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 in R3_{. Let 𝑓(𝑥, 𝑦) = 𝑥}2_{− 5𝑥𝑦 + 𝑦}2_{be a polynomial function}

over the domain 𝑋 = {(𝑥, 𝑦) ∈ [−3, 10] × [−3, 10]}. For the constraint 𝑧 = 𝑥2_{− 5𝑥𝑦 + 𝑦}2_,

the feasible region, denoted by 𝒮, is shown in Figure 5.1a which corresponds to the graph of 𝑓 over domain 𝑋. Recalling our MINLP algorithms, we add linear constraints to strengthen the LP relaxation. For any hyperplane 𝐻 in R3_{defined in the form {(𝑥, 𝑦, 𝑧) | 𝑧 = 𝑎𝑥 + 𝑏𝑦 + 𝑐}}

with constants 𝑎, 𝑏, 𝑐 ∈ R, 𝐻 is said to be a linear underestimator to 𝑓 over 𝑋 if 𝒮 = {(𝑥, 𝑦, 𝑧) | 𝑧 = 𝑓 (𝑥, 𝑦), (𝑥, 𝑦) ∈ 𝑋} ⊂ {(𝑥, 𝑦, 𝑧) | 𝑧 ≥ 𝑎𝑥 + 𝑏𝑦 + 𝑐}

⏟ ⏞

downward closed halfspace to 𝐻

.

Graphically, it means that the corresponding downward closed halfspace completely contains the graph of 𝑓 over 𝑋.

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 hyperplanes 𝐻 that support the graph, otherwise we can move it upwardly until the new generated hyperplane intersects the graph.

In other words, we say a linear underestimators 𝐻 is below (see Definition 5.16) the graph 𝒮. Thus 𝐻 is said to be valid.

(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 underestimators 𝐻, we study the intersection points 𝐻 ∩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_{, 𝑦}0_{, 𝑧}0_{)on the graph is said to be locally convex if there exists 𝐻 ∋ (𝑥}0_{, 𝑦}0_{, 𝑧}0₎

and 𝐻 is below the graph of 𝑓 over a neighborhood of (𝑥0_{, 𝑦}0_{, 𝑧}0_{). A point (𝑥}1_{, 𝑦}1_{, 𝑧}1_{)on the}

graph is said to be globally convex if there exists 𝐻 ∋ (𝑥1_{, 𝑦}1_{, 𝑧}1_{)and 𝐻 is below 𝒮.}

The hyperplane 𝐻𝑡 _{= {(𝑥, 𝑦, 𝑧) | 𝑧 = 9𝑥 − 30𝑦 − 90}, shown as the yellow hyperplane}

in Figure 5.2, can be verified to be a linear underestimator for 𝑓 over 𝑋. The hyperplane

𝐻𝑡intersects 𝒮 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 of 𝒮. Consider further a point (𝑥0_{, 𝑦}0_{, 𝑧}0_{)such that the corresponding domain point (𝑥}0_{, 𝑦}0_{)is an}

interior point of 𝑋. As we will show in Section 5.3.2, to check if (𝑥0_{, 𝑦}0_{, 𝑧}0_{)is globally convex,}

we need only to check if the corresponding tangent plane is below 𝒮. However, in practice, it is quite hard to find those globally convex points such that the corresponding domain points are interior points of 𝑋. 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 above 𝑓 is neither convex nor concave over 𝑋.

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

5.2 Basic ideas of this chapter

(a) Subtangent plane (b) Globally convex boundary point in R2

Figure 5.3:Example for a globally convex boundary point

This subgraph is shown as the red curve in Figure 5.1a. Since 𝑦 = −3 is satisfied for any point in the red subgraph, after projecting the space {(𝑥, −3, 𝑧)} ⊂ R3_{to the space {(𝑥, 𝑧)} ⊂ R}2_,

we get an isomorphic two-dimensional curve in R2

{(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 function ˜𝑓 is a

univariate function. Fortunately, the study of the convexity of univariate functions is much easier than that for bivariate functions. In the example ˜𝑓 has domain [−3, 10] and is a convex

function over [−3, 10].

According to the definition of globally convex points, any point 𝑥* _{∈ [−3, 10]in Figure 5.1b}

is globally convex in the projected space R2_{. Theorem 5.12 implies that for any such 𝑥}*_{, because}

𝑥*is globally convex in the projected space, the boundary point (𝑥*, −3)in 𝑋 is also globally

convex in the original space. This means there exists a hyperplane 𝐻 ∋ (𝑥*_{, −3, 𝑓 (𝑥}*_{, −3))and}

𝐻is below 𝒮. Consider the case 𝑥*= 0. Then (0, −3, 9) is a globally convex point. Figure 5.3b shows that in the projected space R2_{, the tangent plane, shown as the green line, is the unique}

underestimator. The corresponding line in R3_{, 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 underestimator 𝐻 with 𝐻 ∋ (0, −3, 9) satisfies

𝐻 ⊃ {(𝑥, −3, 15𝑥 + 9) | 𝑥 ∈ R},

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.,

𝐻𝑡= {(𝑥, 𝑦, 𝑧) | 𝑧 = 9𝑥 − 30𝑦 − 90} = aff {{(𝑥, −3, 9𝑥) | 𝑥 ∈ R}, {(10, 10, −300)}} . For any point (10, 10, 𝑧1_{)with 𝑧}1_{< −300, we can also verify that}

𝐻𝑙= aff{︁_{{(𝑥, −3, 9𝑥) | 𝑥 ∈ R}, {(10, 10, 𝑧}1)}}︁ is also a linear underestimator. By comparing 𝐻𝑡_{and 𝐻}𝑙_{we have}

𝐻𝑡∩ 𝒮 = {(−3, −3, −27), (10, 10, −300)} ) {(−3, −3, −27)} = 𝐻𝑙∩ 𝒮.

From the intuition, we prefer 𝐻𝑡_{since the resulting relaxation is tighter. For this purpose}

we define tight and loose hyperplanes in Section 5.3.4. In general, a valid hyperplane 𝐻𝑙 _is

definitely loose if there exists another valid hyperplane 𝐻𝑡_{which preserves all intersection}

points and intersects in additional point(s) with 𝒮, which means (𝐻𝑡_{∩ 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 hyperplane 𝐻𝑙_{that there exists a tight}

hyperplane 𝐻𝑡_{that preserves intersection points with}

(𝐻𝑡∩ 𝒮) ⊃ (𝐻𝑙∩ 𝒮).

We call the corresponding halfspaces tight or loose halfspaces. Note that in the example above we have 𝐻𝑡_{∩ 𝐻}𝑙 _{= {(𝑥, −3, 9𝑥) | 𝑥 ∈ R} which is the blue line in Figure 5.2a. Graphically,}

we can rotate 𝐻𝑙_{around the blue line as axis to generate 𝐻}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 of 𝒮 using 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 to 3 over 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 degree 3 over a closed interval in R. 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.