \ldots , h_s)$. Then $p \ge 0$ on $K(h_1, \ldots , h_s)$.

The question addressing the “converse direction”—suppose a polynomial p is non-negative on $K(h_1, \ldots , h_s)$, does $p \in M(h_1, \ldots , h_s)$ hold?—is more difficult to answer. Conditions that guarantee such representations are addressed in Positivstellensätzen. In this article, we use a Positivstellensatz by Putinar. It holds under a technical condition that we outline next.

2.3 The Archimedean Property and Putinar’s Positivstellensatz

The condition needed for the Positivstellensatz to hold is that the quadratic module $M = M(h_1, \ldots , h_s)$ needs to be Archimedean. The quadratic module M is Archimedean if for all p in ${\mathbb {R}}[X_1, \ldots , X_n]$ there exists $k \in {\mathbb {N}}$ with $p + k \in M$. The following equivalent characterization is useful for our purposes.

Theorem 2.1

(see, e.g., Corollary 5.2.4 in [26]) Given $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$ , let $M = M(h_1, \ldots , h_s)$ be the associated quadratic module. Then M is Archimedean if and only if there is a number $k \in {\mathbb {N}}$ such that $k-\sum _{i=1}^n X_i^2 \in M$.

A consequence of $M(h_1, \ldots , h_s)$ being Archimedean, which is straightforward, well-known but nevertheless important, is that the basic closed semi-algebraic set associated with the polynomials $h_i$ is compact.

Corollary 2.1

Let $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$ and suppose $M(h_1, \ldots , h_s)$ is Archimedean. Then $K(h_1, \ldots , h_s)$ is compact.

Proof

This follows from Observation 2.1 and Theorem 2.1. $\square$

On the other hand, if $K(h_1, \ldots , h_s)$ is compact, then $M(h_1, \ldots , h_s)$ need not be Archimedean, see, e.g., Example 7.3.1 in [26]. As one of our main results (Theorem 6.1) requires $M(h_1, \ldots , h_s)$ to be Archimedean, it is a natural question if one can decide whether a given quadratic module satisfies this property.

Remark 2.1

If $S = K(h_1, \ldots , h_s)$, it is possible to enforce the Archimedean property on the associated quadratic module M if we have a known bound $R \ge 0$ such that $\Vert x \Vert _2 \le R$ for all $x \in S$.¹ Specifically, by adding the redundant constraint $h_{s+1} := R^2 - \sum _{i=1}^n X_i^2$ to the description of S, we still have the equality $S = K(h_1, \ldots , h_{s+1})$, but now Theorem 2.1 guarantees that $M(h_1, \ldots , h_{s+1})$ is Archimedean, see, e.g., [33].

We can now give the Positivstellensatz. Note that the theorem requires positivity, a stronger requirement than nonnegativity.

Theorem 2.2

(Putinar’s Positivstellensatz, see, e.g., Theorem 5.6.1 in [26]) Let p, $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$ and $M(h_1, \ldots , h_s)$ be Archimedean. Then $p(x) > 0$ for all $x \in K(h_1, \ldots , h_s)$ implies $p \in M(h_1, \ldots , h_s)$.

2.4 Gauge Functions

A gauge is a function $\gamma : {\mathbb {R}}^n \rightarrow {\mathbb {R}}$ of the form

$$\begin{aligned} \gamma (x) = \gamma (x; A) \displaystyle := \inf \{t \ge 0 : x \in t A \} \end{aligned}$$

for $A \subset {\mathbb {R}}^n$ compact, convex with $0 \in {{\,\mathrm{int}\,}}A$. Note that every norm with unit ball B is a gauge $\gamma (\cdot ;B)$. On the other hand, a gauge $\gamma (\cdot ;A)$ satisfies definiteness, positive homogeneity and the triangle property as norms do. It is a norm if additionally absolute homogeneity holds, equivalently, if A is symmetric, i.e., $-A=A$.

Given $C, D \subset {\mathbb {R}}^n$ and a gauge $\gamma$ on ${\mathbb {R}}^n$, the distance from C to D is

$$\begin{aligned} d(C,D) := \inf \left\{ \gamma (d- c) : c \in C, d \in D \right\} . \end{aligned}$$

For a singleton set $C = \{c \}$ we also write d(c, D). Analogous to norms, the distance measured by gauges between two nonempty sets C, $K \subset {\mathbb {R}}^n$ is attained if C is closed and K compact, i.e., there exist $c^* \in C$ and $k^* \in K$ with

$$\begin{aligned} d(c^*,k^*) = d(C,K) \end{aligned}$$

(5)

in this case.

The polar of a set $A\subset {\mathbb {R}}^n$ is

$$\begin{aligned} A^\circ \displaystyle := \{x \in {\mathbb {R}}^n : x^T y \le 1 \ \forall y \in A \}. \end{aligned}$$

(6)

It can be shown that if A is compact, convex with $0 \in {{\,\mathrm{int}\,}}A$, the same holds for $A^{\circ }$, see, e.g., Corollary 14.5.1 in [34]. For a gauge $\gamma$, the function

$$\begin{aligned} \gamma ^\circ (x) \displaystyle := \sup \{x^T y : y \in {\mathbb {R}}^n, \gamma (y) \le 1 \} \end{aligned}$$

is the polar of $\gamma$. It then holds that

$$\begin{aligned} \gamma ^{\circ }(x; A) = \gamma (x; A^{\circ }), \end{aligned}$$

(7)

that is, the polar of a gauge is again a gauge, see, e.g., Theorem 15.1 in [34].

