In the previous chapters, we dealt with general problems by first formulating all necessary constraints and then passing the problem to an LO or MILO solver, but in a way, we have been oblivious to the problem’s structure. However, it is often advantageous to analyze this structure, as it can enable us to find better solution methods. In this chapter, we consider a very general class of problems with special structure – the network problems. In the following example, we illustrate the key ideas.

In Chapter 5, we claimed that the watershed between easy and difficult problems is their convexity status. Convex optimization problems are, however, a very broad class and one of their downsides is that the dual problem is not always readily available; see the discussion in Section 5.3. In view of the computational benefits of concurrently solving the primal and dual problems, a natural question arises: Is there a subclass of convex optimization problems that are expressive enough to model relevant real-life problems and, at the same time, allow us for a systematic derivation of the dual akin to linear optimization?

The particular feature of linear optimization problems is that as long as the decision variables satisfy all the constraints, they can take any value. However, there are many situations in which it makes sense to restrict the solution space in a way that cannot be expressed using linear (in)equality constraints. For example, some numbers might need to be integers, such as the number of people to be assigned to a task. Another situation is when certain constraints need to hold only if another constraint holds. For example, the amount of power generated by a power plant must not be less than a certain minimum threshold only if that generator is turned on. Neither of these two examples can be expressed using only linear constraints, as we have seen up to this point. In these cases, it is often still possible to formulate the problem as an LO problem, although some additional restrictions may be needed on certain variables, requiring them to take integer values only. We will refer to this type of LO problem in which some variables are constrained to be integers as mixed-integer linear optimization (MILO) problems.

The simplest and most scalable type of optimization problem is one in which the objective function and constraints are formulated using the simplest type of functions – linear functions. We refer to this class of problems as linear optimization (LO) problems.

This chapter introduces robust optimization (RO), where we aim to solve a MILO in which some of the parameters/data can take multiple (possibly infinitely many) values and we want the optimal solution to perform “the best possible,” assuming that the unknown problem parameters can always turn out to be “the worst possible.”

Our aim so far has been to formulate a real-life decision problem as a (mixed integer) linear optimization problem. The reason was clear: Linear functions are simple, so problems formulated with them should also be simple. However, two questions arise. First, is the world of linear functions flexible enough to model all real-life problems? Second, are MILOs the only simple problems we can solve quickly?

In Chapter 5, we look at approaches that belong to heuristic algorithms. These methods are derived from observations nature provide. In our argumentation for specific heuristic optimization algorithms, we discuss the local search and the hill climbing problem. One of the outcomes of this discussion is the argument for attempting to avoid cycling during a search. Tabu search optimization is built on this premise where we avoid cycling. An entirely different class of heuristic optimization algorithms are given by Particle Swarm optimization and Ant Colony optimization algorithms. In contrast to Tabu search and local search, the PSO and AC optimization algorithms utilize a number of agents in order to search for optimality. Another multi agent based algorithm is the Genetic algorithm. GA’s are inspired by Darwin’s survival of the fittest principle and use the terminology found in the field of genetics. Additionally, in this chapter we use heuristic optimization to formulate optimum control concepts, including hybrid control using fuzzy logic-based controllers and Matlab scripts to realize each of the heuristic optimization algorithms.