Basic ideas of this chapter

(a) The graph of a polynomial function and a piece of the boundary (b) A subgraph on the boundary Figure 5.1:Investigating the boundary of the graph of a polynomial function

are presented, respectively. In this chapter, we focus on the characteristics of the convex hull of graphs of general polynomial functions over a polytope.

5.2 Basic ideas of this chapter

Consider the following example inR³. Let𝑓(𝑥, 𝑦) =𝑥²−5𝑥𝑦+𝑦²be a polynomial function over the domain𝑋 ={(𝑥, 𝑦)∈[−3,10]×[−3,10]}. For the constraint𝑧=𝑥²−5𝑥𝑦+𝑦², the feasible region, denoted by𝒮, is shown in Figure 5.1a which corresponds to the graph of𝑓 over domain𝑋. Recalling our MINLP algorithms, we add linear constraints to strengthen the LP relaxation. For any hyperplane𝐻inR³defined in the form{(𝑥, 𝑦, 𝑧)|𝑧=𝑎𝑥+𝑏𝑦+𝑐}

with constants𝑎, 𝑏, 𝑐∈R,𝐻is said to be a linear underestimator to𝑓 over𝑋if 𝒮={(𝑥, 𝑦, 𝑧)|𝑧=𝑓(𝑥, 𝑦),(𝑥, 𝑦)∈𝑋} ⊂ {(𝑥, 𝑦, 𝑧)|𝑧≥𝑎𝑥+𝑏𝑦+𝑐}

⏟ ⏞

downward closed halfspace to𝐻

.

Graphically, it means that the corresponding downward closed halfspace completely contains the graph of𝑓 over𝑋.

In contrast to general MINLP algorithms, we want to find such linear underestimators directly.

They are expected to strengthen the LP relaxation. The intuition is that we only want to consider such hyperplanes𝐻that support the graph, otherwise we can move it upwardly until the new generated hyperplane intersects the graph.

In other words, we say a linear underestimators𝐻isbelow(see Definition 5.16) the graph𝒮. Thus𝐻is said to bevalid.

(a) The view of the graph and a linear underestimator (b) The view from the other side Figure 5.2:A linear underestimator which supports two boundary points of the graph of a polynomial

function

To find linear underestimators𝐻, we study the intersection points𝐻∩𝒮. After a series of pre-liminary definitions in Section 5.3.1, we define locally and globally convex points in Section 5.3.2.

A point(𝑥⁰, 𝑦⁰, 𝑧⁰)on the graph is said to be locally convex if there exists𝐻 ∋ (𝑥⁰, 𝑦⁰, 𝑧⁰) and𝐻is below the graph of𝑓 over a neighborhood of(𝑥⁰, 𝑦⁰, 𝑧⁰). A point(𝑥¹, 𝑦¹, 𝑧¹)on the graph is said to be globally convex if there exists𝐻 ∋(𝑥¹, 𝑦¹, 𝑧¹)and𝐻is below𝒮.

The hyperplane𝐻^𝑡 = {(𝑥, 𝑦, 𝑧) | 𝑧 = 9𝑥−30𝑦−90}, shown as the yellow hyperplane in Figure 5.2, can be verified to be a linear underestimator for𝑓 over𝑋. The hyperplane 𝐻^𝑡intersects𝒮 in two points(−3,−3,−27)and(10,10,−300). Hence(−3,−3,−27)and (10,10,−300)both are globally convex points. Note that they both are boundary points of𝒮. Consider further a point(𝑥⁰, 𝑦⁰, 𝑧⁰)such that the corresponding domain point(𝑥⁰, 𝑦⁰)is an interior point of𝑋. As we will show in Section 5.3.2, to check if(𝑥⁰, 𝑦⁰, 𝑧⁰)is globally convex, we need only to check if the corresponding tangent plane is below𝒮. However, in practice, it is quite hard to find those globally convex points such that the corresponding domain points are interior points of𝑋. In addition, the property of global convexity usually depends on the domain. On the one hand, any locally convex point may become globally convex if the domain size is small enough. On the other hand, a globally convex point with respect to the current domain could be only locally convex for a larger domain. Note that in the example above𝑓 is neither convex nor concave over𝑋.

Now we move our attention to those globally convex points for which the corresponding domain points are on the boundary of𝑋. Consider the subgraph with restriction𝑦 = −3, which is presented as

{(𝑥, 𝑦, 𝑧)|𝑧=𝑓(𝑥, 𝑦),−3≤𝑥≤10, 𝑦=−3}.

5.2 Basic ideas of this chapter

(a) Subtangent plane (b) Globally convex boundary point inR² Figure 5.3:Example for a globally convex boundary point

This subgraph is shown as the red curve in Figure 5.1a. Since𝑦 =−3is satisfied for any point in the red subgraph, after projecting the space{(𝑥,−3, 𝑧)} ⊂R³to the space{(𝑥, 𝑧)} ⊂R², we get an isomorphic two-dimensional curve inR²

{(𝑥, 𝑧)|𝑧=𝑓(𝑥,−3) =𝑥²+ 15𝑥+ 9 =: ˜𝑓(𝑥),−3≤𝑥≤10}.