We also consider gauges $\gamma$ that are polyhedral: A gauge $\gamma (\cdot , A)$ is called polyhedral if A is a polyhedron. As the polar of a polyhedron is a polyhedron (see, e.g., Corollary 19.2.2 in [34]), it is clear in view of (7) that the polar of a polyhedral gauge is again a polyhedral gauge. For a polyhedral gauge $\gamma (\cdot , A)$, the extreme points of A of are called fundamental directions of $\gamma$.

3 A Geometric Reformulation and Its Properties

In this section we formulate the task to find a tight valid linear inequality as the following geometric optimization² problem: Given $q \in S$, find a valid linear inequality for S such that the associated hyperplane has a minimum distance (defined by an arbitrary gauge function³) to q.

Let us interpret solutions of Program $V 1$ geometrically.

Proposition 3.1

For Program $V 1$, it holds:

• 1. Every feasible solution (a, b) yields a valid inequality $(a^Tx \le b)$ for S.

• 2. Every optimal solution (a, b) yields an inequality $(a^Tx \le b)$ that is tight for S.

• 3. Every optimal solution (a, b) with objective value 0 yields an inequality that is tight for S at q.

Proof

Claim 1 is clear. To see Claim 2, let (a, b) be an optimal solution and assume the contrary, i.e., there is $b' < b$ with $(a^Tx \le b')$ valid for S. By (5) there is $u \in H(a,b)$ with $\gamma (u-q)= d(q,H(a,b))$. Using $q \in S$, we get the inequalities

$$\begin{aligned} a^T q \le b' < b = a^T u, \end{aligned}$$

(8)

and hence $a^T(u - q) > 0$. Put

$$\begin{aligned} {\hat{u}} := u + \lambda (q-u) \ \text {with} \ \lambda := \frac{b - b'}{a^T(u - q)}. \end{aligned}$$

Observe that ${\hat{u}}$ is a point on $H(a,b')$:

$$\begin{aligned} b' = a^T u + b' - b = a^T u + \frac{a^T(q-u)}{a^T(q-u)}(b'-b) = a^T u + \lambda a^T (q-u) = a^T {\hat{u}}. \end{aligned}$$

Note that (8) implies $\lambda > 0$ and $\lambda \le \frac{b - b'}{b-b'} = 1$. Since ${\hat{u}} - q = (1 - \lambda )(u-q)$, all our observations combine to

$$\begin{aligned} d\left( q, H(a,b')\right) \le \gamma ( {\hat{u}} - q) = (1 - \lambda ) \gamma (u - q) < \gamma ( u - q) = d(q, H(a,b)). \end{aligned}$$

Hence $(a, b')$ is a feasible solution to $V 1$ with better objective value, contradicting optimality of (a, b).

To see Claim 3, note that if the objective value is 0 at (a, b) we know from (5) that $d(q, u) = 0$ for some $u \in H(a,b)$, hence $q = u$ and we conclude $q \in H(a,b)$. The claim follows. $\square$

It turns out that feasibility of $V 1$ is sufficient for the existence of optimal solutions.

Theorem 3.1

Let $S \subset {\mathbb {R}}^n$ and $q \in S$. Then, the following are equivalent:

• 1. Program $V 1$ is feasible.

• 2. Program $V 1$ has optimal solutions.

• 3. ${{\,\mathrm{conv}\,}}S \subsetneq {\mathbb {R}}^n$.

Proof

To see Claim 1 $\Rightarrow$ Claim 3, let Program $V 1$ be feasible, and hence $S \subset (a^T x \le b)$ for some $a \in {\mathbb {R}}^n$, $a \ne 0$, $b \in {\mathbb {R}}$. Now

$$\begin{aligned} {{\,\mathrm{conv}\,}}S \subset {{\,\mathrm{conv}\,}}(a^T x \le b) = (a^T x \le b) \subsetneq {\mathbb {R}}^n \end{aligned}$$

follows.

For the implication Claim 3 $\Rightarrow$ Claim 1, let $z \in {\mathbb {R}}^n \setminus {{\,\mathrm{conv}\,}}S$. By the Separating Hyperplane Theorem (see, e.g., Theorem 4.4 in [35]), we may separate z from ${{\,\mathrm{conv}\,}}S$ by a hyperplane H(a, b) with $a^T x \le b$ for all $x \in {{\,\mathrm{conv}\,}}S$, and this hyperplane yields a feasible solution to Program $V 1$.

To see Claim 1 $\Rightarrow$ Claim 2, we construct an optimal solution that corresponds to a supporting hyperplane at a suitably chosen point on the boundary of the closure of the convex hull of S. So let $(a^Tx \le b)$ be an inequality that is valid for S and thus ${{\,\mathrm{conv}\,}}S$. Moreover, as half-spaces are closed, $(a^Tx \le b)$ remains valid for $C := {{\,\mathrm{cl}\,}}{{\,\mathrm{conv}\,}}{S}$, and we conclude $C \subsetneq {\mathbb {R}}^n$ . Also, C is convex as it is the closure of a convex set (see, e.g., Corollary 11.5.1 in [34]). As $q \in S \subset C$, C is a nonempty, proper closed subset of ${\mathbb {R}}^n$, so its boundary $B := {{\,\mathrm{bd}\,}}C$ is nonempty. As B is closed, (5) ensures the existence of $x_1 \in B$ with $d(q,x_1) = d(q,B)$. By the Supporting Hyperplane Theorem (see, e.g., Chapter 2.5.2, p. 51 in [36]), there is a half-space $(a_1^T x \le b_1)$ containing C with $a_1^T x_1 = b_1$. We claim that $(a_1,b_1)$ is optimal.

Suppose it is not, so there is $(a_2,b_2)$ such that $(a_2^T x \le b_2)$ is valid for S and the corresponding hyperplane $H_2:=H(a_2,b_2)$ satisfies the inequality $d_2 := d(q, H_2) < d(q, H_1) =: d_1$. Again there is $x_2 \in H_2$ with $d(q, x_2) = d_2$. We now distinguish two possible locations for $x_2$ and derive a contradiction in every case.

