Optimizing with Arrays

Arrays in Optimization Models and Array Abstraction

Arrays in Optimization Problems

The textbook description of LP, QP and NLP formulations in the previous chapter depicted the set of decision variables as a one-dimensional vector, along with a linear vector of constraints. However, array-valued variables and array-valued constraints arise naturally in many optimization problems, and it is often more natural and convenient to formulate specific decision variables as multi-dimensional arrays, with dimensionality differing from decision to decision. Structured optimization allows you to easily define array-valued decision variables, using the dimensionality that is natural to your problem. Additionally, since the variables appearing in a constraint expression may themselves be array-valued when computed, a single inequality expression may expand to be a multi-dimensional array of scalar constraints. Internally, structured optimization takes care of flattening and concatenating all these decision and constraint arrays for solution by the underlying solver engine so that you don’t have to worry about it.

Array Abstraction

As experienced users know, array abstraction is an important feature of Analytica. Array abstraction allows an index to be added to the input of a computation, such that the original computation is carried out repeatedly for each new input value without having to alter the original computation. Array abstraction in Analytica is a univeral concept that derives its power from the fact that it applies at every level, from simple expressions all the way up to entire models. Optimization models are no different. If we have a submodel that solves an optimization problem, we can vary the set of inputs across a new index and repeat the optimization repeatedly for a set of scenario combinations. Thus, array abstraction gives rise to arrays of optimization problems.

A Necessary Distinction Comes About

Analytica with its Structured Optimization feature therefore provides both the ability to incorporate arrays within an optimization, as well as array abstract to obtain arrays of optimizations. This leads to a distinction, in which we can describe indexes as being either intrinsic or extrinsic. Intrinsic indexes lead to arrays within an optimization problem, while extrinsic indexes lead to arrays of optimizations. An intrinsic index is what other optimization environments would simply call an index. It makes all of its elements available to the optimizer during a single optimization run. An extrinsic index is available for array abstraction in Analytica. If you start with model that performs a single optimization, and then add a new dimension to one of the input variables, Analytica will generally treat the new index as extrinsic and abstract over it. You will now have an array of optimizations corresponding to each element of the extrinsic index. The inherent support for both types of arrays is a key differentiator of Analytica’s Structured Optimization. In this chapter we explain how to control array handling in Analytica and work through a detailed example that handles indexes in both ways.

Note

Readers who are not already familiar with the basic concepts of array abstraction will benefit from reading Tutorial Chapter 5 and User Guide Chapter 12.