In general, we show at the beginning of Section 5.3.3 that certain subgraphs on the boundary can be projected to an isomorphic graph in a space with lower dimension. The one-dimensional curve is shown as the red curve in Figure 5.1b. Note that the corresponding function𝑓˜is a univariate function. Fortunately, the study of the convexity of univariate functions is much easier than that for bivariate functions. In the example𝑓˜has domain[−3,10]and is a convex function over[−3,10].

According to the definition of globally convex points, any point𝑥^* ∈[−3,10]in Figure 5.1b is globally convex in the projected spaceR². Theorem 5.12 implies that for any such𝑥^*, because 𝑥^*is globally convex in the projected space, the boundary point(𝑥^*,−3)in𝑋is also globally convex in the original space. This means there exists a hyperplane𝐻 ∋(𝑥^*,−3, 𝑓(𝑥^*,−3))and 𝐻is below𝒮. Consider the case𝑥^*= 0. Then(0,−3,9)is a globally convex point. Figure 5.3b shows that in the projected spaceR², the tangent plane, shown as the green line, is the unique underestimator. The corresponding line inR³, shown as the green line in Figure 5.3a, is then {(𝑥,−3,15𝑥+ 9)|𝑥∈R}which is defined as subtangent plane in Section 5.3.3. Corollary 5.13 implies that every linear underestimator𝐻with𝐻∋(0,−3,9)satisfies

𝐻⊃ {(𝑥,−3,15𝑥+ 9)|𝑥∈R},

which means any valid hyperplane which contains(0,−3,9)always contains the green line.

The blue line{(𝑥,−3,9𝑥)|𝑥∈R}in Figure 5.2a is a subtangent plane on(−3,−3,−27), as defined in Section 5.3.3. We can verify that the yellow hyperplane is the affine hull of the blue line and the point(10,10,−300), i.e.,

𝐻^𝑡={(𝑥, 𝑦, 𝑧)|𝑧= 9𝑥−30𝑦−90}= aff{{(𝑥,−3,9𝑥)|𝑥∈R},{(10,10,−300)}}. For any point(10,10, 𝑧¹)with𝑧¹<−300, we can also verify that

𝐻^𝑙= aff^{︁{(𝑥,−3,9𝑥)|𝑥∈R},{(10,10, 𝑧¹)}^}︁

is also a linear underestimator. By comparing𝐻^𝑡and𝐻^𝑙we have

𝐻^𝑡∩ 𝒮 ={(−3,−3,−27),(10,10,−300)}){(−3,−3,−27)}=𝐻^𝑙∩ 𝒮.

From the intuition, we prefer𝐻^𝑡since the resulting relaxation is tighter. For this purpose we define tight and loose hyperplanes in Section 5.3.4. In general, a valid hyperplane𝐻^𝑙 is definitely loose if there exists another valid hyperplane𝐻^𝑡which preserves all intersection points and intersects in additional point(s) with𝒮, which means

(𝐻^𝑡∩ 𝒮))(𝐻^𝑙∩ 𝒮).

This is a sufficient but not necessary condition for loose hyperplanes. Using Lemma 5.26 in Section 5.3.4 we verify that the yellow hyperplane in Figure 5.2a is a tight hyperplane.

After that, in Section 5.3.5 we prove for every loose hyperplane𝐻^𝑙that there exists a tight hyperplane𝐻^𝑡that preserves intersection points with

(𝐻^𝑡∩ 𝒮)⊃(𝐻^𝑙∩ 𝒮).

We call the corresponding halfspaces tight or loose halfspaces. Note that in the example above we have𝐻^𝑡∩𝐻^𝑙 ={(𝑥,−3,9𝑥) |𝑥∈R}which is the blue line in Figure 5.2a. Graphically, we can rotate𝐻^𝑙around the blue line as axis to generate𝐻^𝑡. The rotation approach is the basic idea of a few proofs in this section.

Finally, in Section 5.3.6, we prove that to form the convex hull of𝒮using halfspaces, we only need tight hyperplanes. In other words, any loose hyperplane is proved to be redundant.

In Section 5.3 we only include theoretical results. We cannot use them to solve MINLP directly. In Section 5.4 we develop algorithms to compute tight hyperplanes for the graph of bivariate polynomial functions with degree up to3over a polygon in R². Note that the domain does not have to be box-constrained. In the algorithms, we first find all globally convex domain points on the boundary. This is very tractable since we only need to find globally convex points in the graph of univariate polynomial functions with degree3over a closed interval inR. Based on those globally convex domain points, the algorithms find a series of tight halfspaces. Computations in Section 5.5 show that these tight halfspaces improve the dual bounds significantly.

5.3 Convex hull of graphs of polynomial functions over a polytope

5.3 Convex hull of graphs of polynomial functions over a