• 1. $x_2 \in {\mathbb {R}}^n \setminus \displaystyle {{\,\mathrm{int}\,}}C$. As $q \in C$, the line segment from $x_2$ to q crosses the boundary B of C at a point $x_3$. But then $d(x_3, q) \le d_2 < d_1$, contradicting the optimality of $x_1$.

• 2. $x_2 \in {{\,\mathrm{int}\,}}C$. Hence there is $\varepsilon > 0$ with $x_2 + \varepsilon \displaystyle a_2 \in C$, and we conclude $a_2^T(x_2 +\varepsilon a_2) = b_2 + \varepsilon a_2^Ta_2 > b_2$ as $x_2 \in H(a_2,b_2)$. Consequently, $(a_2^T x \le b_2)$ is not a valid inequality for C. On the other hand, $S \subset \{ x \in {\mathbb {R}}^n: a_2^T x \le b_2\}$ by assumption on $(a_2^T x \le b_2)$. Since half-spaces are convex and closed, we may conclude that also $C = {{\,\mathrm{cl}\,}}{{\,\mathrm{conv}\,}}S \subset (a_2^T x \le b_2 )$, contradicting our observation that $(a_2^T x \le b_2)$ is not valid for $x_2 + \varepsilon a_2 \in C$.

We conclude that $x_2$ cannot exist, so neither can $(a_2,b_2)$. Hence $(a_1, b_1)$ is an optimal solution to Program $V 1$.

There is nothing to prove for Claim 2 $\Rightarrow$ Claim 1. $\square$

4 Linearizing the Objective

In this section, we linearize the objective function $d(q, H(a,b))$ in Program $V 1$. As a first step, we use an analytic expression for the objective from the literature.

Theorem 4.1

(Theorem 1.1 in [30]) Let $\gamma$ be a gauge on ${\mathbb {R}}^n$ and denote its polar by $\gamma ^\circ$. Furthermore, let $0 \ne a \in {\mathbb {R}}^n$ and $b \in {\mathbb {R}}$. Then

$$\begin{aligned} d(q, H(a,b)) = {\left\{ \begin{array}{ll} (b - a^T q)/\gamma ^\circ (a), &{} a^T q \le b, \\ (a^T q - b)/\gamma ^\circ (-a), &{} a^T q > b. \end{array}\right. } \end{aligned}$$

(9)

Let us first note that $\gamma ^{\circ }(a) > 0$: Since $\gamma$ is a gauge, there is $A \subset {\mathbb {R}}^n$ closed, convex with $0 \in {{\,\mathrm{int}\,}}A$ such that $\gamma (\cdot ) = \gamma (\cdot ;A)$. By (7), we have $\gamma ^{\circ }(\cdot ) = \gamma (\cdot ;A^{\circ })$, and we saw in Sect. 2.4 that $A^{\circ }$ is also closed, convex with $0 \in {{\,\mathrm{int}\,}}A^{\circ }$. Thus $\gamma ^{\circ }(x) =0$ if and only if $x = 0$, hence $\gamma ^{\circ }(a) > 0$. The variable a enters the fractions in (9) in a nonlinear fashion. Moreover, the constraint $a \ne 0$ is not closed. Now compare Program $V 1$ with the following program with linear objective that avoids a constraint of the form $a \ne 0$: R1T3

It turns out that Programs $V 1$ and $V 2$ are closely related. To this end let us introduce the following notion: Two solutions (a, b) and $(a',b')$ are geometrically equivalent if

$$\begin{aligned} H(a,b) = H(a',b'). \end{aligned}$$

We say that two programs have geometrically equivalent feasible/optimal solutions if for every feasible/optimal solution (a, b) of the first program there is a geometrically equivalent feasible/optimal solution $(a',b')$ of the second program and vice versa.

Proposition 4.1

Let $q \in S \subset {\mathbb {R}}^n$ and $\gamma$ be a gauge on ${\mathbb {R}}^n$. Then, the following hold:

• 1. Programs V1 and V2 have geometrically equivalent feasible solutions.

• 2. The optimal values of both programs coincide.

In particular, both programs have geometrically equivalent optimal solutions.

Proof

By (9) and using the fact that $a^Tq \le b$, Program $V 1$ is the same as

For the first claim, let (a, b) be feasible for $\text {V1}'$. Then $\gamma ^\circ (a)^{-1}\cdot (a,b)$ is feasible for $V 2$ and geometrically equivalent to (a, b). On the other hand, if (a, b) is feasible for $V 2$, it is also feasible for $\text {V1}'$.

For the second claim, we note that the programs  $\text {V1}'$ and $V 2$ are either both feasible or both infeasible, so in the following we may assume they are feasible. Let $z_1'$ be the optimal value of  $\text {V1}'$ and $z_2$ be the optimal value of  $V 2$. Let (a, b) be feasible for  $\text {V1}'$. Then $\gamma ^{\circ }(a)^{-1} \cdot (a, b)$ is feasible for  $V 2$ and geometrically equivalent to (a, b). Furthermore, this shows that $z_2 \le z_1'$. Now, let (a, b) be feasible for  $V 2$. Then (a, b) is feasible for  $\text {V1}'$. Since $\gamma ^\circ (a) \ge 1$, we have

$$\begin{aligned} \displaystyle z_1' \le (b - a^T q)/ \gamma ^\circ (a) \le (b - a^T q). \end{aligned}$$

As (a, b) was an arbitrary feasible solution to  $V 2$, we have shown that $z_1' \le z_2$. The claim about geometrically equivalent optimal solutions follows immediately from the first two statements. $\square$

To summarize, instead of solving $V 1$ we may solve $V 2$.

