Combinatorial Online Optimization: Elevators & Yellow Angels

In classical optimization it is assumed that full information about the problem to be solved is given. This, in particular, includes that all data are at hand. The real world may not be so “nice” to optimizers. Some problem constraints may not be known, the data may be corrupted, or some data may not be available at the moments when decisions have to be made. The last issue is the subject of online optimization which will be addressed here. We explain some theory that has been developed to cope with such situations and provide examples from practice where unavailable information is not the result of bad data handling but an inevitable phenomenon. We begin with some informal definitions concerning online optimization, refraining from giving formal definitions to avoid “technical overkill”. Consider the following situation: we have to make immediate decisions, we have some data of the past on hand but information about future activities is unavailable, and we would like to make “good” decisions. In the classical optimization world we would study the problem to be considered, invent a mathematical model supposed to catch the key aspects of the problem, collect all data needed and use mathematical theory and algorithms to solve the problem. We call this approach in this paper offline from now on since the data collection phase is totally separated from the solution phase. In online optimization these phases are intertwined and there are many situations where this is not due to data acquisition difficulties but due to the nature of the problem itself. A little more formal, in an online problem, data arrive in a sequence (that we will call request sequence), and each time when a new request arrives, a decision has to be made. If there are costs or some other objectives involved, we face an online optimization problem since we would like to make decisions in such a way that “at the end of the day” (when the last request has arrived and has been processed) the total cost is as small as possible. There are lots of variants of this “basic online framework”, and in each case one has to exactly state what the side constraints are, what online precisely means, how costs are calculated, etc. We outline a few of the possibilities that come up in practice. For those familiar with optimal control the issues indicated here probably sound very familiar, and of course, online optimization is not a topic invented in combinatorial optimization or computer science. Online problems occur, for instance, when steel is cast continuously, chemical reactors are to be controlled, or a space shuttle reenters the atmosphere. There are some differences between these control problems and the combinatorial online problems we describe here. In the continuous control case, one usually has a mathematical model of the process. This is typically given in the form of a ordinary and/or partial differential equations. These describe the behavior of the real system under parameter changes over time. One precomputes a solution (often called optimal control) and the online algorithm has the task to adapt the running system in such a way that it follows the optimal trajectory. For instance, when a space craft reenters the atmosphere it is clear where it has to land. It should follow a predetermined trajectory to the landing place. The measuring instruments determine the deviations from this trajectory, and the algorithms steering the space craft make sure that the craft follows the trajectory using the available control mechanisms on board. In contrast, the combinatorial online optimization problems we consider have no “trajectory” that one could precompute. Decisions are discrete (yes or no), can only be made at certain points in time and not continuously, and in most cases, decisions once made cannot be revoked. In other words, although optimal control and