Nested Pricing

Although much of the focus of this research has been on the generation of strong lower bounds forz_IP, the performance of branch-and-bound algorithms also depends highly on the ability to find good feasible solutions, yielding strong upper bounds. Moreover, in practice, users of optimization solvers often wish to find the best feasible solution possible in some fixed amount of time. We can tell whether a feasible solution is of high quality by looking at the gap between the lower and upper bounds onz_IP.

As described in Chapter 2, integrated decomposition methods improve upon traditional methods by implicitly building an innerandan outer approximation. By intersecting these two polyhedra, one can find better approximations of the convex hull of the original problem. In Section 2.3.1, we discussed the template paradigm and the idea of a polytope defined by some class of valid in-equalities. Generally, one thinks of the collection of valid inequalities astheouter approximation generating during the cutting-plane method. However, one can also view the closures defined by each class of inequalities as separate polyhedral regions. From this point of view, the outer ap-proximation is the intersection of the set of these closures. Computationally, it is well-known that generating cuts from various classes of inequalities can be beneficial in that different classes of valid inequalities contribute to describing different areas of the feasible region; or, in the context of a specific application, different cuts correspond to different aspects of the constraint system. This is especially true for, integer programs that arise as the combination of two (or more) integer programs

defined over the same set of variables. For example, the classical Vehicle Routing Problem is often seen as the intersection between a routing problem (TSP) and a bin packing problem. Each problem, by itself, is relatively easy to solve, while the combination is often extremely difficult. This line of thinking is the motivating factor behind the idea presented below.

In every study we are aware of, inner approximation methods generate an approximation based ononefixed relaxation (or polytope). If one considers inner approximations simply as the dual of outer approximations, then this motivates us to consider generating extreme points from multiple polyhedra, in a fashion similar to that in which facets are generated from multiple polyhedra in outer methods. In fact, any polyhedronP_C⁰ ⊆ P⁰ is a candidate for inclusion in the pricing phase. For example, in Step 3 of the Dantzig-Wolfe method, when calling the subroutine

OPT

³

P⁰, c^>−¡ u^t_DW¢_>

A⁰⁰, α^t_DW

´ ,

we might also call

OPT

³

P_C⁰ , c^>−¡ u^t_DW¢_>

A⁰⁰, α^t_DW

´ ,

producing another extreme point. It appears that this simple but intriguing idea has been overlooked until now. To illustrate, let us again revisit theVehicle Routing Problem.

Example 4: VRP (continued) Recall the formulation for VRP described in Section 2.3.1. Let’s assume we choose the perfectb-matching relaxation for our decomposition. Then, the relaxed poly-hedronP⁰ is defined by the constraints(2.35), which enforce that each customer must be serviced by exactly one vehicle, and(2.34), which enforce that each of thekvehicles must depart and return to the depot exactly once. Now, let us also consider again the multiple traveling salesman relaxation.

Feasible solutions to this relaxation also satisfy the same constraints as in the case ofb-matching. In addition, solutions tok-TSP also satisfy the subtour elimination constraints(2.40). The two nested relaxations are defined as follows:

P^bMatch = conv{x∈R^E |xsatisfies(2.34),(2.35),(2.37),(2.38)}, P^kTSP = conv{x∈R^E |xsatisfies(2.34),(2.35),(2.37),(2.38),(2.40)}.

where P^bMatch ⊃ P^kTSP. So, by choosingP⁰ = P^bMatch, we can generate extreme points from polytopes. As usual, in order to attain valid bounds, we have to solve the optimization subproblem for P^bMatch exactly. Then, since P^kTSP is contained in P^bMatch, we can solve the optimization subproblem forP^bMatchheuristically.

Let us now return to our motivating principle—using different classes of valid inequalities when generating cutting planes. Analogously, it would be beneficial if the extreme points used for the in-ner approximation had some diversity with respect to which parts of the feasible region are required to be satisfied. To this end, let us now consider an additional relaxation to use in our nested pricing scheme for VRP. Recall that(2.36), thegeneralized subtour elimination constraints(GSECs), have exponential size. In the standard cutting-plane algorithm for VRP, these GSECs would be generated dynamically as needed. Assume we have run several iterations of a cutting-plane method and define Gas the set of subsets of nodes that represent the GSECs that were found. Now, define the following class of inequalities, a subset of the full set of GSECs:

x(δ(S))≥2b(S)∀S∈ G. (3.19)

Now, combining these inequalities with(2.34)and(2.35), let us define a new relaxation

P^bMatch+=conv{x∈R^E |xsatisfies(2.34),(2.35),(3.19),(2.37),(2.38)},

which now incorporates some of the capacity restrictions from the original problem. Now, since P^bMatch⊃ P^bMatch+, we can use all three polyhedra,P^bMatch,P^bMatch+, andP^kTSPwhen generating extreme points, thereby improving our diversity and increasing our chances of finding feasible so-lutions quickly.

One of the key drawbacks for methods like price-and-cut is that, in order to derive a valid bound, one mustprice outexactly over the chosen inner relaxationP⁰. Although this is still true here, the idea is that, one can also generate points from tighter relaxations heuristically. Since these extreme points are in some sensecloserto feasible for the original problem, the probability of finding upper

bounds earlier in the search tree is higher. Typically pricing over more restrictive polytopes can be more expensive computationally. So, there is a tradeoff between the time spent generating the extreme points and the improvement gained by better upper bounds. In Section 5.1, we will present computational results to show the effectiveness of this idea.