5 A Linear Semi-Infinite Program for Polyhedral Gauges

Program $V 2$ contains the non-convex constraint $\gamma ^\circ (a) \ge 1$. This constraint can be linearized if we restrict ourselves to polyhedral gauges. This is not a hard restriction since due to [37] every norm can be approximated arbitrarily closely by a block norm, and similarly every gauge by a polyhedral gauge.

We need the following characterization of the facets of a unit ball in terms of the extreme points of the polar polyhedron defined in (6).

Theorem 5.1

(see, e.g., Proposition 3.2, and Theorems 5.3 and 5.5 in Chapter I.4 in [38]) If P is a full-dimensional and bounded polyhedron and $0\in {{\,\mathrm{int}\,}}P$ then

$$\begin{aligned} P = \bigcap _{k \in K} \left\{ x \in {\mathbb {R}}^n : \pi _k^T x \le 1 \right\} \end{aligned}$$

where $\{\pi _k\}_{k \in K}$ are the extreme points of $P^{\circ }$. The inequalities $\pi ^T_k x\le 1$ describe exactly the facets of P.

Now suppose $\gamma$ is a polyhedral gauge. Denote its fundamental directions by $v_1, \ldots , v_l \in {\mathbb {R}}^n$, and the unit ball of the polar gauge $\gamma ^\circ$ by $B^\circ$. As $\gamma$ is a polyhedral gauge, so is the polar gauge $\gamma ^\circ$, cf. Sect. 2. This implies that $B^\circ$ is a polyhedron which is full-dimensional, bounded, with 0 in its interior. Thus, $B^\circ$ satisfies the assumptions of Theorem 5.1, and hence

$$\begin{aligned} B^{\circ }= \bigcap _{k \in {\hat{K}}} \{ x \in {\mathbb {R}}^n : {\hat{\pi }}_k ^T x \le 1\} \end{aligned}$$

where $\{{\hat{\pi }}_k\}_{{\hat{K}}}$ are the extreme points of $B^{\circ \circ }:=(B^{\circ })^{\circ }$. Using the fact that $B^{\circ \circ } = B$—this holds for all closed, convex sets containing the origin, see, e.g., Theorem 14.5 in [34]—we have $\{{\hat{\pi }}_k\}_{k \in {\hat{K}}} = \{v_1, \ldots , v_l\}$, and hence

$$\begin{aligned} B^{\circ }= \bigcap _{j \in [l]} \{ x \in {\mathbb {R}}^n : v_j^T x \le 1\}. \end{aligned}$$

(10)

We use this characterization as follows.

Corollary 5.1

Let $\gamma$ be a polyhedral gauge and denote its fundamental directions by $v_1, \ldots , v_l \in {\mathbb {R}}^n$. Then

$$\begin{aligned} \left\{ x \in {\mathbb {R}}^n : \gamma ^{\circ }(x) \ge 1 \right\}&= \bigcup _{j \in [l]} \left\{ x \in {\mathbb {R}}^n : v_j^T x \ge 1 \right\} , \end{aligned}$$

(11)

and

$$\begin{aligned} \left\{ x \in {\mathbb {R}}^n : \gamma ^{\circ }(x) = 1 \right\}&= \bigcup _{j \in [l]} \left\{ x \in {\mathbb {R}}^n : v_i^T x \le 1 \ \forall i \in [l],\ v_j^T x \ge 1 \right\} . \end{aligned}$$

(12)

Proof

For the interior of the unit ball $B^\circ$ of $\gamma ^\circ$ it holds that

$$\begin{aligned} {{\,\mathrm{int}\,}}B^\circ = \left\{ x \in {\mathbb {R}}^n : \gamma ^{\circ }(x) < 1 \right\} . \end{aligned}$$

Since $B^\circ$ is polyhedral, we have from (10)

$$\begin{aligned} {{\,\mathrm{int}\,}}B^{\circ }= \bigcap _{j \in [l]} \{ x \in {\mathbb {R}}^n : v_j^T x < 1\}. \end{aligned}$$

This means that the set $\left\{ x \in {\mathbb {R}}^n : \gamma ^{\circ }(x) \ge 1 \right\}$ equals

$$\begin{aligned} {\mathbb {R}}^n \setminus {{\,\mathrm{int}\,}}B^\circ = {\mathbb {R}}^n \setminus \bigcap _{j \in [l]} \{ x \in {\mathbb {R}}^n : v_j^T x < 1\} = \bigcup _{j \in [l]} \left\{ x \in {\mathbb {R}}^n : v_j^T x \ge 1 \right\} , \end{aligned}$$

proving the first equality. The second equality follows from the fact that

$$\begin{aligned} \{x \in {\mathbb {R}}^n: \gamma ^\circ (x) = 1\} =\{x \in {\mathbb {R}}^n: \gamma ^\circ (x) \le 1\} \cap \{x \in {\mathbb {R}}^n: \gamma ^\circ (x) \ge 1\}, \end{aligned}$$

(13)

and then using distributivity for union and intersection of sets on the explicit representations (10) and (12) of the two sets on the right hand side in (13). $\square$

The idea now is to use Corollary 5.1 to decompose the nonlinear program $V 2$ into a set of l linear programs, one for each fundamental direction $v_j$, $j \in [l]$ of the polyhedral gauge $\gamma$. These programs are given as

The relation between the programs $\text {V}3_j$ and $V 2$ is described next.

Proposition 5.1

Let $q \in S \subset {\mathbb {R}}^n$ and $\gamma$ be a polyhedral gauge on ${\mathbb {R}}^n$ with fundamental directions $v_1, \ldots , v_l \in {\mathbb {R}}^n$. Denote the optimal value of Program $V 2$ by $z^*$ and for $\text {V}3_j$ by $z_j^*$. Let $(a,b) \in {\mathbb {R}}^n \times {\mathbb {R}}$. Then the following hold:

• 1. (a, b) is a feasible solution of $V 2$ if and only if it is a feasible solution of $\text {V}3_j$ for some $j \in [l]$.

