New Approaches to in silico Design of Epitope-Based Vaccines

3.1. COMBINATORIAL OPTIMIZATION 19
where c ∈ Rn , A ∈ Rm,n and b ∈ Rm . Typically, the optimization variables are binary,
i.e., xi ∈ {0, 1}. While the exact solution to convex optimization problems and thus LPs
can be found efficiently, ILPs are NP-complete. An example for an ILP is given in the
following:
maximize x1 + x2
subject to 2.5x1 + x2 ≤ 10,
−0.5x1 + x2 ≤ 2,
x1 ≤ 3.5,
x1, x2 ∈ Z + 0 .
(3.4)
It corresponds to the LP in (3.2) with integrality constraints on all optimization variables.
The geometric interpretation of this problem is shown in Figure 3.1B.
3.1.3 Methods
A well-known problem in combinatorial optimization is the Travelling Salesman Problem
(TSP) [49]: Starting from and returning to his home town, a travelling salesman has to
visit a given set of cities. The task is to find a shortest possible tour that visits each
city exactly once. While there are only 12 feasible solutions for five cities, the number of
solutions grows rapidly to 181,440 for 10 cities and to 6 × 10 16 for 20 cities. Assuming
that a computer requires 0.5 µs to evaluate one tour, an exhaustive search for 20 cities
would take 964 years. This combinatorial explosion, which is typical for combinatorial
optimization problems, calls for efficient optimization methods.
Optimization methods can be divided into two classes: heuristic methods and exact
methods. The former find one or more good but not necessarily optimal solutions in
reasonable time. If an optimal solution exists, the latter guarantee to find it at the cost of
potentially longer run times.
In the following, we will briefly introduce a small selection of algorithms for solving
ILPs: branch-and-bound algorithms [50], cutting plane algorithms [51], and branch-and-cut
algorithms [52]. Approaches to optimally solving an ILP are generally based on solving the
LP relaxation, i.e., the ILP without integrality constraints. This can be done efficiently.
The corresponding objective value is called dual bound of the ILP. If the dual bound is
integral, it corresponds to the solution of the ILP. Otherwise, further steps are required to
solve the ILP.
Branch-and-bound algorithms. If the solution x ′ of the LP relaxation is not integral,
branch-and-bound algorithms split the optimization problem into two subproblems. This
step is called branching. Different branching strategies have been proposed. Often, an
optimization variable that is not integral in the current solution is used: if x ′ 1 = 1.8, then
the constraint x1 ≤ 1 is added to one subproblem and the constraint x1 ≥ 2 to the other.
In a bounding step upper and lower bounds of the current subproblem are determined.
Every subproblem will continuously be solved and subdivided resulting in a search tree
until either the global optimal solution is found or the bounding step reveals that the
current branch can be pruned since it does not contain the optimal solution.