• 2. $z^* = \min _{j \in [l]} z_j^*$.

• 3. (a, b) is an optimal solution of $V 2$ if and only if there is $j_0 \in [l]$ such that (a, b) is an optimal solution to $\text {V}3_j$ for $j=j_0$ with $z_{j_0}^* = \min _{j \in [l]} z^*_{j}$.

Proof

Denote the feasible set of $V 2$ by $F'$ and of $\text {V}3_j$ by $F_j$. From (11) we then have $F' = \bigcup _{j \in [l]} F_j$ and all claims follow easily. $\square$

Remark 5.1

Note that $\text {V}3_j$ has a single linear constraint involving the fundamental directions. This is the reason why in $V 2$, we did not use the constraint $\gamma ^\circ (a) = 1$ instead of $\gamma ^{\circ }(a) \ge 1$: In view of Corollary 5.1, we would have l additional constraints involving the fundamental directions.

Remark 5.2

For practical purposes let us note that the number l of fundamental directions of a gauge, and therefore the number of programs  $\text {V}3_j$, can vary tremendously. For example, the gauge given by the 1-norm on ${\mathbb {R}}^n$ has $l=2n$ fundamental directions, i.e., is linear in the dimension n, whilst the gauge given by the $\infty$-norm on ${\mathbb {R}}^n$ has $l=2^n$ fundamental directions, i.e., is exponential in the dimension, see, e.g., p. 5 in [39].

To summarize, instead of solving $V 1$ we may solve the linear and semi-infinite programs $\text {V}3_j$ for all $j \in [l]$. How the semi-infinite constraint can be circumvented is shown in the next section.

6 An Approximating Hierarchy for Polynomial Constraints and Polyhedral Gauges

In this section we approximate Program $\text {V}3_j$ by a hierarchy of sos programs. The main reason is that the constraint

$$\begin{aligned} a^T x \le b \quad \text {for all}\ x \in S \end{aligned}$$

is semi-infinite if S contains infinitely many points. There is much literature on semi-infinite programming problems. Classical overview articles are, e.g., [40, 41]; a more recent survey is [42]. A bi-level approach is explored in [43]. Also, several numerical solution methods exist, for an overview, we refer to [44,45,46].

However, in this article we take a different route. Let us explore how semi-infinite constraints can be sidestepped by the requirement of semi-algebraic S and a polyhedral gauge $\gamma$. For example when considering MIPP, the set $S = F$ is semi-algebraic. In the following, we use an arbitrary basic closed semi-algebraic set $S = K(h_1, \ldots , h_s)$.

With this in mind, we consider the following hierarchy of programs, where $k \in {\mathbb {N}}$, $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$, $S := K(h_1, \ldots , h_s)$ a semi-algebraic set with $q \in S$, and $v_1, \ldots , v_l \in {\mathbb {R}}^n$.

The number k is called the truncation order of program $\text {VR}_{j,k}$. Next we show that $\text {VR}_{j,k}$ is an sos program, i.e., that it has the form

for $c \in {\mathbb {R}}^m$ and fixed polynomials $p_{i0}, p_{ij} \in {\mathbb {R}}[X_1, \ldots , X_n]$, $i \in [r]$, $j \in [m]$, and decision variables $y \in {\mathbb {R}}^m$. This is helpful since it is possible to solve sos programs. For a detailed introduction to sos programming, we refer to [24, 25].

Proposition 6.1

Program $\text {VR}_{j,k}$ is an sos program.

Proof

As is common in sos programming, a constraint of the form

$$\begin{aligned} \begin{aligned} p_0 + y_1 p_1 + \cdots + y_m p_m&\in M(h_1, \ldots , h_s)[k] \\ y&\in {\mathbb {R}}^m \end{aligned} \end{aligned}$$

for some $p_i \in {\mathbb {R}}[X_1, \ldots , X_n]$ and $k \in {\mathbb {N}}$ translates to a classical sos programming constraint as follows: The statement

$$\begin{aligned} p_0 + y_1 p_1 + \cdots + y_m p_m&\in M(h_1, \ldots , h_s)[k] \end{aligned}$$

is, using the fact that $h_0 = 1$ in the defining equation (4) of M[k], equivalent to

$$\begin{aligned} p_0 + y_1 p_1 +\cdots + y_m p_m - \sum _{j=1}^s \sigma _j h_j&\in \varSigma _n \\ \deg \sigma _i&\le k - \deg h_i,&\quad i \in [s], \\ \sigma _i&\in \varSigma _n,&\quad i \in [s]. \end{aligned}$$

The degree bounds ensure that only finitely many real decision variables appear in the $\sigma _i$, and thus they can be rewritten by constraints of the form SOSP. Note that we tacitly assume $h_i \ne 0$, otherwise we may remove the constraint. Also, linear programming constraints can be used in sos programming, since for $c \in {\mathbb {R}}$, the requirement $c \ge 0$ is equivalent to $c \in \varSigma _n$. Finally, we note that the objective of Program $\text {VR}_{j,k}$ is linear. $\square$

The next proposition shows that feasible solutions to Program $\text {VR}_{j,k}$ yield feasible solutions to Program $\text {V}3_j$.

Proposition 6.2

Let $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$ and $S := K(h_1, \ldots , h_s)$. Furthermore, let $v_1, \ldots , v_l \in {\mathbb {R}}^n$ be given. If (a, b) is feasible to $\text {VR}_{j,k}$ for $j \in [l]$, $k \in {\mathbb {N}}$, then (a, b) is feasible to $\text {V}3_j$.

Proof

Let (a, b) feasible to $\text {VR}_{j,k}$. Feasibility implies that

$$\begin{aligned} b - \sum _{i=1}^n a_iX_i \in M\left( h_1, \ldots , h_s \right) [k], \end{aligned}$$

which imposes $b-a^Tx \ge 0$ on S by Observation 2.1. Hence (a, b) is feasible to $\text {V}3_j$. $\square$

The next theorem shows that, if $M(h_1, \ldots , h_s)$ is Archimedean, we get a hierarchy of sos programs indexed by the truncation order k, producing a sequence of valid inequalities for S. Also, as $k\rightarrow \infty$, we can show that the distance of the hyperplanes to the point q is monotonically decreasing and converges to the optimal solution of $V 1$.

Theorem 6.1

Let $h_1, \ldots , h_s \in {\mathbb {R}}[X_1, \ldots , X_n]$ and $q \in S := K(h_1, \ldots , h_s)$. Suppose further that $M(h_1, \ldots , h_s)$ is Archimedean. Also, let a polyhedral gauge $\gamma$ on ${\mathbb {R}}^n$ with fundamental directions $v_1, \ldots , v_l \in {\mathbb {R}}^n$ be given. Then, it holds:

• 1. For every $j \in [l]$, Program $\text {VR}_{j,k}$ is feasible for large enough values of k.

• 2. Denote the optimal value of $V 1$ by $z^*$. For $j \in [l]$, $k \in {\mathbb {N}}$ denote the optimal value of $\text {VR}_{j,k}$ by $z_{j,k}$ and put $z_k = \min _{j \in [l]} z_{j,k}$. Then $z_k\searrow z^*$ for $k \rightarrow \infty$.

Proof

Let $M: = M(h_1, \ldots , h_s)$. To see Claim 1, fix $j \in [l]$. Put $a^*: = v_j/\Vert v_j \Vert _2$ and note that $v_j^T a^* \ge 1$. By Corollary 2.1, S is compact, hence the map

$$\begin{aligned} \mu : {\mathbb {R}}^n \rightarrow {\mathbb {R}}, \ x \mapsto (a^*)^T x, \end{aligned}$$

attains its maximum $b^*$ on S. Define the family of polynomials

$$\begin{aligned} p(a,b) := b - \sum _{i=1}^n a_iX_i \in {\mathbb {R}}[X_1, \ldots , X_n], \quad a \in {\mathbb {R}}^n, b \in {\mathbb {R}}. \end{aligned}$$

Now fix some $\varepsilon > 0$, and note that $p(a^*, b^* + \varepsilon )$ is positive on S. The Positivstellensatz (Theorem 2.2) ensures that $p(a^*, b^* + \varepsilon ) \in M$. There is $k_\varepsilon \in {\mathbb {N}}$ with $p(a^*, b^* + \varepsilon ) \in M[k_\varepsilon ]$, and by definition of M[k], $p(a^*, b^* + \varepsilon ) \in M[k]$ for $k \ge k_\varepsilon$. In other words, $(a^*, b^* + \varepsilon )$ is feasible for $\text {VR}_{j,k}$ for $k \ge k_\varepsilon$.

Concerning Claim 2 we note that compactness of S implies feasibility of $V 1$, and hence $z^* < +\infty$. By Theorem 3.1, $V 1$ has an optimal solution (a, b). By Proposition 4.1 and rescaling if necessary, we may further assume that (a, b) solves $V 2$ to optimality, that is,

$$\begin{aligned} z^* = b - a^T q. \end{aligned}$$

By Proposition 5.1, there is $j_0 \in [l]$ such that (a, b) solves $V 3_{j_0}$ to optimality. Let $\varepsilon > 0$. Hence, the linear polynomial $p(a,b+\varepsilon )$ is positive on S and by the Positivstellensatz lies in $M[k_\varepsilon ]$ for some $k_\varepsilon \in {\mathbb {N}}$. Put differently, $(a, b+ \varepsilon )$ is feasible for $VR _{{j_0},{k_\varepsilon }}$ with objective value

$$\begin{aligned} b + \varepsilon - \displaystyle v_{j_0}^T a = z^* + \varepsilon . \end{aligned}$$

Note that $z^* \le \displaystyle z_{{j_0},{k_\varepsilon }}$, and our estimates combine to

$$\begin{aligned} z^* \le z_{j_0, k_\varepsilon } \le b + \varepsilon - \displaystyle v_{j_0}^T a = z^* + \varepsilon , \end{aligned}$$

and we conclude $z_{j_0, k_\varepsilon } \rightarrow z^*$ for $\varepsilon \rightarrow 0$. On the other hand, we have

$$\begin{aligned} \displaystyle z^* \le z_{j,k+1} \le z_{j,k}, \quad j\in [l], k \in {\mathbb {N}}, \end{aligned}$$

since $M[k] \subset M[k+1]$ by definition of M[k], which yields $z_{j_0,k} \searrow z^*$ for $k \rightarrow +\infty$. The claim $\min _{j \in [l]} z_{j,k} = z_k \searrow z^*$ follows. $\square$

7 Illustration

We have implemented the hierarchy using SOSTOOLS and SeDuMi and illustrate our results on some examples.⁴ Our implementation is published as open-source software [49].

In our first example, we consider the polynomials

$$\begin{aligned} h_1 = 1 - X_2^2 - X_1^2, \quad h_2 = X_1 + X_2^3 \in {\mathbb {R}}[X_1, X_2]\displaystyle . \end{aligned}$$

(14)

Thus, the set $S=\{x \in {\mathbb {R}}^2: h_1(x) \ge 0, h_2(x) \ge 0\} \subset {\mathbb {R}}^2$ is thus the compact set given by the intersection of the Euclidean unit norm ball and the epigraph of the function $x_1 \mapsto x_2(x_1) = - \root 3 \of {x_1}$.

Note that $1-X_1^2-X_2^2 = 0\cdot 1 + 1 \cdot h_1 + 0 \cdot h_2 \in M(h_1,h_2)$, hence by Theorem 2.1 the associated quadratic module $M(h_1, h_2)$ is Archimedean, and the hierarchy converges by Theorem 6.1. The point $q = (0.4,-0.5)$ lies in S. We have solved our hierarchy for the polyhedral gauge $\gamma = \Vert \cdot \Vert _1$ (in this case a block norm) with fundamental directions $\{(1, 0), (0, 1), (-1, 0), (0,-1)\}$ and two different truncation orders, which we report in Table 1.

Table 1 Example configurations, computed values and computation times

Figures 1 and 2 show the vanishing sets $V(h_1)$ and $V(h_2)$ of $h_1$ and $h_2$, that is, $V(h_i) = \{ x \in {\mathbb {R}}^2 : h_i(x) = 0\}$, and the point q. The figures show a computed optimal hyperplane for a low ($k=2$) and a high ($k=5$) truncation order k. The optimal solutions and optimal values along with the computation times can be found in Table 1. Allowing for a higher truncation order of $k=6$ did not improve the result further. We infer from these first examples that a low truncation order of, say, $k=2$ cannot be expected to give an optimal hyperplane—however, for $k=2$, we get a valid inequality that already can be used as approximation in a very short computation time. The examples also show that an optimal solution (a, b) for $\text {VR}_{j,k}$ does not necessarily yield a tight inequality for S.

Fig. 1

Bounded example, k low

Fig. 2

Bounded example, k high

An unbounded example (Fig. 3) is given by

$$\begin{aligned} h_1 = X_2 - X_1^2, \quad h_2 = X_2^2 - X_1. \end{aligned}$$

(15)

Fig. 3

Unbounded example

The set $S=\{x \in {\mathbb {R}}^2: h_1(x) \ge 0, h_2(x) \ge 0\}$ is the intersection of the filled unit parabola given by $x_2 \ge x_1^2$ and the outside of the rotated unit parabola given by $x_1 \le x_2^2$. The set S is thus indeed unbounded, hence $M(h_1, h_2)$ cannot be Archimedean. Nevertheless we can apply our approach (now without convergence guarantee). We choose $q = (0.25, 0.5)$ on the boundary of S. We report the computed values for $k=4$ in Table 1. This example shows that, even though the Archimedean condition does not hold, it can still be possible to obtain a valid inequality that is close to q and nearly tight. For lower orders ($k=3$), no solution was found. This can also be concluded directly from (4), as we would have to express a nontrivial linear polynomial as a linear combination of $h_1$ and $h_2$, which is impossible. The objective did not improve by increasing to $k=5$ or $k=6$.

8 Modifications and Extensions

In this section we consider some modifications of Program $V 1$. Namely, we consider the case that a point $q \in S$ is not known, and the case that the normal a is fixed. We will state the modifications as general optimization problems (similar to $V 1$) along with a reformulation using sos programming (similar to $\text {VR}_{j,k}$).

8.1 Finding Valid inequalities Without a known $q \in S$

As a first modification, we search for a tight valid inequality for S without knowing a point $q \in S$. We formulate the program and its sos variant with the constraint $\gamma ^{\circ }(a) = 1$ that leads to more constraints—cf. Remark 5.1—as follows. As before, let $S\subset {\mathbb {R}}^n$, $M = M(h_1, \ldots , h_s)$ for $h_i \in {\mathbb {R}}[X_1, \ldots , X_n]$, and $v_j \in {\mathbb {R}}^n$.

We require an equation in $\text {Vb}$ as opposed to $V 2$ because otherwise the program would be unbounded from below whenever the optimal objective is negative. Let us again state some observations.

Proposition 8.1

Let $S \subset {\mathbb {R}}^n$ and $\gamma$ be a gauge on ${\mathbb {R}}^n$.

• 1. Every feasible solution (a, b) of $\text {Vb}$ yields a valid inequality $(a^Tx \le b)$ for S. If (a, b) is optimal, the inequality is tight.

• 2. Suppose further $S = K(h_1, \ldots , h_s)$ for $h_i \in {\mathbb {R}}[X_1, \ldots , X_n]$ and let $\gamma$ be a polyhedral gauge with fundamental directions $v_1, \ldots , v_l$. If (a, b) is feasible for $\text {VbR}_{j,k}$, then (a, b) is feasible for $\text {Vb}$.

• 3. Additionally, let $M = M(h_1, \ldots , h_s)$ be Archimedean. Then, for $j \in [l]$, $\text {VbR}_{j,k}$ is feasible for eventually all k. Denote the optimal value of $\text {Vb}$ by $z^*$ and the optimal value of $\text {VbR}_{j,k}$ by $z_{j,k}$ and put $z_k = \min _j z_{j,k}$. Then $z_k \searrow z^*$ for $k \rightarrow \infty$.

Proof

Statements 1 and 3 are shown analogously to Proposition 3.1 and Theorem 6.1, respectively. We prove Statement 2. Let (a, b) be a feasible solution for $\text {VbR}_{j,k}$. The constraint $b - \sum _{i=1}^n a_i X_i \in M[k]$ implies $b - a^T x \ge 0$ on $S = K(h_1, \ldots , h_s)$ by Observation 2.1. By Corollary 5.1, the constraint $\gamma ^{\circ }(a) = 1$ is of the form $v_i^T a \le 1$ for $i \in [l]$ and $v_j^T a \ge 1$ for some $j \in [l]$. The claim follows. $\square$

For this modification, we consider the following example (Fig. 4):

$$\begin{aligned} h_1 = 8X_1 X_2 - 1, \quad h_2 = \frac{1}{16} - \left( X_1 + \frac{1}{2} \right) ^2. \end{aligned}$$

(16)

The set $S=\{x \in {\mathbb {R}}^2: h_1(x) \ge 0, h_2(x) \ge 0\}$ is the intersection of a branch of a hyperbola $\{(x_1,x_2) \in {\mathbb {R}}^2: x_1 < 0 \text{ and } x_2 \le \frac{1}{8 x_1}\}$ and a strip given by $\{(x_1, x_2) \in {\mathbb {R}}^2: |x_ 1 + \frac{1}{2} | \le 4\}$. Hence, S is unbounded. We ran the hierarchy $\text {VbR}_{j,k}$ for $k=4$ (let us stress that we did not specify a point $q \in S$). The values of the computed tight inequality are shown in Table 1. No feasible solution could be found for $k=3$ and the objective did not improve for $k=5$ or $k=6$, which is to be expected since Fig. 4 reveals that the hyperplane is tight.

Fig. 4

Example without q, ...

Fig. 5

...with a fixed, optimized for b

8.2 Fixed Normal

The second modification we consider is the variant where a fixed normal $a \in {\mathbb {R}}^n$, $a \ne 0$, is given and we want to find $b \in {\mathbb {R}}$ such that $(a^T x \le b)$ is valid for S and as tight as possible, i.e., $b \in {\mathbb {R}}$ is the only decision variable.⁵ The programs read

In the next result we show that for a fixed normal and provided some $q \in S$ is known, the optimal solutions do not change by replacing the objective by d(q, H(a, b)).

Observation 8.1

Let $q \in S \subset {\mathbb {R}}^n$, a gauge $\gamma$ and $0 \ne a \in {\mathbb {R}}^n$ be given. Consider the program

$$\begin{aligned} \begin{aligned} \min \quad&d(q, H(a,b)) \\ \text {s.t.}\quad&a^T x \le b \quad \text {for all} \ x \in S \\&b \in {\mathbb {R}}. \end{aligned} \end{aligned}$$

(17)

Then, Vn and  (17) have the same feasible and optimal solutions.

Proof

The claim regarding feasibility is trivial. From Theorem 4.1 we know that $d(q, H(a,b)) = (b-a^Tq)/\gamma ^{\circ }(a)$. Since a is fixed, both objectives only differ by a constant positive scaling and a constant translation, hence optimal solutions coincide. $\square$

Let us state some properties of Vn and $\text {VnR}_{k}$ . We omit the proof since it is similar to the proof for the corresponding statements of $V 2$ and $\text {VR}_{j,k}$ .

Proposition 8.2

Let $S \subset {\mathbb {R}}^n$.

• 1. Every feasible solution b of Vn yields a valid inequality $(a^Tx \le b)$ for S. If b is optimal, the inequality is tight.

• 2. Suppose further $S = K(h_1, \ldots , h_s)$ for $h_i \in {\mathbb {R}}[X_1, \ldots , X_n]$ and let $\gamma$ be a polyhedral gauge with fundamental directions $v_1, \ldots , v_l$. If b is feasible for $\text {VnR}_{k}$, then b is feasible for Vn.

• 3. Additionally, let $M = M(h_1, \ldots , h_s)$ be Archimedean. Then, $\text {VnR}_{k}$ is feasible for eventually all k. Denote the optimal value of Vn by $z^*$ and the optimal value of $\text {VnR}_{k}$ by $z_{j,k}$ and put $z_k = \min _j z_{j,k}$. Then $z_k \searrow z^*$ for $k \rightarrow \infty$.

We illustrate this modification in our last example (Fig. 5), where we use $h_1$ and $h_2$ from (16) but fix the normal to $a = (-1,1)$. We report the computed optimal value of b in Table 1. Again, no solution was found for $k=3$ and the objective did not improve for $k=5$ and $k=6$, which is again to be expected since the figure reveals that the hyperplane is tight in this example, too.

9 Conclusions

To summarize, we have shown that the problem to find a tight valid inequality for a subset S of ${\mathbb {R}}^n$, using a polyhedral gauge $\gamma$, can be approximated with sos programming if the set S is semi-algebraic, i.e., if S is given as $K(h_1, \ldots , h_s)$ for some polynomials $h_1, \ldots , h_s$. The approximating hierarchy is guaranteed to converge if the quadratic module $M(h_1, \ldots , h_s)$ is Archimedean. In view of Remark 2.1, this is the case if S is a bounded set.

Sos programs like ours are computationally tractable on current SDP solvers for small instances (few variables and low degrees of the polynomials). We hence can find cuts for MIPP instances in reasonable time in this case. If the corresponding semidefinite programs become too large for current state-of-the-art solvers, there are promising ideas that keep larger instances tractable, e.g., restrictions to subsets of sos polynomials that translate to linear or second-order cone programs  [50, 51] as well as column generation [52]. Still, our sos programs translate to SDPs which can, leaving technical details aside, essentially be solved in polynomial time [53]. Note that we cannot expect much more, since MIPP is known to be NP-hard. In the continuous case, this can be seen since deciding nonnegativity of a polynomial of degree 4 is NP-hard [54] and thus minimization of a polynomial of degree 4 is NP-hard. In the integer case, it can be shown that no algorithm for integer programming with quadratic constraints exists [55].

Further research includes to identify situations when a cut may be derived from a given tight valid inequality. That is, given a tight valid inequality for S, are there assumptions that ensure that there is a way to derive a related inequality which is valid for the integer points in S but violated at (a non-integer) point q in S, say? First ideas in this direction are given in [56].